Optical soliton perturbation in magneto-optic waveguides by extended $G^{\prime }/G$–expansion

Abstract

This paper retrieves a spectrum of soliton solutions propagating through magneto-optic waveguides that maintain a wide range of nonlinear refractive index structures. The implemented algorithm is the extended version of $G^{\prime }/G$–expansion scheme. The existence criteria for such solitons are also presented.

1 Introduction

One of the most detrimental features in soliton transmission across intercontinental distances is soliton clutter. In order to address this, a very clever approach has been adopted in several forms of fibers all across the board. This is the introduction of magnetic field in a variety of fibers. Such a situation would definitely curtail this clutter and consequently the possible loss of information during soliton propagation would be controlled and mitigated. Therefore, it is imperative to address this important and essential feature, in its comprehensive form, for a wide range of optical fibers with a diverse form of nonlinear refractive index. This current work addresses soliton dynamics with a wide range of optical fibers by the aid of an extended version of the popular $G^{\prime }/G$–expansion approach. This would lead to the retrieval of a range of soliton solutions that would be otherwise not possible to reveal by its original form of this integration architecture. The existence criteria for such solitons is also exhibited in the work. The details now follow through after a succinct intro to the governing model.

1.1 Mathematical model

The mathematical model that describes the dynamics of soliton propagation through optical fibers in presence of magneto-optic field is given by the following coupled system of nonlinear Schrödinger’s equation (NLSE) (Biswas 2004; Biswas and Konar 2006; Biswas et al. 2018; Guzman et al. 2014; Savescu et al. 2014):

$$\begin{aligned}&iq_{t}+a_1q_{xx}+b_1q_{xt}+ \left\{ \xi _1 F\left( \left| q \right| ^{2}\right) + \eta _1 F\left( \left| r \right| ^{2}\right) \right\} q \nonumber \\&\quad = Q_1 r + i \left\{ \alpha _1 q_{x} + \beta _1 \left( F\left( \left| q \right| ^{2}\right) q\right) _{x}\right. \nonumber \\&\quad \quad \left. + \nu _1 \left( F\left( \left| q \right| ^{2}\right) \right) _{x}q + \theta _1 F\left( \left| q \right| ^{2}\right) q_{x} \right\} \end{aligned}$$

(1)

$$\begin{aligned}&ir_{t}+a_2r_{xx}+b_2r_{xt}+ \left\{ \xi _2 F\left( \left| r \right| ^{2}\right) + \eta _2 F\left( \left| q \right| ^{2}\right) \right\} r \nonumber \\&\quad = Q_2 q + i \left\{ \alpha _2 r_{x} + \beta _2 \left( F\left( \left| r \right| ^{2}\right) r\right) _{x}\right. \nonumber \\&\qquad \left. + \nu _2 \left( F\left( \left| r \right| ^{2}\right) \right) _{x}r + \theta _2 F\left( \left| r \right| ^{2}\right) r_{x} \right\} . \end{aligned}$$

(2)

Here, in (1) and (2), $a_j$ stands for the coefficients of group velocity dispersion (GVD) while $b_j$ for $j=1, 2$ are the coefficients of spatio-temporal dispersion (STD). The functional F is the type of nonlinearity that will be considered. On the right hand side $Q_j$ stands for the magnetic field effect that avoids the formation of soliton clutter. From the perturbation terms, $\alpha _j$ are the coefficients of intermodal dispersion. Also, $\beta _j$ represents the coefficients of self-steepening terms, $\nu _j$ are the coefficients of nonlinear dispersion, while $\theta _j$ also gives nonlinear dispersion. The next sections detail the integration algorithm for nine types of nonlinearity.

2 Kerr law

For Kerr law nonlinearity, $F(s)=s$. Then, Eqs. (1) and (2) reduce to

$$\begin{aligned}&iq_{t}+a_1q_{xx}+b_1q_{xt}+ \left\{ \xi _1 \left| q\right| ^{2} + \eta _1 \left| r\right| ^{2} \right\} q \nonumber \\&\quad = Q_1 r + i \left\{ \alpha _1 q_{x} + \beta _1 \left( \left| q\right| ^{2}q\right) _{x} \right. \left. + \nu _1 \left( \left| q\right| ^{2}\right) _{x}q + \theta _1 \left| q\right| ^{2} q_{x} \right\} \end{aligned}$$

(3)

$$\begin{aligned}&ir_{t}+a_2r_{xx}+b_2r_{xt}+ \left\{ \xi _2 \left| r\right| ^{2} + \eta _2 \left| q\right| ^{2} \right\} r \nonumber \\&\quad = Q_2 q + i \left\{ \alpha _2 r_{x} + \beta _2 \left( \left| r\right| ^{2}r\right) _{x} \right. + \left. \nu _2 \left( \left| r\right| ^{2}\right) _{x}r + \theta _2 \left| r\right| ^{2} r_{x} \right\} . \end{aligned}$$

(4)

In order for integrating the the systems (3)–(4), a solution structure of the form is picked as

$$\begin{aligned} q(x,t)=&P_1(\zeta )e^{i\phi \left( x,t\right) } \end{aligned}$$

(5)

$$\begin{aligned} r(x,t)=&P_2(\zeta )e^{i\phi \left( x,t\right) } \end{aligned}$$

(6)

where $\zeta$ that represents the wave variable is given by

$$\begin{aligned} \zeta =x-v t . \end{aligned}$$

(7)

Here, $P_l(\zeta )$ for $l=1, 2$ stand for the amplitude component of the soliton and v is the soliton speed, while phase factor is structured as

$$\begin{aligned} \phi \left( x,t\right) =-\kappa x + \omega t + \theta \end{aligned}$$

(8)

where $\kappa$ is the frequency while $\omega$ is the wave number and $\theta$ is the phase constant. Inserting (5) and (6) into Eqs. (3) and (4) real parts give

$$\begin{aligned} (a_l-b_lv)P_l''+ \left( b_l\omega \kappa -\omega -a_l\kappa ^2 -\alpha _l\kappa \right) P_l -\left( \kappa \left( \beta _l + \theta _l\right) - \xi _l \right) P_l^{3} + \eta _lP_lP_{{\bar{l}}}^{2} = Q_lP_{{\bar{l}}} \end{aligned}$$

(9)

with ${\bar{l}}=3-l$, and imaginary parts read

$$\begin{aligned} -v(1-b_l\kappa )P_l'+ \left( b_l\omega -2a_l\kappa -\alpha _l\right) P_l'= \left( 3 \beta _l + 2\nu _l + \theta _l\right) P_l^{2}P_l' . \end{aligned}$$

(10)

From Eq. (10) is possible to recover the velocity of the soliton

$$\begin{aligned} \begin{aligned} v=\frac{b_l\omega - 2a_l\kappa -\alpha _l }{1-b_l \kappa }, \end{aligned}&\quad \begin{aligned} b_l \kappa \ne 1 \end{aligned} \end{aligned}$$

(11)

as long as the constraint

$$\begin{aligned} 3 \beta _l + 2\nu _l + \theta _l =0 \end{aligned}$$

(12)

remains valid. Now, (11) implies that

$$\begin{aligned} a_1=a_2, \quad b_1=b_2, \quad \alpha _1=\alpha _2 \end{aligned}$$

(13)

and then, (11) modifies to

$$\begin{aligned} v=\frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \end{aligned}$$

(14)

where it is assumed that $a_l=a$, $b_l=b$ and $\alpha _l=\alpha$ for $l=1, 2.$ So the coupled NLSE having Kerr law nonlinearity for the perturbed magneto-optic waveguide is re-casted as

$$\begin{aligned}&iq_{t}+aq_{xx}+bq_{xt}+ \left\{ \xi _1 \left| q\right| ^{2} + \eta _1 \left| r\right| ^{2} \right\} q \nonumber \\&\quad = Q_1 r + i \left\{ \alpha q_{x} + \beta _1 \left( \left| q\right| ^{2}q\right) _{x} \right. \left. + \nu _1 \left( \left| q\right| ^{2}\right) _{x}q + \theta _1 \left| q\right| ^{2} q_{x} \right\} \end{aligned}$$

(15)

$$\begin{aligned}&ir_{t}+ar_{xx}+br_{xt}+ \left\{ \xi _2 \left| r\right| ^{2} + \eta _2 \left| q\right| ^{2} \right\} r \nonumber \\&\quad = Q_2 q + i \left\{ \alpha r_{x} + \beta _2 \left( \left| r\right| ^{2}r\right) _{x} \right. \left. + \nu _2 \left( \left| r\right| ^{2}\right) _{x}r + \theta _2 \left| r\right| ^{2} r_{x} \right\} \end{aligned}$$

(16)

and as a results the real part given by (9) changes to

$$\begin{aligned} (a-bv)P_l''+ \left( b\omega \kappa -\omega -a\kappa ^2 -\alpha \kappa \right) P_l -\left( \kappa \left( \beta _l + \theta _l\right) - \xi _l \right) P_l^{3} + \eta _lP_lP_{{\bar{l}}}^{2}-Q_l P_{{\bar{l}}}=0. \end{aligned}$$

(17)

Utilizing the balancing principle in Eq. (17) brings about

$$\begin{aligned} P_{ {\bar{l}}}=P_l. \end{aligned}$$

(18)

Therefore, Eq. (17) modifies to

$$\begin{aligned} (a-b v)P_l^{\prime \prime }+\left( b \omega \kappa - \omega -a \kappa ^2 - \alpha \kappa -Q_l\right) P_l +\left( \eta _l+\xi _l- \kappa \left( \beta _l+\theta _l\right) \right) P_l^3=0. \end{aligned}$$

(19)

2.1 Extended $G^{\prime }/G-$expansion approach

This subsection will apply extended $G^{\prime }/G-$expansion scheme (Ekici et al. 2016; Guo and Zhou 2010; Hayek 2010; Zhou et al. 2016) in order to address the systems (1) and (2). To kick off, the assumption for the soliton structure is taken to be

$$\begin{aligned} U_l(\zeta )=&\alpha _0^{(l)}+ \sum _{i=1}^{M}\left\{ \alpha _i^{(l)}\left( \frac{G'}{G}\right) ^i + \beta _i^{(l)}\left( \frac{G'}{G}\right) ^{i-1}\sqrt{\sigma \left( 1+\frac{1}{\mu }\left( \frac{G'}{G}\right) ^2 \right) }\right. \nonumber \\&\left. + \gamma _i^{(l)}\left( \frac{G'}{G}\right) ^{-i} + \delta _i^{(l)}\frac{\left( \frac{G'}{G}\right) ^{-i+1}}{\sqrt{\sigma \left( 1+\frac{1}{\mu }\left( \frac{G'}{G}\right) ^2 \right) }}\right\} \end{aligned}$$

(20)

where $\alpha _0^{(l)}$, $\alpha _i^{(l)}$, $\beta _i^{(l)}$, $\gamma _i^{(l)}$, $\delta _i^{(l)}$ $(i=1,\ldots ,M)$ are constants to be fixed later, $\sigma =\pm 1$, and M is a positive integer, and $G=G(\zeta )$ ensures

$$\begin{aligned} G''+\mu G=0 \end{aligned}$$

(21)

where $\mu$ is a constant to be identified later. Also, Eq. (21) has the following solutions according to the state of parameter $\mu$.

• (i) When $\mu < 0$, solutions are of hyperbolic-type as

  $$\begin{aligned} \frac{G'}{G} = \sqrt{-\mu } \left[ \frac{A_1\sinh (\sqrt{-\mu }\zeta )+A_2\cosh (\sqrt{-\mu }\zeta )}{A_1\cosh (\sqrt{-\mu }\zeta )+A_2\sinh (\sqrt{-\mu }\zeta )}\right] \end{aligned}$$

  (22)

  where $A_j$ for $j=1, 2$ are arbitrary constants and $A_1\ne A_2$.

• (ii) For $\mu >0$, solutions are of trigonometric-type as

  $$\begin{aligned} \frac{G'}{G} = \sqrt{\mu } \left[ \frac{A_1\cos (\sqrt{\mu }\zeta )-A_2\sin (\sqrt{\mu }\zeta )}{A_1\sin (\sqrt{\mu }\zeta )+A_2\cos (\sqrt{\mu }\zeta )}\right] \end{aligned}$$

  (23)

  where $A_j$ for $j=1, 2$ are arbitrary constants.

• (iii) Finally, $\mu =0$, solutions are of rational-type as

  $$\begin{aligned} \frac{G'}{G} = \frac{A_1}{A_1\zeta +A_2} \end{aligned}$$

  (24)

where $A_j$ for $j=1, 2$ are arbitrary constants.

Next, balancing $P_l^{\prime \prime }$ with $P_l^3$ in (19) gives $M=1$. Then, the solution of Eq. (19) becomes

$$\begin{aligned} P_l(\zeta )=&\alpha _0^{(l)}+\alpha _1^{(l)}\left( \frac{G'}{G}\right) + \beta _1^{(l)}\sqrt{\sigma \left( 1+\frac{1}{\mu }\left( \frac{G'}{G}\right) ^2 \right) }\nonumber \\&+ \gamma _1^{(l)}\left( \frac{G'}{G}\right) ^{-1} +\delta _1^{(l)}\frac{1}{\sqrt{\sigma \left( 1+\frac{1}{\mu }\left( \frac{G'}{G}\right) ^2 \right) }}. \end{aligned}$$

(25)

Plugging (25) with (21) into (19) and handing the resulting system, the following solution sets are derived:

Set-1

$$\begin{aligned} \alpha _0^{(l)}=&\beta _1^{(l)}= \gamma _1^{(l)}=\delta _1^{(l)}=0, \quad \alpha _1^{(l)}= \displaystyle \alpha _1^{(l)} , \nonumber \\ \omega =&\displaystyle \frac{2 \kappa (\alpha +b v \kappa )+2Q_l + \left( \alpha _1^{(l)}\right) ^2 \left( \kappa ^2-2 \mu \right) \left( \kappa \left( \beta _l+\theta _l\right) -\eta _l-\xi _l\right) }{2 (b \kappa -1)} , \nonumber \\ a=&\displaystyle \frac{2 b v + \left( \alpha _1^{(l)}\right) ^2 \left( \kappa \left( \beta _l+\theta _l\right) -\eta _l-\xi _l\right) }{2} . \end{aligned}$$

(26)

Set-2

$$\begin{aligned} \alpha _0^{(l)}=&\alpha _1^{(l)}= \gamma _1^{(l)}=\delta _1^{(l)}=0, \quad \beta _1^{(l)}= \displaystyle \beta _1^{(l)} , \nonumber \\ \omega =&\displaystyle \frac{2 \kappa \mu (\alpha +b v \kappa )+ 2 \mu Q_l + \sigma \left( \beta _1^{(l)}\right) ^2 \left( \kappa ^2+\mu \right) \left( \kappa \left( \beta _l+\theta _l\right) -\eta _l-\xi _l\right) }{2 \mu (b \kappa -1)} , \nonumber \\ a=&\displaystyle \frac{2 b v \mu +\sigma \left( \beta _1^{(l)}\right) ^2 \left( \kappa \left( \beta _l+\theta _l\right) -\eta _l-\xi _l\right) }{2 \mu } . \end{aligned}$$

(27)

Set-3

$$\begin{aligned} \alpha _0^{(l)}=&\alpha _1^{(l)}=\beta _1^{(l)}= \delta _1^{(l)}=0, \quad \gamma _1^{(l)}= \displaystyle \gamma _1^{(l)} , \nonumber \\ \omega =&\displaystyle \frac{2 \kappa \mu ^2 (\alpha +b v \kappa ) +2 \mu ^2 Q_l +\left( \gamma _1^{(l)}\right) ^2 \left( \kappa ^2-2 \mu \right) \left( \kappa \left( \beta _l+\theta _l\right) -\eta _l-\xi _l\right) }{2 \mu ^2 (b \kappa -1)} , \nonumber \\ a=&\displaystyle \frac{2 b v \mu ^2 +\left( \gamma _1^{(l)}\right) ^2 \left( \kappa \left( \beta _l+\theta _l\right) -\eta _l-\xi _l\right) }{2 \mu ^2} . \end{aligned}$$

(28)

Set-4

$$\begin{aligned} \alpha _0^{(l)}=&\gamma _1^{(l)}=\delta _1^{(l)}=0, \quad \alpha _1^{(l)}= \displaystyle \pm \frac{ \beta _1^{(l)} \sqrt{\sigma }}{\sqrt{\mu }} , \quad \beta _1^{(l)}=\beta _1^{(l)}, \nonumber \\ \omega =&\displaystyle \frac{\kappa \mu (\alpha +b v \kappa ) +\mu Q_l + \sigma \left( \beta _1^{(l)}\right) ^2 \left( 2 \kappa ^2-\mu \right) \left( \kappa \left( \beta _l+\theta _l\right) -\eta _l-\xi _l\right) }{\mu (b \kappa -1)} , \nonumber \\ a=&\displaystyle \frac{b v \mu -2 \sigma \left( \beta _1^{(l)}\right) ^2 \left( \eta _l+\xi _l-\kappa \left( \beta _l+\theta _l\right) \right) }{\mu } . \end{aligned}$$

(29)

Set-5

$$\begin{aligned} \alpha _0^{(l)}=&\beta _1^{(l)}= \delta _1^{(l)}=0, \quad \alpha _1^{(l)}= \displaystyle \alpha _1^{(l)} , \quad \gamma _1^{(l)}= \alpha _1^{(l)} \mu , \nonumber \\ \omega =&\displaystyle \frac{2 \kappa (\alpha +b v \kappa )+2Q_l + \left( \alpha _1^{(l)}\right) ^2 \left( \kappa ^2+4 \mu \right) \left( \kappa \left( \beta _l+\theta _l\right) -\eta _l-\xi _l\right) }{2 (b \kappa -1)} , \nonumber \\ a=&\displaystyle \frac{2 b v + \left( \alpha _1^{(l)}\right) ^2 \left( \kappa \left( \beta _l+\theta _l\right) -\eta _l-\xi _l\right) }{2} . \end{aligned}$$

(30)

Here $\kappa$ and $\mu$ are arbitrary constants.

As a results, inserting the sets (26)–(30) along with the solutions of (21) into (20), and then considering the wave transformations (5) and (6), solutions to the systems (3) and (4) are revealed as below:

For $\mu <0$, hyperbolic function solutions procured are:

$$\begin{aligned} q(x,t)&= \displaystyle \alpha _1^{(1)} \sqrt{-\mu } \left[ \frac{A_1\sinh (\sqrt{-\mu }\zeta )+A_2\cosh (\sqrt{-\mu }\zeta )}{A_1\cosh (\sqrt{-\mu }\zeta )+A_2\sinh (\sqrt{-\mu }\zeta )}\right] \exp \left[ i \left( - \kappa x + \omega t + \theta \right) \right] \end{aligned}$$

(31)

$$\begin{aligned} r(x,t)&= \displaystyle \alpha _1^{(2)} \sqrt{-\mu } \left[ \frac{A_1\sinh (\sqrt{-\mu }\zeta )+A_2\cosh (\sqrt{-\mu }\zeta )}{A_1\cosh (\sqrt{-\mu }\zeta )+A_2\sinh (\sqrt{-\mu }\zeta )}\right] \exp \left[ i \left( - \kappa x + \omega t + \theta \right) \right] \end{aligned}$$

(32)

$$\begin{aligned} q(x,t)&= \displaystyle \beta _1^{(1)} \sqrt{\sigma \left( 1-\left[ \displaystyle \frac{A_1\sinh (\sqrt{-\mu }\zeta )+A_2\cosh (\sqrt{-\mu }\zeta )}{A_1\cosh (\sqrt{-\mu }\zeta )+A_2\sinh (\sqrt{-\mu }\zeta )}\right] ^{2}\right) }\nonumber \\&\quad \times \exp \left[ i \left( - \kappa x + \omega t + \theta \right) \right] \end{aligned}$$

(33)

$$\begin{aligned} r(x,t)&= \displaystyle \beta _1^{(2)} \sqrt{\sigma \left( 1-\left[ \displaystyle \frac{A_1\sinh (\sqrt{-\mu }\zeta )+A_2\cosh (\sqrt{-\mu }\zeta )}{A_1\cosh (\sqrt{-\mu }\zeta )+A_2\sinh (\sqrt{-\mu }\zeta )}\right] ^{2}\right) }\nonumber \\&\quad \times \exp \left[ i \left( - \kappa x + \omega t + \theta \right) \right] \end{aligned}$$

(34)

$$\begin{aligned} q(x,t)&= \displaystyle \frac{\gamma _1^{(1)}}{\sqrt{-\mu }} \left[ \frac{A_1\sinh (\sqrt{-\mu }\zeta )+A_2\cosh (\sqrt{-\mu }\zeta )}{A_1\cosh (\sqrt{-\mu }\zeta )+A_2\sinh (\sqrt{-\mu }\zeta )}\right] ^{-1}\nonumber \\&\quad \times \exp \left[ i \left( - \kappa x + \omega t + \theta \right) \right] \end{aligned}$$

(35)

$$\begin{aligned} r(x,t)&= \displaystyle \frac{\gamma _1^{(2)}}{\sqrt{-\mu }} \left[ \frac{A_1\sinh (\sqrt{-\mu }\zeta )+A_2\cosh (\sqrt{-\mu }\zeta )}{A_1\cosh (\sqrt{-\mu }\zeta )+A_2\sinh (\sqrt{-\mu }\zeta )}\right] ^{-1}\nonumber \\&\quad \times \exp \left[ i \left( - \kappa x + \omega t + \theta \right) \right] \end{aligned}$$

(36)

$$\begin{aligned} q(x,t)&= \displaystyle \beta _1^{(1)} \sqrt{\sigma } \left( \pm i \frac{A_1\sinh (\sqrt{-\mu }\zeta )+A_2\cosh (\sqrt{-\mu }\zeta )}{A_1\cosh (\sqrt{-\mu }\zeta )+A_2\sinh (\sqrt{-\mu }\zeta )} \right. \nonumber \\&\quad \left. + \sqrt{1-\left[ \displaystyle \frac{A_1\sinh (\sqrt{-\mu }\zeta )+A_2\cosh (\sqrt{-\mu }\zeta )}{A_1\cosh (\sqrt{-\mu }\zeta )+A_2\sinh (\sqrt{-\mu }\zeta )}\right] ^{2}} \right) \nonumber \\&\quad \times \exp \left[ i \left( - \kappa x + \omega t + \theta \right) \right] \end{aligned}$$

(37)

$$\begin{aligned} r(x,t)&= \displaystyle \beta _1^{(2)} \sqrt{\sigma } \left( \pm i \frac{A_1\sinh (\sqrt{-\mu }\zeta )+A_2\cosh (\sqrt{-\mu }\zeta )}{A_1\cosh (\sqrt{-\mu }\zeta )+A_2\sinh (\sqrt{-\mu }\zeta )}\right. \nonumber \\&\quad \left. + \sqrt{1-\left[ \displaystyle \frac{A_1\sinh (\sqrt{-\mu }\zeta )+A_2\cosh (\sqrt{-\mu }\zeta )}{A_1\cosh (\sqrt{-\mu }\zeta )+A_2\sinh (\sqrt{-\mu }\zeta )}\right] ^{2}} \right) \nonumber \\&\quad \times \exp \left[ i \left( - \kappa x + \omega t + \theta \right) \right] \end{aligned}$$

(38)

$$\begin{aligned} q(x,t)&= \displaystyle \alpha _1^{(1)} \sqrt{-\mu } \left( \frac{A_1\sinh (\sqrt{-\mu }\zeta )+A_2\cosh (\sqrt{-\mu }\zeta )}{A_1\cosh (\sqrt{-\mu }\zeta )+A_2\sinh (\sqrt{-\mu }\zeta )}\right. \nonumber \\&\quad \left. - \left[ \frac{A_1\sinh (\sqrt{-\mu }\zeta )+A_2\cosh (\sqrt{-\mu }\zeta )}{A_1\cosh (\sqrt{-\mu }\zeta )+A_2\sinh (\sqrt{-\mu }\zeta )}\right] ^{-1} \right) \nonumber \\&\quad \times \exp \left[ i \left( - \kappa x + \omega t + \theta \right) \right] \end{aligned}$$

(39)

$$\begin{aligned} r(x,t)&=\displaystyle \alpha _1^{(2)} \sqrt{-\mu } \left( \frac{A_1\sinh (\sqrt{-\mu }\zeta )+A_2\cosh (\sqrt{-\mu }\zeta )}{A_1\cosh (\sqrt{-\mu }\zeta )+A_2\sinh (\sqrt{-\mu }\zeta )}\right. \nonumber \\&\quad \left. - \left[ \frac{A_1\sinh (\sqrt{-\mu }\zeta )+A_2\cosh (\sqrt{-\mu }\zeta )}{A_1\cosh (\sqrt{-\mu }\zeta )+A_2\sinh (\sqrt{-\mu }\zeta )}\right] ^{-1} \right) \nonumber \\&\quad \times \exp \left[ i \left( - \kappa x + \omega t + \theta \right) \right] \end{aligned}$$

(40)

where $\zeta = x- \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t$ and $\omega$ are given by the sets (26)–(30), respectively.

If, however, $\mu >0$, trigonometric function solutions acquired are:

$$\begin{aligned} q(x,t)=&\displaystyle \alpha _1^{(1)} \sqrt{\mu } \left[ \frac{A_1\cos (\sqrt{\mu }\zeta )-A_2\sin (\sqrt{\mu }\zeta )}{A_1\sin (\sqrt{\mu }\zeta )+A_2\cos (\sqrt{\mu }\zeta )}\right] \nonumber \\& \times \exp \left[ i \left( - \kappa x + \omega t + \theta \right) \right] \end{aligned}$$

(41)

$$\begin{aligned} r(x,t)=&\displaystyle \alpha _1^{(2)} \sqrt{\mu } \left[ \frac{A_1\cos (\sqrt{\mu }\zeta )-A_2\sin (\sqrt{\mu }\zeta )}{A_1\sin (\sqrt{\mu }\zeta )+A_2\cos (\sqrt{\mu }\zeta )}\right] \nonumber \\& \times \exp \left[ i \left( - \kappa x + \omega t + \theta \right) \right] \end{aligned}$$

(42)

$$\begin{aligned} q(x,t)=&\displaystyle \beta _1^{(1)} \sqrt{\sigma \left( 1+\left[ \frac{A_1\cos (\sqrt{\mu }\zeta )-A_2\sin (\sqrt{\mu }\zeta )}{A_1\sin (\sqrt{\mu }\zeta )+A_2\cos (\sqrt{\mu }\zeta )} \right] ^{2}\right) }\nonumber \\& \times \exp \left[ i \left( - \kappa x + \omega t + \theta \right) \right] \end{aligned}$$

(43)

$$\begin{aligned} r(x,t)=&\displaystyle \beta _1^{(2)} \sqrt{\sigma \left( 1+\left[ \frac{A_1\cos (\sqrt{\mu }\zeta )-A_2\sin (\sqrt{\mu }\zeta )}{A_1\sin (\sqrt{\mu }\zeta )+A_2\cos (\sqrt{\mu }\zeta )} \right] ^{2}\right) }\nonumber \\& \times \exp \left[ i \left( - \kappa x + \omega t + \theta \right) \right] \end{aligned}$$

(44)

$$\begin{aligned} q(x,t)=&\displaystyle \frac{\gamma _1^{(1)}}{\sqrt{\mu }} \left[ \frac{A_1\cos (\sqrt{\mu }\zeta )-A_2\sin (\sqrt{\mu }\zeta )}{A_1\sin (\sqrt{\mu }\zeta )+A_2\cos (\sqrt{\mu }\zeta )}\right] ^{-1}\nonumber \\& \times \exp \left[ i \left( - \kappa x + \omega t + \theta \right) \right] \end{aligned}$$

(45)

$$\begin{aligned} r(x,t)=&\displaystyle \frac{\gamma _1^{(2)}}{\sqrt{\mu }} \left[ \frac{A_1\cos (\sqrt{\mu }\zeta )-A_2\sin (\sqrt{\mu }\zeta )}{A_1\sin (\sqrt{\mu }\zeta )+A_2\cos (\sqrt{\mu }\zeta )}\right] ^{-1} \nonumber \\& \times \exp \left[ i \left( - \kappa x + \omega t + \theta \right) \right] \end{aligned}$$

(46)

$$\begin{aligned} q(x,t)=&\displaystyle \beta _1^{(1)} \sqrt{\sigma } \left( \pm \frac{A_1\cos (\sqrt{\mu }\zeta )-A_2\sin (\sqrt{\mu }\zeta )}{A_1\sin (\sqrt{\mu }\zeta )+A_2\cos (\sqrt{\mu }\zeta )}\right. \left. + \sqrt{1+\left[ \frac{A_1\cos (\sqrt{\mu }\zeta )-A_2\sin (\sqrt{\mu }\zeta )}{A_1\sin (\sqrt{\mu }\zeta )+A_2\cos (\sqrt{\mu }\zeta )} \right] ^{2}} \right) \nonumber \\&\times \exp \left[ i \left( - \kappa x + \omega t + \theta \right) \right] \end{aligned}$$

(47)

$$\begin{aligned} r(x,t)=&\displaystyle \beta _1^{(2)} \sqrt{\sigma } \left( \pm \frac{A_1\cos (\sqrt{\mu }\zeta )-A_2\sin (\sqrt{\mu }\zeta )}{A_1\sin (\sqrt{\mu }\zeta )+A_2\cos (\sqrt{\mu }\zeta )}\right. \left. + \sqrt{1+\left[ \frac{A_1\cos (\sqrt{\mu }\zeta )-A_2\sin (\sqrt{\mu }\zeta )}{A_1\sin (\sqrt{\mu }\zeta )+A_2\cos (\sqrt{\mu }\zeta )} \right] ^{2}} \right) \nonumber \\&\times \exp \left[ i \left( - \kappa x + \omega t + \theta \right) \right] \end{aligned}$$

(48)

$$\begin{aligned} q(x,t)=&\displaystyle \alpha _1^{(1)} \sqrt{\mu } \left( \frac{A_1\cos (\sqrt{\mu }\zeta )-A_2\sin (\sqrt{\mu }\zeta )}{A_1\sin (\sqrt{\mu }\zeta )+A_2\cos (\sqrt{\mu }\zeta )} + \left[ \frac{A_1\cos (\sqrt{\mu }\zeta )-A_2\sin (\sqrt{\mu }\zeta )}{A_1\sin (\sqrt{\mu }\zeta )+A_2\cos (\sqrt{\mu }\zeta )}\right] ^{-1} \right) \nonumber \\&\times \exp \left[ i \left( - \kappa x + \omega t + \theta \right) \right] \end{aligned}$$

(49)

$$\begin{aligned} r(x,t)=&\displaystyle \alpha _1^{(2)} \sqrt{\mu } \left( \frac{A_1\cos (\sqrt{\mu }\zeta )-A_2\sin (\sqrt{\mu }\zeta )}{A_1\sin (\sqrt{\mu }\zeta )+A_2\cos (\sqrt{\mu }\zeta )} + \left[ \frac{A_1\cos (\sqrt{\mu }\zeta )-A_2\sin (\sqrt{\mu }\zeta )}{A_1\sin (\sqrt{\mu }\zeta )+A_2\cos (\sqrt{\mu }\zeta )}\right] ^{-1} \right) \nonumber \\&\times \exp \left[ i \left( - \kappa x + \omega t + \theta \right) \right] \end{aligned}$$

(50)

where $\zeta = x- \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t$ and $\omega$ are given by the sets (26)–(30), respectively.

Finally, whenever $\mu =0$, plane wave solutions secured are: By Eqs. (26) and (30), one gets

$$\begin{aligned} q(x,t)=&\displaystyle \alpha _1^{(1)} \left( \frac{A_1}{A_1\zeta +A_2} \right) \exp \left[ i \left( - \kappa x + \omega t + \theta \right) \right] \end{aligned}$$

(51)

$$\begin{aligned} r(x,t)=&\displaystyle \alpha _1^{(2)} \left( \frac{A_1}{A_1\zeta +A_2} \right) \exp \left[ i \left( - \kappa x + \omega t + \theta \right) \right] \end{aligned}$$

(52)

while, by the help of Eq. (28), one has

$$\begin{aligned} q(x,t)=&\displaystyle \gamma _1^{(1)} \left( \frac{A_1}{A_1\zeta +A_2} \right) ^{-1} \exp \left[ i \left( - \kappa x + \omega t + \theta \right) \right] \end{aligned}$$

(53)

$$\begin{aligned} r(x,t)=&\displaystyle \gamma _1^{(2)} \left( \frac{A_1}{A_1\zeta +A_2} \right) ^{-1} \exp \left[ i \left( - \kappa x + \omega t + \theta \right) \right] \end{aligned}$$

(54)

where $\zeta = x- \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t$ and $\omega$ is given by the sets (26) and (28).

The special cases are listed as:

For $\mu <0$ and $A_1^2>A_2^2$, the solutions from (31) to (40) turn into the following optical soliton solutions, respectively:

Dark solitons

$$\begin{aligned} q(x,t)=&\displaystyle \alpha _1^{(1)} \sqrt{-\mu } \tanh \left[ \sqrt{-\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \textstyle \frac{2 \kappa (\alpha +b v \kappa )+2Q_l + \left( \alpha _1^{(l)}\right) ^2 \left( \kappa ^2-2 \mu \right) \left( \kappa \left( \beta _l+\theta _l\right) -\eta _l-\xi _l\right) }{2 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(55)

$$\begin{aligned} r(x,t)=&\displaystyle \alpha _1^{(2)} \sqrt{-\mu } \tanh \left[ \sqrt{-\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \textstyle \frac{2 \kappa (\alpha +b v \kappa )+2Q_l + \left( \alpha _1^{(l)}\right) ^2 \left( \kappa ^2-2 \mu \right) \left( \kappa \left( \beta _l+\theta _l\right) -\eta _l-\xi _l\right) }{2 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(56)

bright solitons

$$\begin{aligned} q(x,t)=&\displaystyle \beta _1^{(1)} \sqrt{\sigma } {{\,{\mathrm{sech}}\,}}\left[ \sqrt{-\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \textstyle \frac{2 \kappa \mu (\alpha +b v \kappa )+ 2 \mu Q_l + \sigma \left( \beta _1^{(l)}\right) ^2 \left( \kappa ^2+\mu \right) \left( \kappa \left( \beta _l+\theta _l\right) -\eta _l-\xi _l\right) }{2 \mu (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(57)

$$\begin{aligned} r(x,t)=&\displaystyle \beta _1^{(2)} \sqrt{\sigma } {{\,{\mathrm{sech}}\,}}\left[ \sqrt{-\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \textstyle \frac{2 \kappa \mu (\alpha +b v \kappa )+ 2 \mu Q_l + \sigma \left( \beta _1^{(l)}\right) ^2 \left( \kappa ^2+\mu \right) \left( \kappa \left( \beta _l+\theta _l\right) -\eta _l-\xi _l\right) }{2 \mu (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(58)

singular solitons

$$\begin{aligned} q(x,t)=&\displaystyle \frac{\gamma _1^{(1)}}{\sqrt{-\mu }} \coth \left[ \sqrt{-\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \textstyle \frac{2 \kappa \mu ^2 (\alpha +b v \kappa ) +2 \mu ^2 Q_l +\left( \gamma _1^{(l)}\right) ^2 \left( \kappa ^2-2 \mu \right) \left( \kappa \left( \beta _l+\theta _l\right) -\eta _l-\xi _l\right) }{2 \mu ^2 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(59)

$$\begin{aligned} r(x,t)=&\displaystyle \frac{\gamma _1^{(2)}}{\sqrt{-\mu }} \coth \left[ \sqrt{-\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \textstyle \frac{2 \kappa \mu ^2 (\alpha +b v \kappa ) +2 \mu ^2 Q_l +\left( \gamma _1^{(l)}\right) ^2 \left( \kappa ^2-2 \mu \right) \left( \kappa \left( \beta _l+\theta _l\right) -\eta _l-\xi _l\right) }{2 \mu ^2 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(60)

complexiton solutions

$$\begin{aligned} q(x,t)=&\displaystyle \beta _1^{(1)} \sqrt{\sigma }\left( \pm i \tanh \left[ \sqrt{-\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right. \nonumber \\&\left. + {{\,{\mathrm{sech}}\,}}\left[ \sqrt{-\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \textstyle \frac{\kappa \mu (\alpha +b v \kappa ) +\mu Q_l + \sigma \left( \beta _1^{(l)}\right) ^2 \left( 2 \kappa ^2-\mu \right) \left( \kappa \left( \beta _l+\theta _l\right) -\eta _l-\xi _l\right) }{\mu (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(61)

$$\begin{aligned} r(x,t)=&\displaystyle \beta _1^{(2)} \sqrt{\sigma } \left( \pm i \tanh \left[ \sqrt{-\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right. \nonumber \\&\left. + {{\,{\mathrm{sech}}\,}}\left[ \sqrt{-\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \textstyle \frac{\kappa \mu (\alpha +b v \kappa ) +\mu Q_l + \sigma \left( \beta _1^{(l)}\right) ^2 \left( 2 \kappa ^2-\mu \right) \left( \kappa \left( \beta _l+\theta _l\right) -\eta _l-\xi _l\right) }{\mu (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(62)

and singular solitons

$$\begin{aligned} q(x,t)=&\displaystyle -2 \alpha _1^{(1)} \sqrt{-\mu } {{\,{\mathrm{csch}}\,}}\left[ 2\sqrt{-\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + 2\zeta _0 \right] \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \textstyle \frac{2 \kappa (\alpha +b v \kappa )+2Q_l + \left( \alpha _1^{(l)}\right) ^2 \left( \kappa ^2+4 \mu \right) \left( \kappa \left( \beta _l+\theta _l\right) -\eta _l-\xi _l\right) }{2 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(63)

$$\begin{aligned} r(x,t)=&\displaystyle -2 \alpha _1^{(2)} \sqrt{-\mu } {{\,{\mathrm{csch}}\,}}\left[ 2\sqrt{-\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + 2\zeta _0 \right] \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \textstyle \frac{2 \kappa (\alpha +b v \kappa )+2Q_l + \left( \alpha _1^{(l)}\right) ^2 \left( \kappa ^2+4 \mu \right) \left( \kappa \left( \beta _l+\theta _l\right) -\eta _l-\xi _l\right) }{2 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(64)

where $\zeta _0=\tanh ^{-1}(A_2/A_1)$. Also, taking $A_1=0$, $A_2\ne 0$ or $A_2=0$, $A_1\ne 0$ in the solutions form (31) to (40), more solitary wave solutions can be recovered. However, these are omitted.

If, however, $\mu >0$, the solutions from (41) to (50) turn into periodic waves, respectively

$$\begin{aligned} q(x,t)=&\displaystyle - \alpha _1^{(1)} \sqrt{\mu } \tan \left[ \sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} - \zeta _0 \right] \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \textstyle \frac{2 \kappa (\alpha +b v \kappa )+2Q_l + \left( \alpha _1^{(l)}\right) ^2 \left( \kappa ^2-2 \mu \right) \left( \kappa \left( \beta _l+\theta _l\right) -\eta _l-\xi _l\right) }{2 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(65)

$$\begin{aligned} r(x,t)=&\displaystyle - \alpha _1^{(2)} \sqrt{\mu } \tan \left[ \sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} - \zeta _0 \right] \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \textstyle \frac{2 \kappa (\alpha +b v \kappa )+2Q_l + \left( \alpha _1^{(l)}\right) ^2 \left( \kappa ^2-2 \mu \right) \left( \kappa \left( \beta _l+\theta _l\right) -\eta _l-\xi _l\right) }{2 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(66)

$$\begin{aligned} q(x,t)=&\displaystyle \beta _1^{(1)} \sqrt{\sigma } \sec \left[ \sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} - \zeta _0 \right] \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \textstyle \frac{2 \kappa \mu (\alpha +b v \kappa )+ 2 \mu Q_l + \sigma \left( \beta _1^{(l)}\right) ^2 \left( \kappa ^2+\mu \right) \left( \kappa \left( \beta _l+\theta _l\right) -\eta _l-\xi _l\right) }{2 \mu (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(67)

$$\begin{aligned} r(x,t)=&\displaystyle \beta _1^{(2)} \sqrt{\sigma } \sec \left[ \sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} - \zeta _0 \right] \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \textstyle \frac{2 \kappa \mu (\alpha +b v \kappa )+ 2 \mu Q_l + \sigma \left( \beta _1^{(l)}\right) ^2 \left( \kappa ^2+\mu \right) \left( \kappa \left( \beta _l+\theta _l\right) -\eta _l-\xi _l\right) }{2 \mu (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(68)

$$\begin{aligned} q(x,t)=&\displaystyle - \frac{\gamma _1^{(1)}}{\sqrt{\mu }} \cot \left[ \sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} - \zeta _0 \right] \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \textstyle \frac{2 \kappa \mu ^2 (\alpha +b v \kappa ) +2 \mu ^2 Q_l +\left( \gamma _1^{(l)}\right) ^2 \left( \kappa ^2-2 \mu \right) \left( \kappa \left( \beta _l+\theta _l\right) -\eta _l-\xi _l\right) }{2 \mu ^2 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(69)

$$\begin{aligned} r(x,t)=&\displaystyle - \frac{\gamma _1^{(2)}}{\sqrt{\mu }} \cot \left[ \sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} - \zeta _0 \right] \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \textstyle \frac{2 \kappa \mu ^2 (\alpha +b v \kappa ) +2 \mu ^2 Q_l +\left( \gamma _1^{(l)}\right) ^2 \left( \kappa ^2-2 \mu \right) \left( \kappa \left( \beta _l+\theta _l\right) -\eta _l-\xi _l\right) }{2 \mu ^2 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(70)

$$\begin{aligned} q(x,t)=&\displaystyle \beta _1^{(1)} \sqrt{\sigma } \left( \sec \left[ \sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} - \zeta _0 \right] \right. \nonumber \\&\left. \mp \tan \left[ \sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} - \zeta _0 \right] \right) \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \textstyle \frac{\kappa \mu (\alpha +b v \kappa ) +\mu Q_l + \sigma \left( \beta _1^{(l)}\right) ^2 \left( 2 \kappa ^2-\mu \right) \left( \kappa \left( \beta _l+\theta _l\right) -\eta _l-\xi _l\right) }{\mu (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(71)

$$\begin{aligned} r(x,t)=&\displaystyle \beta _1^{(2)} \sqrt{\sigma } \left( \sec \left[ \sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} - \zeta _0 \right] \right. \nonumber \\&\left. \mp \tan \left[ \sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} - \zeta _0 \right] \right) \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \textstyle \frac{\kappa \mu (\alpha +b v \kappa ) +\mu Q_l + \sigma \left( \beta _1^{(l)}\right) ^2 \left( 2 \kappa ^2-\mu \right) \left( \kappa \left( \beta _l+\theta _l\right) -\eta _l-\xi _l\right) }{\mu (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(72)

$$\begin{aligned} q(x,t)=&\displaystyle -2 \alpha _1^{(1)} \sqrt{\mu } \csc \left[ 2\sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} - 2\zeta _0 \right] \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \textstyle \frac{2 \kappa (\alpha +b v \kappa )+2Q_l + \left( \alpha _1^{(l)}\right) ^2 \left( \kappa ^2+4 \mu \right) \left( \kappa \left( \beta _l+\theta _l\right) -\eta _l-\xi _l\right) }{2 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(73)

$$\begin{aligned} r(x,t)=&\displaystyle -2 \alpha _1^{(2)} \sqrt{\mu } \csc \left[ 2\sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} - 2\zeta _0 \right] \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \textstyle \frac{2 \kappa (\alpha +b v \kappa )+2Q_l + \left( \alpha _1^{(l)}\right) ^2 \left( \kappa ^2+4 \mu \right) \left( \kappa \left( \beta _l+\theta _l\right) -\eta _l-\xi _l\right) }{2 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(74)

where $\zeta _0=\tan ^{-1}(A_1/A_2)$. Also, setting $A_1=0$, $A_2\ne 0$ or $A_2=0$, $A_1\ne 0$ in the solutions from (31) to (40), more periodic waves can be procured. However, these are ignored.

A few numerical simulations which stand for bright and dark soliton profile are exhibited as the following. The picked parameter values are also indicated respectively.

The profiles of dark solitons are represented by Fig. 1. The parameter values chosen are $a= 0.25, \ b= 1.5, \ \alpha = 0.15, \ \zeta _0 = 0, \ \kappa = 0.1, \ \mu = -1, \ \omega = 0.7, \ \alpha _1^{(1)} =1, \ \alpha _1^{(2)} =2.$

That of bright solitons are given by Fig. 2. In this case, the picked parameter values are $a= 0.25, \ b= 1.5, \ \alpha = 0.15, \ \zeta _0 = 0, \ \kappa = 0.1, \ \mu = -1, \ \omega = 0.7, \ \beta _1^{(1)} =1, \ \beta _1^{(2)} =2.$

Fig. 1

Numerical simulations of dark solitons (55) and (56)

Fig. 2

Numerical simulations of bright solitons (57) and (58)

3 Power law

For power law nonlinearity, $F(s)=s^n$, where n stands for the power law nonlinearity parameter. Here the parameter n is in the range $0<n<2$ and also $n\ne 2$ for avoiding self-focusing singularity. Therefore, the system given by (1) and (2) collapses to

$$\begin{aligned}&iq_{t}+a_1q_{xx}+b_1q_{xt}+ \left\{ \xi _1 \left| q\right| ^{2n} + \eta _1 \left| r\right| ^{2n} \right\} q \nonumber \\&\quad = Q_1 r + i \left\{ \alpha _1 q_{x} + \beta _1 \left( \left| q\right| ^{2n}q\right) _{x}\right. \nonumber \\&\qquad \left. + \nu _1 \left( \left| q\right| ^{2n}\right) _{x}q + \theta _1 \left| q\right| ^{2n} q_{x} \right\} \end{aligned}$$

(75)

$$\begin{aligned}&ir_{t}+a_2r_{xx}+b_2r_{xt}+ \left\{ \xi _2 \left| r\right| ^{2n} + \eta _2 \left| q\right| ^{2n} \right\} r\nonumber \\&\quad = Q_2 q + i \left\{ \alpha _2 r_{x} + \beta _2 \left( \left| r\right| ^{2n}r\right) _{x}\right. \nonumber \\&\qquad \left. + \nu _2 \left( \left| r\right| ^{2n}\right) _{x}r + \theta _2 \left| r\right| ^{2n} r_{x} \right\} . \end{aligned}$$

(76)

Insert (5) and (6) into (75) and (76). The real parts lead to

$$\begin{aligned}&(a_l-b_lv)P_l''+ \left( b_l\omega \kappa -\omega -a_l\kappa ^2 -\alpha _l\kappa \right) P_l +\left( \xi _l P_l^{2n} + \eta _lP_{{\bar{l}}}^{2n} \right) P_l = Q_lP_{{\bar{l}}} + \kappa \left( \beta _l + \theta _l\right) P_l^{2n+1} \end{aligned}$$

(77)

while the imaginary parts imply

$$\begin{aligned} v(b_l\kappa - 1)P_l'+ \left( b_l\omega -2a_l\kappa -\alpha _l\right) P_l'= \left\{ (2n+1) \beta _l + 2 n \nu _l + \theta _l\right\} P_l^{2n}P_l' . \end{aligned}$$

(78)

From (78) is possible to obtain the solitons speed (11) as long as the constraint

$$\begin{aligned} (2n+1) \beta _l + 2 n \nu _l + \theta _l = 0 \end{aligned}$$

(79)

remains valid. As a consequence (13) and (14) are also satisfied in this case, and the real part (77) simplifies to

$$\begin{aligned}&(a-bv)P_l''+ \left( b\omega \kappa -\omega -a\kappa ^2 -\alpha \kappa \right) P_l +\left( \xi _l P_l^{2n} + \eta _lP_{{\bar{l}}}^{2n} \right) P_l = Q_lP_{{\bar{l}}} + \kappa \left( \beta _l + \theta _l\right) P_l^{2n+1}. \end{aligned}$$

(80)

From the balancing principle

$$\begin{aligned} P_{{ {\bar{l}}}}=P_l \end{aligned}$$

(81)

and then Eq. (80) changes to

$$\begin{aligned} (a-b v)P_l^{\prime \prime }+\left( b \omega \kappa - \omega -a \kappa ^2 - \alpha \kappa -Q_l\right) P_l+ \left( \xi _l+\eta _l- \kappa \left( \beta _l+\theta _l\right) \right) P_l^{2n+1}=0. \end{aligned}$$

(82)

Set

$$\begin{aligned} P_l=U_l^{\frac{1}{2n}} \end{aligned}$$

(83)

so that (82) turns into

$$\begin{aligned}&(a-b v) \left( 2 n U_l U_l''-(2n-1)\left( U_l'\right) ^2 \right) -4 n^2 \left( \alpha \kappa + a \kappa ^2 + \omega -b \kappa \omega +Q_l \right) U_l^2 \nonumber \\&\qquad +4n^2 \left( \eta _l +\xi _l - \kappa \left( \beta _l+\theta _l\right) \right) U_l^3=0. \end{aligned}$$

(84)

3.1 Extended $G^{\prime }/G-$Expansion approach

Balancing $U_l U_l^{\prime \prime }$ or $(U_l')^2$ with $U_l^3$ in Eq. (84) gives $M = 2$. Thus, one reaches

$$\begin{aligned} U_l(\zeta )=&\alpha _0^{(l)}+ \sum _{i=1}^{2}\left\{ \alpha _i^{(l)}\left( \frac{G'}{G}\right) ^i + \beta _i^{(l)}\left( \frac{G'}{G}\right) ^{i-1}\sqrt{\sigma \left( 1+\frac{1}{\mu }\left( \frac{G'}{G}\right) ^2 \right) } \right. \nonumber \\&+ \gamma _i^{(l)}\left( \frac{G'}{G}\right) ^{-i}+ \left. \delta _i^{(l)}\frac{\left( \frac{G'}{G}\right) ^{-i+1}}{\sqrt{\sigma \left( 1+\frac{1}{\mu }\left( \frac{G'}{G}\right) ^2 \right) }}\right\} . \end{aligned}$$

(85)

Proceeding as in the previous section, the following solutions are recovered:

For $\mu <0$ and $A_1^2>A_2^2$, one secures bright solitons

$$\begin{aligned} q(x,t)=&\left\{ \displaystyle -\frac{ \mu (n+1) (a-b v)}{n^2 \left( \eta _1+\xi _1-\kappa \left( \beta _1+\theta _1\right) \right) } {{\,{\mathrm{sech}}\,}}^2\left[ \sqrt{-\mu } \left\{ x - \textstyle \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right\} ^\frac{1}{2n} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{ \mu (a-b v)+ n^2 (a \kappa ^2 +\alpha \kappa +Q_l) }{n^2 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(86)

$$\begin{aligned} r(x,t)=&\left\{ \displaystyle -\frac{ \mu (n+1) (a-b v)}{n^2 \left( \eta _2+\xi _2-\kappa \left( \beta _2+\theta _2\right) \right) } {{\,{\mathrm{sech}}\,}}^2\left[ \sqrt{-\mu } \left\{ x - \textstyle \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right\} ^\frac{1}{2n} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{ \mu (a-b v)+ n^2 (a \kappa ^2 +\alpha \kappa +Q_l) }{n^2 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(87)

singular solitons

$$\begin{aligned} q(x,t)=&\left\{ \displaystyle \frac{ \mu (n+1) (a-b v)}{ n^2\left( \eta _1+\xi _1-\kappa \left( \beta _1+\theta _1\right) \right) } {{\,{\mathrm{csch}}\,}}^2\left[ \sqrt{-\mu } \left\{ x - \textstyle \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right\} ^\frac{1}{2n} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{ \mu (a-b v)+ n^2 (a \kappa ^2 +\alpha \kappa +Q_l) }{n^2 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(88)

$$\begin{aligned} r(x,t)=&\left\{ \displaystyle \frac{ \mu (n+1) (a-b v)}{ n^2\left( \eta _2+\xi _2-\kappa \left( \beta _2+\theta _2\right) \right) } {{\,{\mathrm{csch}}\,}}^2\left[ \sqrt{-\mu } \left\{ x \textstyle - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right\} ^\frac{1}{2n} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{ \mu (a-b v)+ n^2 (a \kappa ^2 +\alpha \kappa +Q_l) }{n^2 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(89)

complexiton solutions

$$\begin{aligned} q(x,t)=&\left\{ \displaystyle -\frac{ \mu (n+1) (a-b v)}{2 n^2 \left( \eta _1+\xi _1-\kappa \left( \beta _1+\theta _1\right) \right) } {{\,{\mathrm{sech}}\,}}\left[ \textstyle \sqrt{-\mu } \left\{ x - \textstyle \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right. \nonumber \\&\times \left. \left( {{\,{\mathrm{sech}}\,}}\left[ \scriptstyle \sqrt{-\mu } \left\{ x - \textstyle \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \pm i \tanh \left[ \scriptstyle \sqrt{-\mu } \left\{ x - \textstyle \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \right\} ^\frac{1}{2n} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{\mu (a-b v)+4 n^2 (a \kappa ^2 +\alpha \kappa + Q_l)}{4 n^2 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(90)

$$\begin{aligned} r(x,t)=&\left\{ \displaystyle -\frac{ \mu (n+1) (a-b v)}{2 n^2 \left( \eta _2+\xi _2-\kappa \left( \beta _2+\theta _2\right) \right) } {{\,{\mathrm{sech}}\,}}\left[ \sqrt{-\mu } \left\{ x - \textstyle \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right. \nonumber \\&\times \left. \left( {{\,{\mathrm{sech}}\,}}\left[ \scriptstyle \sqrt{-\mu } \left\{ x - \textstyle \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \pm i \tanh \left[ \scriptstyle \sqrt{-\mu } \left\{ x - \textstyle \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \right\} ^\frac{1}{2n} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{\mu (a-b v)+4 n^2 (a \kappa ^2 +\alpha \kappa + Q_l)}{4 n^2 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(91)

and singular solitons

$$\begin{aligned} q(x,t)=&\left\{ \displaystyle \frac{ 4\mu (n+1) (a-b v)}{n^2 \left( \eta _1+\xi _1-\kappa \left( \beta _1+\theta _1\right) \right) } {{\,{\mathrm{csch}}\,}}^2\left[ 2\sqrt{-\mu } \left\{ x - \textstyle \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + 2\zeta _0 \right] \right\} ^\frac{1}{2n} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{4 \mu (a-b v)+n^2 (a \kappa ^2 +\alpha \kappa +Q_l)}{n^2 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(92)

$$\begin{aligned} r(x,t)=&\left\{ \displaystyle \frac{ 4\mu (n+1) (a-b v)}{n^2 \left( \eta _2+\xi _2-\kappa \left( \beta _2+\theta _2\right) \right) } {{\,{\mathrm{csch}}\,}}^2\left[ 2\sqrt{-\mu } \left\{ x - \textstyle \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + 2\zeta _0 \right] \right\} ^\frac{1}{2n} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{4 \mu (a-b v)+n^2 (a \kappa ^2 +\alpha \kappa +Q_l)}{n^2 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(93)

where $\zeta _0=\tanh ^{-1}(A_2/A_1).$ If, however, $\mu >0$, one gets periodic waves as:

$$\begin{aligned} q(x,t)=&\left\{ \displaystyle -\frac{ \mu (n+1) (a-b v)}{n^2 \left( \eta _1+\xi _1-\kappa \left( \beta _1+\theta _1\right) \right) } \sec ^2\left[ \sqrt{\mu } \left\{ x - \textstyle \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} - \zeta _0 \right] \right\} ^\frac{1}{2n} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{ \mu (a-b v)+ n^2 (a \kappa ^2 +\alpha \kappa +Q_l) }{n^2 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(94)

$$\begin{aligned} r(x,t)=&\left\{ \displaystyle -\frac{ \mu (n+1) (a-b v)}{n^2 \left( \eta _2+\xi _2-\kappa \left( \beta _2+\theta _2\right) \right) } \sec ^2\left[ \sqrt{\mu } \left\{ x - \textstyle \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} - \zeta _0 \right] \right\} ^\frac{1}{2n} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{ \mu (a-b v)+ n^2 (a \kappa ^2 +\alpha \kappa +Q_l) }{n^2 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(95)

$$\begin{aligned} q(x,t)=&\left\{ \displaystyle -\frac{ \mu (n+1) (a-b v)}{ n^2\left( \eta _1+\xi _1-\kappa \left( \beta _1+\theta _1\right) \right) } \csc ^2\left[ \sqrt{\mu } \left\{ x - \textstyle \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} - \zeta _0 \right] \right\} ^\frac{1}{2n} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{ \mu (a-b v)+ n^2 (a \kappa ^2 +\alpha \kappa +Q_l) }{n^2 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(96)

$$\begin{aligned} r(x,t)=&\left\{ \displaystyle -\frac{ \mu (n+1) (a-b v)}{ n^2\left( \eta _2+\xi _2-\kappa \left( \beta _2+\theta _2\right) \right) } \csc ^2\left[ \sqrt{\mu } \left\{ x - \textstyle \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} - \zeta _0 \right] \right\} ^\frac{1}{2n} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{ \mu (a-b v)+ n^2 (a \kappa ^2 +\alpha \kappa +Q_l) }{n^2 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(97)

$$\begin{aligned} q(x,t)=&\left\{ \displaystyle -\frac{ \mu (n+1) (a-b v)}{2 n^2 \left( \eta _1+\xi _1-\kappa \left( \beta _1+\theta _1\right) \right) } \sec \left[ \sqrt{\mu } \left\{ x - \textstyle \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} - \zeta _0 \right] \right. \nonumber \\&\times \left. \left( \sec \left[ \sqrt{\mu } \left\{ x - \textstyle \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} - \zeta _0 \right] \pm \tan \left[ \sqrt{\mu } \left\{ x - \textstyle \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} - \zeta _0 \right] \right) \right\} ^\frac{1}{2n} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{\mu (a-b v)+4 n^2 (a \kappa ^2 +\alpha \kappa + Q_l)}{4 n^2 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(98)

$$\begin{aligned} r(x,t)=&\left\{ \displaystyle -\frac{ \mu (n+1) (a-b v)}{2 n^2 \left( \eta _2+\xi _2-\kappa \left( \beta _2+\theta _2\right) \right) } \sec \left[ \sqrt{\mu } \left\{ x - \textstyle \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} - \zeta _0 \right] \right. \nonumber \\&\times \left. \left( \sec \left[ \sqrt{\mu } \left\{ x - \textstyle \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} - \zeta _0 \right] \pm \tan \left[ \sqrt{\mu } \left\{ x - \textstyle \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} - \zeta _0 \right] \right) \right\} ^\frac{1}{2n} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{\mu (a-b v)+4 n^2 (a \kappa ^2 +\alpha \kappa + Q_l)}{4 n^2 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(99)

$$\begin{aligned} q(x,t)=&\left\{ \displaystyle - \frac{ 4\mu (n+1) (a-b v)}{n^2 \left( \eta _1+\xi _1-\kappa \left( \beta _1+\theta _1\right) \right) } \csc ^2\left[ 2\sqrt{\mu } \left\{ x - \textstyle \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} - 2\zeta _0 \right] \right\} ^\frac{1}{2n} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{4 \mu (a-b v)+n^2 (a \kappa ^2 +\alpha \kappa +Q_l)}{n^2 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(100)

$$\begin{aligned} r(x,t)=&\left\{ \displaystyle - \frac{ 4\mu (n+1) (a-b v)}{n^2 \left( \eta _2+\xi _2-\kappa \left( \beta _2+\theta _2\right) \right) } \csc ^2\left[ 2\sqrt{\mu } \left\{ x - \textstyle \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} - 2\zeta _0 \right] \right\} ^\frac{1}{2n} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{4 \mu (a-b v)+n^2 (a \kappa ^2 +\alpha \kappa +Q_l)}{n^2 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(101)

where $\zeta _0=\tan ^{-1}(A_1/A_2).$

The profiles of bright solitons are stood for by Fig. 3. The parameter values picked are $n=0.8, \kappa = -1, \ a= 1, \ b= 0.5, \xi _1 = 0.1, \xi _2 = 0.2, \beta _1 = 0.5, \beta _2 = 0.4, \theta _1 = -0.3, \theta _2 = -0.1, \eta _1 = 0.4, \eta _2 = 0.4, \mu = -0.5 , \alpha = 1, \zeta _0 = 0, \omega = 1.$

Fig. 3

Numerical simulations of bright solitons (86) and (87)

4 Quadratic-cubic law

In this case, $F(s)=c_{1}\sqrt{s}+c_{2}s$ where $c_{1}$ and $c_{2}$ are constants. The system (1)–(2) therefore is:

$$\begin{aligned}&iq_{t}+a_1q_{xx}+b_1q_{xt}+ \left\{ \xi _1 \left( c_{1} \left| q \right| +c_{2} \left| q \right| ^{2} \right) + \eta _1 \left( c_{1} \left| r \right| +c_{2} \left| r \right| ^{2} \right) \right\} q \ \nonumber \\&\quad =Q_1 r + i \left\{ \alpha _1 q_{x} + \beta _1 \left( \left( c_{1} \left| q \right| +c_{2} \left| q \right| ^{2} \right) q\right) _{x} + \nu _1 \left( c_{1} \left| q \right| +c_{2} \left| q \right| ^{2}\right) _{x}q + \theta _1 \left( c_{1} \left| q \right| +c_{2} \left| q \right| ^{2} \right) q_{x} \right\} \end{aligned}$$

(102)

$$\begin{aligned}&ir_{t}+a_2r_{xx}+b_2r_{xt}+ \left\{ \xi _2 \left( c_{1} \left| r\right| +c_{2} \left| r\right| ^{2} \right) + \eta _2 \left( c_{1} \left| q\right| +c_{2} \left| q\right| ^{2} \right) \right\} r \ \nonumber \\&\quad =Q_2 q + i \left\{ \alpha _2 r_{x} + \beta _2 \left( \left( c_{1} \left| r\right| +c_{2} \left| r\right| ^{2} \right) r\right) _{x} + \nu _2 \left( c_{1} \left| r\right| +c_{2} \left| r\right| ^{2}\right) _{x}r + \theta _2 \left( c_{1} \left| r\right| +c_{2} \left| r\right| ^{2} \right) r_{x} \right\} . \end{aligned}$$

(103)

Substitute (5) and (6) into (102) and (103). The obtained real and imaginary parts are

$$\begin{aligned}&\left( a_l-b_l v\right) P_l''+ \left( b_l\kappa \omega - a_l \kappa ^2 - \alpha _l \kappa - \omega \right) P_l + c_1\left( \xi _l-\kappa \left( \beta _l+\theta _l\right) \right) P_l^2 \nonumber \\&\quad \quad +c_2 \left( \xi _l-\kappa \left( \beta _l+\theta _l\right) \right) P_l^3 + \left( c_1 + c_2P_{{\bar{l}}}\right) \eta _lP_l P_{{\bar{l}}} -Q_l P_{{\bar{l}}} = 0 \end{aligned}$$

(104)

and

$$\begin{aligned}&\left( 2 a_l \kappa +\alpha _l-b_l (\kappa v+\omega )+v\right) P_l' +c_1 \left( 2 \beta _l+\theta _l+\nu _l\right) P_l'P_l +c_2 \left( 3 \beta _l+\theta _l + 2 \nu _l\right) P_l'P_l^2 =0 \end{aligned}$$

(105)

respectively. From (105) is possible to get the solitons speed (11) as long as the constraints

$$\begin{aligned} 2 \beta _l+\theta _l+\nu _l =&0 \end{aligned}$$

(106)

$$\begin{aligned} 3 \beta _l+\theta _l + 2 \nu _l=&0 \end{aligned}$$

(107)

remain valid. Next, from (106) and (107)

$$\begin{aligned}&\beta _l + \nu _l = 0 \end{aligned}$$

(108)

$$\begin{aligned}&\beta _l + \theta _l = 0 \end{aligned}$$

(109)

$$\begin{aligned}&\nu _l = \theta _l. \end{aligned}$$

(110)

As a result (13) and (14) are also satisfied in this case, and the real part (104) simplifies to

$$\begin{aligned}&\left( a-b v\right) P_l''+ \left( b\kappa \omega - a \kappa ^2 - \alpha \kappa - \omega \right) P_l + c_1\left( \xi _l-\kappa \left( \beta _l+\theta _l\right) \right) P_l^2 \nonumber \\&\quad \quad +c_2 \left( \xi _l-\kappa \left( \beta _l+\theta _l\right) \right) P_l^3 + \left( c_1 + c_2 P_{{\bar{l}}}\right) \eta _lP_l P_{{\bar{l}}} -Q_l P_{{\bar{l}}} = 0. \end{aligned}$$

(111)

Employing the balancing principle in Eq. (111) gives rise to

$$\begin{aligned} P_{ {\bar{l}}}=P_l. \end{aligned}$$

(112)

Then, Eq. (111) modifies to

$$\begin{aligned}&\left( a-b v\right) P_l''+ \left( b\kappa \omega - a \kappa ^2 - \alpha \kappa - \omega -Q_l \right) P_l + c_1\left( \xi _l + \eta _l -\kappa \left( \beta _l+\theta _l\right) \right) P_l^2 \nonumber \\&\qquad +c_2 \left( \xi _l + \eta _l -\kappa \left( \beta _l+\theta _l\right) \right) P_l^3 = 0. \end{aligned}$$

(113)

4.1 Extended $G^{\prime }/G-$expansion approach

Balancing $P_l''$ with $P_l^3$ in Eq. (113), yields $M = 1$. Thus, one has

$$\begin{aligned} P_l(\zeta )=&\alpha _0^{(l)}+\alpha _1^{(l)}\left( \frac{G'}{G}\right) + \beta _1^{(l)}\sqrt{\sigma \left( 1+\frac{1}{\mu }\left( \frac{G'}{G}\right) ^2 \right) }+ \gamma _1^{(l)}\left( \frac{G'}{G}\right) ^{-1}\nonumber \\&+\delta _1^{(l)}\frac{1}{\sqrt{\sigma \left( 1+\frac{1}{\mu }\left( \frac{G'}{G}\right) ^2 \right) }}. \end{aligned}$$

(114)

Proceeding as in previous sections, one derives the following solutions:

If $\mu <0$ and $A_1^2>A_2^2$, one secures dark solitons

$$\begin{aligned} q(x,t)=&\displaystyle \alpha _0^{(1)} \left( 1 \pm \tanh \left[ \sqrt{- \mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{\kappa (a \kappa +\alpha )+4 \mu (a-b v)+Q_l}{b \kappa -1}\right) t + \theta \right) \right] \end{aligned}$$

(115)

$$\begin{aligned} r(x,t)=&\displaystyle \alpha _0^{(2)} \left( 1 \pm \tanh \left[ \sqrt{- \mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{\kappa (a \kappa +\alpha )+4 \mu (a-b v)+Q_l}{b \kappa -1}\right) t + \theta \right) \right] \end{aligned}$$

(116)

for

$$\begin{aligned} c_1=-3 \alpha _0^{(l)} c_2, \quad \xi _l=\displaystyle \frac{2 \mu (a-b v)+(\alpha _0^{(l)})^2 c_2 \left( \kappa \left( \beta _l+\theta _l\right) -\eta _l\right) }{(\alpha _0^{(l)})^2 c_2} \end{aligned}$$

(117)

bright solitons

$$\begin{aligned} q(x,t)=&\displaystyle \alpha _0^{(1)} \left( 1 \pm \sqrt{2} {{\,{\mathrm{sech}}\,}}\left[ \sqrt{-\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{a \kappa ^2+\alpha \kappa -2 a \mu +2 b \mu v+Q_l}{b \kappa -1} \right) t + \theta \right) \right] \end{aligned}$$

(118)

$$\begin{aligned} r(x,t)=&\displaystyle \alpha _0^{(2)} \left( 1 \pm \sqrt{2} {{\,{\mathrm{sech}}\,}}\left[ \sqrt{-\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{a \kappa ^2+\alpha \kappa -2 a \mu +2 b \mu v+Q_l}{b \kappa -1} \right) t + \theta \right) \right] \end{aligned}$$

(119)

for

$$\begin{aligned} c_1=-3 \alpha _0^{(l)} c_2, \quad \xi _l= \displaystyle \frac{-a \mu +(\alpha _0^{(l)})^2 c_2 \left( \kappa \left( \beta _l+\theta _l\right) -\eta _l\right) +b \mu v}{(\alpha _0^{(l)} )^2 c_2} \end{aligned}$$

(120)

singular solitons

$$\begin{aligned} q(x,t)=&\displaystyle \alpha _0^{(1)} \left( 1 \mp \coth \left[ \sqrt{-\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{\kappa (a \kappa +\alpha )+4 \mu (a-b v)+Q_l}{b \kappa -1} \right) t + \theta \right) \right] \end{aligned}$$

(121)

$$\begin{aligned} r(x,t)=&\displaystyle \alpha _0^{(2)} \left( 1 \mp \coth \left[ \sqrt{-\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{\kappa (a \kappa +\alpha )+4 \mu (a-b v)+Q_l}{b \kappa -1} \right) t + \theta \right) \right] \end{aligned}$$

(122)

for

$$\begin{aligned} c_1=-3 \alpha _0^{(l)} c_2, \quad \xi _l=\displaystyle \frac{2 \mu (a-b v)+(\alpha _0^{(l)} )^2 c_2 \left( \kappa \left( \beta _l+\theta _l\right) -\eta _l\right) }{(\alpha _0^{(l)} )^2 c_2} \end{aligned}$$

(123)

complexiton solutions

$$\begin{aligned} q(x,t)=&\displaystyle \alpha _0^{(1)} \left( 1 \pm \tanh \left[ \sqrt{-\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right. \nonumber \\&\left. +i {{\,{\mathrm{sech}}\,}}\left[ \sqrt{-\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{\kappa (a \kappa +\alpha )+\mu (a-b v)+Q_l}{b \kappa -1}\right) t + \theta \right) \right] \end{aligned}$$

(124)

$$\begin{aligned} r(x,t)=&\displaystyle \alpha _0^{(2)} \left( 1 \pm \tanh \left[ \sqrt{-\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right. \nonumber \\&\left. +i {{\,{\mathrm{sech}}\,}}\left[ \sqrt{-\mu }\left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{\kappa (a \kappa +\alpha )+\mu (a-b v)+Q_l}{b \kappa -1}\right) t + \theta \right) \right] \end{aligned}$$

(125)

for

$$\begin{aligned} c_1=-3 \alpha _0^{(l)} c_2, \quad \xi _l=\displaystyle \frac{\mu (a-b v)+2 (\alpha _0^{(l)} )^2 c_2 \left( \kappa \left( \beta _l+\theta _l\right) -\eta _l\right) }{2 (\alpha _0^{(l)} )^2 c_2} \end{aligned}$$

(126)

singular solitons

$$\begin{aligned} q(x,t)=&\displaystyle \alpha _0^{(1)} \left( 1 \mp {{\,{\mathrm{csch}}\,}}\left[ 2\sqrt{-\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + 2\zeta _0 \right] \right) \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{\kappa (a \kappa +\alpha )+16 \mu (a-b v)+Q_l}{b \kappa -1}\right) t + \theta \right) \right] \end{aligned}$$

(127)

$$\begin{aligned} r(x,t)=&\displaystyle \alpha _0^{(2)} \left( 1 \mp {{\,{\mathrm{csch}}\,}}\left[ 2\sqrt{-\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + 2\zeta _0 \right] \right) \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{\kappa (a \kappa +\alpha )+16 \mu (a-b v)+Q_l}{b \kappa -1}\right) t + \theta \right) \right] \end{aligned}$$

(128)

for

$$\begin{aligned} c_1=-3 \alpha _0^{(l)} c_2, \quad \xi _l=\displaystyle \frac{8 \mu (a-b v)+(\alpha _0^{(l)} )^2 c_2 \left( \kappa \left( \beta _l+\theta _l\right) -\eta _l\right) }{(\alpha _0^{(l)} )^2 c_2} \end{aligned}$$

(129)

and complexiton solutions

$$\begin{aligned} q(x,t)=&\displaystyle \alpha _0^{(1)} \frac{\sqrt{2}}{2} \left( \sqrt{2} \pm 2i {{\,{\mathrm{csch}}\,}}\left[ 2\sqrt{-\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + 2\zeta _0 \right] \right) \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{a \kappa ^2+\alpha \kappa -8 a \mu +8 b \mu v+Q_l}{b \kappa -1}\right) t + \theta \right) \right] \end{aligned}$$

(130)

$$\begin{aligned} r(x,t)=&\displaystyle \alpha _0^{(2)} \frac{\sqrt{2}}{2} \left( \sqrt{2} \pm 2i {{\,{\mathrm{csch}}\,}}\left[ 2\sqrt{-\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + 2\zeta _0 \right] \right) \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{a \kappa ^2+\alpha \kappa -8 a \mu +8 b \mu v+Q_l}{b \kappa -1}\right) t + \theta \right) \right] \end{aligned}$$

(131)

for

$$\begin{aligned} c_1=-3 \alpha _0^{(l)} c_2, \quad \xi _l=\displaystyle \frac{(\alpha _0^{(l)} )^2 c_2 \left( \kappa \left( \beta _l+\theta _l\right) -\eta _l\right) -4 \mu (a-b v)}{(\alpha _0^{(l)} )^2 c_2} \end{aligned}$$

(132)

where $\zeta _0=\tanh ^{-1}(A_2/A_1).$ If, however, $\mu >0$, one obtains periodic waves as follows:

$$\begin{aligned} q(x,t)=&\displaystyle \alpha _0^{(1)} \left( 1 \mp i \tan \left[ \sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{\kappa (a \kappa +\alpha )+4 \mu (a-b v)+Q_l}{b \kappa -1}\right) t + \theta \right) \right] \end{aligned}$$

(133)

$$\begin{aligned} r(x,t)=&\displaystyle \alpha _0^{(2)} \left( 1 \mp i \tan \left[ \sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{\kappa (a \kappa +\alpha )+4 \mu (a-b v)+Q_l}{b \kappa -1}\right) t + \theta \right) \right] \end{aligned}$$

(134)

$$\begin{aligned} q(x,t)=&\displaystyle \alpha _0^{(1)} \left( 1 \pm \sqrt{2} \sec \left[ \sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{a \kappa ^2+\alpha \kappa -2 a \mu +2 b \mu v+Q_l}{b \kappa -1} \right) t + \theta \right) \right] \end{aligned}$$

(135)

$$\begin{aligned} r(x,t)=&\displaystyle \alpha _0^{(2)} \left( 1 \pm \sqrt{2} \sec \left[ \sqrt{\mu }\left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{a \kappa ^2+\alpha \kappa -2 a \mu +2 b \mu v+Q_l}{b \kappa -1} \right) t + \theta \right) \right] \end{aligned}$$

(136)

$$\begin{aligned} q(x,t)=&\displaystyle \alpha _0^{(1)} \left( 1 \mp i \cot \left[ \sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{\kappa (a \kappa +\alpha )+4 \mu (a-b v)+Q_l}{b \kappa -1} \right) t + \theta \right) \right] \end{aligned}$$

(137)

$$\begin{aligned} r(x,t)=&\displaystyle \alpha _0^{(2)} \left( 1 \mp i \cot \left[ \sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{\kappa (a \kappa +\alpha )+4 \mu (a-b v)+Q_l}{b \kappa -1} \right) t + \theta \right) \right] \end{aligned}$$

(138)

$$\begin{aligned} q(x,t)=&\displaystyle \alpha _0^{(1)} \left( 1 \mp i \tan \left[ \sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right. \nonumber \\&\left. + i \sec \left[ \sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{\kappa (a \kappa +\alpha )+\mu (a-b v)+Q_l}{b \kappa -1}\right) t + \theta \right) \right] \end{aligned}$$

(139)

$$\begin{aligned} r(x,t)=&\displaystyle \alpha _0^{(2)} \left( 1 \mp i \tan \left[ \sqrt{\mu }\left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right. \nonumber \\&\left. + i \sec \left[ \sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{\kappa (a \kappa +\alpha )+\mu (a-b v)+Q_l}{b \kappa -1}\right) t + \theta \right) \right] \end{aligned}$$

(140)

$$\begin{aligned} q(x,t)=&\displaystyle \alpha _0^{(1)}\left( 1 \mp i \csc \left[ 2\sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + 2\zeta _0 \right] \right) \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{\kappa (a \kappa +\alpha )+16 \mu (a-b v)+Q_l}{b \kappa -1}\right) t + \theta \right) \right] \end{aligned}$$

(141)

$$\begin{aligned} r(x,t)=&\displaystyle \alpha _0^{(2)}\left( 1 \mp i \csc \left[ 2\sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + 2\zeta _0 \right] \right) \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{\kappa (a \kappa +\alpha )+16 \mu (a-b v)+Q_l}{b \kappa -1}\right) t + \theta \right) \right] \end{aligned}$$

(142)

$$\begin{aligned} q(x,t)=&\alpha _0^{(1)} \frac{\sqrt{2}}{2} \left( \sqrt{2} \mp 2 \csc \left[ 2\sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + 2\zeta _0 \right] \right) \nonumber \\&\times \exp \left[ i \left( - \kappa x +\left( \displaystyle \frac{a \kappa ^2+\alpha \kappa -8 a \mu +8 b \mu v+Q_l}{b \kappa -1}\right) t + \theta \right) \right] \end{aligned}$$

(143)

$$\begin{aligned} r(x,t)=&\alpha _0^{(2)} \frac{\sqrt{2}}{2} \left( \sqrt{2} \mp 2 \csc \left[ 2\sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + 2\zeta _0 \right] \right) \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{a \kappa ^2+\alpha \kappa -8 a \mu +8 b \mu v+Q_l}{b \kappa -1}\right) t + \theta \right) \right] \end{aligned}$$

(144)

where $\zeta _0=\tan ^{-1}(A_1/A_2)$. These trigonometric function solution pairs exist as long as Eqs. (117), (120), (123), (126), (129) and (132) remain valid, respectively.

The profiles of dark solitons are represented by Fig. 4. The parameter values chosen are $\kappa = 1, \ a= 1, \ b= 2, \ \mu = -1 , \ \alpha = 1, \ \alpha _0^{(1)}=1, \ \alpha _0^{(2)}=2, \ \zeta _0 = 0, \ \omega = 1 .$

That of bright solitons are given by Fig. 5. In this case, the picked parameter values are $\kappa = 1, \ a= 1, \ b= 2, \ \mu = -1 , \ \alpha = 1, \ \alpha _0^{(1)}=1, \ \alpha _0^{(2)}=2, \ \zeta _0 = 0, \ \omega = 1 .$

Fig. 4

Numerical simulations of dark solitons (115) and (116)

Fig. 5

Numerical simulations of bright solitons (118) and (119)

5 Parabolic law

For parabolic law nonlinearity, $F(s)=c_{1}s+c_{2}s^2$ where $c_{1}$ and $c_{2}$ are constants. The system (1)–(2) therefore takes the form

$$\begin{aligned}&iq_{t}+a_1q_{xx}+b_1q_{xt}+ \left\{ \xi _1 \left( c_{1} \left| q\right| ^{2} +c_{2} \left| q \right| ^{4} \right) + \eta _1 \left( c_{1} \left| r \right| ^{2}+c_{2} \left| r \right| ^{4} \right) \right\} q \nonumber \\&\quad =Q_1 r + i \left\{ \alpha _1 q_{x} + \beta _1 \left( \left( c_{1} \left| q \right| ^{2}+c_{2} \left| q \right| ^{4} \right) q\right) _{x} + \nu _1 \left( c_{1} \left| q \right| ^{2}+c_{2} \left| q \right| ^{4}\right) _{x}q \nonumber \right. \\&\left. \qquad + \theta _1 \left( c_{1} \left| q \right| ^{2}+c_{2} \left| q \right| ^{4} \right) q_{x}\right\} \end{aligned}$$

(145)

$$\begin{aligned}&ir_{t}+a_2r_{xx}+b_2r_{xt}+ \left\{ \xi _2 \left( c_{1} \left| r \right| ^{2} +c_{2} \left| r \right| ^{4} \right) + \eta _2 \left( c_{1} \left| q \right| ^{2}+c_{2} \left| q \right| ^{4} \right) \right\} r \nonumber \\&\quad =Q_2 q + i \left\{ \alpha _2 r_{x} + \beta _2 \left( \left( c_{1} \left| r \right| ^{2}+c_{2} \left| r \right| ^{4} \right) r\right) _{x} + \nu _2 \left( c_{1} \left| r \right| ^{2}+c_{2} \left| r \right| ^{4}\right) _{x}r \nonumber \right. \\&\left. \qquad + \theta _2 \left( c_{1} \left| r \right| ^{2}+c_{2} \left| r \right| ^{4} \right) r_{x}\right\} . \end{aligned}$$

(146)

Put (5) and (6) into (145) and (146). The real part gives

$$\begin{aligned}&\left( a_l-b_lv\right) P_l'' - \left( \omega +\kappa \left( a_l \kappa +\alpha _l-b_l \omega \right) \right) P_l +c_1 \left( \xi _l-\kappa \left( \beta _l+\theta _l\right) \right) P_l^3 \nonumber \\&\quad +c_2 \left( \xi _l-\kappa \left( \beta _l+\theta _l\right) \right) P_l^5 + c_1\eta _l P_l P_{{\bar{l}}}^2 + c_2 \eta _l P_l P_{{\bar{l}}}^4 -Q_l P_{{\bar{l}}} = 0 \end{aligned}$$

(147)

while the imaginary part leads to

$$\begin{aligned}&\left( v+2a_l \kappa +\alpha _l - b_l (v \kappa +\omega ) \right) P_l' +c_1 \left( 3 \beta _l+\theta _l+2 \nu _l\right) P_l^2 P_l' +c_2 \left( 5 \beta _l+\theta _l+4 \nu _l\right) P_l^4 P_l' = 0. \end{aligned}$$

(148)

From (148) is possible to retrieve the solitons speed (11) as long as the constraints

$$\begin{aligned} 3 \beta _l+\theta _l+2 \nu _l =&0 \end{aligned}$$

(149)

$$\begin{aligned} 5 \beta _l+\theta _l+4 \nu _l =&0 \end{aligned}$$

(150)

remain valid. Then, from (149) and (150)

$$\begin{aligned}&\beta _l + \nu _l = 0 \end{aligned}$$

(151)

$$\begin{aligned}&\beta _l + \theta _l = 0 \end{aligned}$$

(152)

$$\begin{aligned}&\nu _l = \theta _l. \end{aligned}$$

(153)

As a results, (13) and (14) are also satisfied in this case, and the real part (147) changes to

$$\begin{aligned}&\left( a-bv\right) P_l'' - \left( \omega +\kappa \left( a \kappa +\alpha -b \omega \right) \right) P_l +c_1 \left( \xi _l-\kappa \left( \beta _l+\theta _l\right) \right) P_l^3 \nonumber \\&\quad \quad +c_2 \left( \xi _l-\kappa \left( \beta _l+\theta _l\right) \right) P_l^5 + c_1\eta _l P_l P_{{\bar{l}}}^2 + c_2 \eta _l P_l P_{{\bar{l}}}^4 -Q_l P_{{\bar{l}}} = 0. \end{aligned}$$

(154)

The balancing principle implies that

$$\begin{aligned} P_{ {\bar{l}}}=P_l. \end{aligned}$$

(155)

In this case, Eq. (154) becomes to

$$\begin{aligned}&\left( a-bv\right) P_l'' - \left( \omega +\kappa \left( a \kappa +\alpha -b \omega \right) + Q_l \right) P_l +c_1 \left( \xi _l + \eta _l -\kappa \left( \beta _l+\theta _l\right) \right) P_l^3 \nonumber \\&\qquad +c_2 \left( \xi _l + \eta _l -\kappa \left( \beta _l+\theta _l\right) \right) P_l^5 = 0. \end{aligned}$$

(156)

Set

$$\begin{aligned} P_l=U_l^\frac{1}{2} \end{aligned}$$

(157)

so that (156) transform to

$$\begin{aligned}&2 (a-b v) U_l U_l'' -(a-b v) \left( U_l' \right) ^2 -4 \left( \omega + \kappa (a \kappa +\alpha -b \omega )+Q_l \right) U_l^2 \nonumber \\&\quad \quad +4 c_1 \left( \xi _l+\eta _l-\kappa \left( \beta _l+\theta _l\right) \right) U_l^3 +4 c_2 \left( \xi _l+\eta _l-\kappa \left( \beta _l+\theta _l\right) \right) U_l^4 = 0. \end{aligned}$$

(158)

5.1 extended $G^{\prime }/G-$expansion approach

Balancing $U_l U_l''$ or $\left( U_l' \right) ^2$ with $U_l^4$ in Eq. (158) causes $M = 1$. Thus, one discovers

$$\begin{aligned} U_l(\zeta )=&\alpha _0^{(l)}+\alpha _1^{(l)}\left( \frac{G'}{G}\right) + \beta _1^{(l)}\sqrt{\sigma \left( 1+\frac{1}{\mu }\left( \frac{G'}{G}\right) ^2 \right) }+ \gamma _1^{(l)}\left( \frac{G'}{G}\right) ^{-1} +\delta _1^{(l)}\frac{1}{\sqrt{\sigma \left( 1+\frac{1}{\mu }\left( \frac{G'}{G}\right) ^2 \right) }}. \end{aligned}$$

(159)

In this case, solutions are:

When $\mu <0$ and $A_1^2>A_2^2$, one attains optic soliton solutions as

$$\begin{aligned} q(x,t)= \displaystyle&\left\{ - \frac{2 \mu (a-b v)}{c_1 \chi _1} \left( 1 \mp \tanh \left[ \sqrt{-\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \right\} ^{\frac{1}{2}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{\kappa (a \kappa +\alpha )+\mu (a-b v)+Q_l}{b \kappa -1} \right) t + \theta \right) \right] \end{aligned}$$

(160)

$$\begin{aligned} r(x,t)= \displaystyle&\left\{ - \frac{2 \mu (a-b v)}{c_1 \chi _2} \left( 1 \mp \tanh \left[ \sqrt{-\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \right\} ^{\frac{1}{2}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{\kappa (a \kappa +\alpha )+\mu (a-b v)+Q_l}{b \kappa -1} \right) t + \theta \right) \right] \end{aligned}$$

(161)

for

$$\begin{aligned} c_2=&\displaystyle \frac{3 c_1^2 \chi _l}{ 16 \mu ( a - b v )} \end{aligned}$$

(162)

$$\begin{aligned} q(x,t)=&\displaystyle \left\{ \frac{2 \mu (a-b v)}{c_1 \chi _1} \left( 1 \pm {{\,{\mathrm{sech}}\,}}\left[ \sqrt{-\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \right\} ^{\frac{1}{2}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \displaystyle \left( \frac{4 \kappa (a \kappa +\alpha )-5 \mu (a-b v)+4 Q_l}{4 (b \kappa -1)}\right) t + \theta \right) \right] \end{aligned}$$

(163)

$$\begin{aligned} r(x,t)= \displaystyle&\left\{ \displaystyle \frac{2 \mu (a-b v)}{c_1 \chi _2} \left( 1 \pm {{\,{\mathrm{sech}}\,}}\left[ \sqrt{-\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \right\} ^{\frac{1}{2}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \displaystyle \left( \frac{4 \kappa (a \kappa +\alpha )-5 \mu (a-b v)+4 Q_l}{4 (b \kappa -1)}\right) t + \theta \right) \right] \end{aligned}$$

(164)

for

$$\begin{aligned} c_2=&\displaystyle -\frac{3 c_1^2 \chi _l}{16 \mu ( a - b v )} \end{aligned}$$

(165)

$$\begin{aligned} q(x,t)= \displaystyle&\left\{ -\frac{2 \mu (a-b v)}{c_1 \chi _1} \left( 1 \pm \coth \left[ \sqrt{-\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \right\} ^{\frac{1}{2}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \displaystyle \left( \frac{\kappa (a \kappa +\alpha )+\mu (a-b v)+Q_l}{b \kappa -1}\right) t + \theta \right) \right] \end{aligned}$$

(166)

$$\begin{aligned} r(x,t)= \displaystyle&\left\{ -\frac{2 \mu (a-b v)}{c_1 \chi _2} \left( 1 \pm \coth \left[ \sqrt{-\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \right\} ^{\frac{1}{2}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \displaystyle \left( \frac{\kappa (a \kappa +\alpha )+\mu (a-b v)+Q_l}{b \kappa -1}\right) t + \theta \right) \right] \end{aligned}$$

(167)

for

$$\begin{aligned} c_2=&\displaystyle \frac{3 c_1^2 \chi _l}{16 \mu ( a - b v)} \end{aligned}$$

(168)

$$\begin{aligned} q(x,t)=&\displaystyle \left\{ -\frac{\mu (a-b v)}{2 c_1 \chi _1} \left( 1 \pm \tanh \left[ \sqrt{-\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right. \right. \nonumber \\&\left. \left. - i {{\,{\mathrm{sech}}\,}}\left[ \sqrt{-\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \right\} ^{\frac{1}{2}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \displaystyle \left( \frac{4 \kappa (a \kappa +\alpha )+\mu (a-b v)+4 Q_l}{4 (b \kappa -1)}\right) t + \theta \right) \right] \end{aligned}$$

(169)

$$\begin{aligned} r(x,t)=\displaystyle&\left\{ -\frac{\mu (a-b v)}{2 c_1 \chi _2} \left( 1 \pm \tanh \left[ \sqrt{-\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right. \right. \nonumber \\&\left. \left. - i {{\,{\mathrm{sech}}\,}}\left[ \sqrt{-\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \right\} ^{\frac{1}{2}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \displaystyle \left( \frac{4 \kappa (a \kappa +\alpha )+\mu (a-b v)+4 Q_l}{4 (b \kappa -1)}\right) t + \theta \right) \right] \end{aligned}$$

(170)

for

$$\begin{aligned} c_2=&\displaystyle \frac{3 c_1^2 \chi _l}{4 \mu (a - b v)} \end{aligned}$$

(171)

$$\begin{aligned} q(x,t)=\displaystyle&\left\{ \frac{4 \mu (a-b v)}{c_1 \chi _1} \left( 2 \pm 2i {{\,{\mathrm{csch}}\,}}\left[ 2\sqrt{-\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + 2\zeta _0 \right] \right) \right\} ^{\frac{1}{2}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \displaystyle \left( \frac{\kappa ( a \kappa +\alpha ) -5 \mu ( a - b v )+Q_l}{b \kappa -1}\right) t + \theta \right) \right] \end{aligned}$$

(172)

$$\begin{aligned} r(x,t)= \displaystyle&\left\{ \frac{4 \mu (a-b v)}{c_1 \chi _2} \left( 2 \pm 2i {{\,{\mathrm{csch}}\,}}\left[ 2\sqrt{-\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + 2\zeta _0 \right] \right) \right\} ^{\frac{1}{2}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \displaystyle \left( \frac{\kappa ( a \kappa +\alpha ) -5 \mu ( a - b v )+Q_l}{b \kappa -1}\right) t + \theta \right) \right] \end{aligned}$$

(173)

for

$$\begin{aligned} c_2=&\displaystyle -\frac{3 c_1^2 \chi _l}{64 \mu (a - b v)} \end{aligned}$$

(174)

$$\begin{aligned} q(x,t)= \displaystyle&\left\{ -\frac{4 \mu (a-b v)}{c_1 \chi _1} \left( 2 \mp \coth \left[ 2\sqrt{-\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + 2\zeta _0 \right] \right) \right\} ^{\frac{1}{2}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \displaystyle \left( \frac{\kappa (a \kappa +\alpha )+4 \mu (a-b v)+Q_l}{b \kappa -1}\right) t + \theta \right) \right] \end{aligned}$$

(175)

$$\begin{aligned} r(x,t)= \displaystyle&\left\{ -\frac{4 \mu (a-b v)}{c_1 \chi _2} \left( 2 \mp \coth \left[ 2\sqrt{-\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + 2\zeta _0 \right] \right) \right\} ^{\frac{1}{2}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \displaystyle \left( \frac{\kappa (a \kappa +\alpha )+4 \mu (a-b v)+Q_l}{b \kappa -1}\right) t + \theta \right) \right] \end{aligned}$$

(176)

for

$$\begin{aligned} c_2=\displaystyle \frac{3 c_1^2 \chi _l}{64 \mu ( a - b v)} \end{aligned}$$

(177)

where $\zeta _0=\tanh ^{-1}(A_2/A_1)$ and

$$\begin{aligned} \chi _{l} = \eta _l+\xi _l -\kappa \left( \beta _l+\theta _l\right) . \end{aligned}$$

(178)

If, however, $\mu >0$, one gains periodic waves as:

$$\begin{aligned} q(x,t)= \displaystyle&\left\{ - \frac{2 \mu (a-b v)}{c_1 \chi _1} \left( 1 \pm i \tan \left[ \sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \right\} ^{\frac{1}{2}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{\kappa (a \kappa +\alpha )+\mu (a-b v)+Q_l}{b \kappa -1} \right) t + \theta \right) \right] \end{aligned}$$

(179)

$$\begin{aligned} r(x,t)= \displaystyle&\left\{ - \frac{2 \mu (a-b v)}{c_1 \chi _2} \left( 1 \pm i \tan \left[ \sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \right\} ^{\frac{1}{2}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{\kappa (a \kappa +\alpha )+\mu (a-b v)+Q_l}{b \kappa -1} \right) t + \theta \right) \right] \end{aligned}$$

(180)

$$\begin{aligned} q(x,t)= \displaystyle&\left\{ \frac{2 \mu (a-b v)}{c_1 \chi _1} \left( 1 \pm \sec \left[ \sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \right\} ^{\frac{1}{2}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \displaystyle \left( \frac{4 \kappa (a \kappa +\alpha )-5 \mu (a-b v)+4 Q_l}{4 (b \kappa -1)}\right) t + \theta \right) \right] \end{aligned}$$

(181)

$$\begin{aligned} r(x,t)= \displaystyle&\left\{ \frac{2 \mu (a-b v)}{c_1 \chi _2} \left( 1 \pm \sec \left[ \sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \right\} ^{\frac{1}{2}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \displaystyle \left( \frac{4 \kappa (a \kappa +\alpha )-5 \mu (a-b v)+4 Q_l}{4 (b \kappa -1)}\right) t + \theta \right) \right] \end{aligned}$$

(182)

$$\begin{aligned} q(x,t)= \displaystyle&\left\{ -\frac{2 \mu (a-b v)}{c_1 \chi _1} \left( 1 \pm i \cot \left[ \sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \right\} ^{\frac{1}{2}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \displaystyle \left( \frac{\kappa (a \kappa +\alpha )+\mu (a-b v)+Q_l}{b \kappa -1}\right) t + \theta \right) \right] \end{aligned}$$

(183)

$$\begin{aligned} r(x,t)= \displaystyle&\left\{ -\frac{2 \mu (a-b v)}{c_1 \chi _2}\left( 1 \pm i \cot \left[ \sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \right\} ^{\frac{1}{2}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \displaystyle \left( \frac{\kappa (a \kappa +\alpha )+\mu (a-b v)+Q_l}{b \kappa -1}\right) t + \theta \right) \right] \end{aligned}$$

(184)

$$\begin{aligned} q(x,t)= \displaystyle&\left\{ -\frac{\mu (a-b v)}{2 c_1 \chi _1} \left( 1 \pm i \tan \left[ \sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right. \right. \nonumber \\&\left. \left. - i \sec \left[ \sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \right\} ^{\frac{1}{2}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \displaystyle \left( \frac{4 \kappa (a \kappa +\alpha )+\mu (a-b v)+4 Q_l}{4 (b \kappa -1)}\right) t + \theta \right) \right] \end{aligned}$$

(185)

$$\begin{aligned} r(x,t)= \displaystyle&\left\{ -\frac{\mu (a-b v)}{2 c_1 \chi _2} \left( 1 \pm i \tan \left[ \sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right. \right. \nonumber \\&\left. \left. - i \sec \left[ \sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \right\} ^{\frac{1}{2}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \displaystyle \left( \frac{4 \kappa (a \kappa +\alpha )+\mu (a-b v)+4 Q_l}{4 (b \kappa -1)}\right) t + \theta \right) \right] \end{aligned}$$

(186)

$$\begin{aligned} q(x,t)= \displaystyle&\left\{ \frac{4 \mu (a-b v)}{c_1 \chi _1} \left( 2 \mp 2\csc \left[ 2\sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + 2\zeta _0 \right] \right) \right\} ^{\frac{1}{2}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \displaystyle \left( \frac{\kappa ( a \kappa +\alpha ) -5 \mu ( a - b v )+Q_l}{b \kappa -1}\right) t + \theta \right) \right] \end{aligned}$$

(187)

$$\begin{aligned} r(x,t)= \displaystyle&\left\{ \frac{4 \mu (a-b v)}{c_1 \chi _2} \left( 2 \mp 2 \csc \left[ 2\sqrt{\mu }\left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + 2\zeta _0 \right] \right) \right\} ^{\frac{1}{2}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \displaystyle \left( \frac{\kappa ( a \kappa +\alpha ) -5 \mu ( a - b v )+Q_l}{b \kappa -1}\right) t + \theta \right) \right] \end{aligned}$$

(188)

$$\begin{aligned} q(x,t)= \displaystyle&\left\{ -\frac{4 \mu (a-b v)}{c_1 \chi _1} \left( 2 \mp 2i \cot \left[ 2\sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + 2\zeta _0 \right] \right) \right\} ^{\frac{1}{2}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \displaystyle \left( \frac{\kappa (a \kappa +\alpha )+4 \mu (a-b v)+Q_l}{b \kappa -1}\right) t + \theta \right) \right] \end{aligned}$$

(189)

$$\begin{aligned} r(x,t)= \displaystyle&\left\{ -\frac{4 \mu (a-b v)}{c_1 \chi _2} \left( 2 \mp 2i \cot \left[ 2\sqrt{\mu }\left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + 2\zeta _0 \right] \right) \right\} ^{\frac{1}{2}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \displaystyle \left( \frac{\kappa (a \kappa +\alpha )+4 \mu (a-b v)+Q_l}{b \kappa -1}\right) t + \theta \right) \right] \end{aligned}$$

(190)

where $\zeta _0=\tan ^{-1}(A_1/A_2).$ These trigonometric function solution pairs exist as long as Eqs. (162), (165), (168), (171), (174) and (177) remain valid, respectively.

The profiles of dark solitons are given by Fig. 6. The chosen parameter values are $\kappa = 1, \ a= 1, \ b= 2, \ \xi _1 = 1.2, \ \xi _2 = 0.7, \ \beta _1 = 0.4, \ \beta _2 = 0.6, \ \theta _1 = 0.3, \ \theta _2 = 0.1, \ \eta _1 = -0.3, \ \eta _2 = 0.4, \ \mu = -1 , \ \alpha = 1, \ \zeta _0 = 0, \ \omega = 1, \ c_1=1.$

That of bright solitons are represented by Fig. 7. In this case, the parameter values picked are $\kappa = 1, \ a= 1, \ b= 2, \ \xi _1 = 1.3, \ \xi _2 = 1.6, \ \beta _1 = 0.6, \ \beta _2 = 0.2, \ \theta _1 = 1.3, \ \theta _2 = 0.7, \ \eta _1 = 1.2, \ \eta _2 = 0.9, \ \mu = -1 , \ \alpha = 1, \ \zeta _0 = 0, \ \omega = 1, \ c_1=1.$

Fig. 6

Numerical simulations of dark solitons (160) and (161)

Fig. 7

Numerical simulations of bright solitons (163) and (164)

6 Dual-power law

For dual-power law nonlinearity, $F(s)=c_{1}s^n+c_{2}s^{2n}$ where $c_{1}$ and $c_{2}$ are constants. Therefore, the governing model reduces to

$$\begin{aligned}&iq_{t}+a_1q_{xx}+b_1q_{xt}+ \left\{ \xi _1 \left( c_{1} \left| q\right| ^{2n} +c_{2} \left| q\right| ^{4n}\right) + \eta _1 \left( c_{1} \left| r\right| ^{2n}+c_{2} \left| r\right| ^{4n}\right) \right\} q \nonumber \\&\quad =Q_1 r + i \left\{ \alpha _1 q_{x} + \beta _1 \left( \left( c_{1} \left| q\right| ^{2n}+c_{2} \left| q\right| ^{4n}\right) q\right) _{x}\right. \left. \nonumber \right. \\&\left. \qquad + \nu _1 \left( c_{1} \left| q\right| ^{2n}+c_{2} \left| q\right| ^{4n}\right) _{x}q + \theta _1 \left( c_{1}\left| q\right| ^{2n}+c_{2}\left| q\right| ^{4n}\right) q_{x} \right\} \end{aligned}$$

(191)

$$\begin{aligned}&ir_{t}+a_2r_{xx}+b_2r_{xt}+ \left\{ \xi _2 \left( c_{1}\left| r\right| ^{2n} +c_{2}\left| r\right| ^{4n}\right) + \eta _2 \left( c_{1}\left| q\right| ^{2n}+c_{2}\left| q\right| ^{4n} \right) \right\} r \nonumber \\&\quad =Q_2 q + i \left\{ \alpha _2 r_{x} + \beta _2 \left( \left( c_{1}\left| r\right| ^{2n}+c_{2}\left| r\right| ^{4n}\right) r\right) _{x}\right. \nonumber \\&\qquad \left. + \nu _2 \left( c_{1}\left| r\right| ^{2n}+c_{2}\left| r\right| ^{4n}\right) _{x}r + \theta _2 \left( c_{1}\left| r\right| ^{2n}+c_{2}\left| r\right| ^{4n}\right) r_{x} \right\} . \end{aligned}$$

(192)

Plug (5) and (6) into (191) and (192). The real part obtained is

$$\begin{aligned}&\left( a_l-b_lv\right) P_l'' - \left( \omega +\kappa \left( a_l \kappa +\alpha _l-b_l \omega \right) \right) P_l +c_1 \left( \xi _l-\kappa \left( \beta _l+\theta _l\right) \right) P_l^{2n+1} \nonumber \\&\quad \quad +c_2 \left( \xi _l-\kappa \left( \beta _l+\theta _l\right) \right) P_l^{4n+1} + c_1\eta _l P_l P_{{\bar{l}}}^{2n} + c_2 \eta _l P_l P_{{\bar{l}}}^{4n} -Q_l P_{{\bar{l}}} = 0 \end{aligned}$$

(193)

while the imaginary part reads

$$\begin{aligned}&\left( v+2a_l \kappa +\alpha _l - b_l (v \kappa +\omega ) \right) P_l' +c_1 \left( \beta _l+\theta _l+2n(\beta _l+ \nu _l)\right) P_l^{2n} P_l' \nonumber \\&\qquad +c_2 \left( \beta _l+\theta _l+4n( \beta _l+\nu _l)\right) P_l^{4n} P_l' = 0. \end{aligned}$$

(194)

From (194) is possible to retrieve the solitons speed (11) as long as the constraints

$$\begin{aligned} \beta _l+\theta _l+2n(\beta _l+ \nu _l) =&0 \end{aligned}$$

(195)

$$\begin{aligned} \beta _l+\theta _l+4n( \beta _l+\nu _l) =&0 \end{aligned}$$

(196)

remain valid. Then, form (195) and (196)

$$\begin{aligned}&\beta _l + \theta _l = 0 \end{aligned}$$

(197)

$$\begin{aligned}&\beta _l + \nu _l = 0 \end{aligned}$$

(198)

$$\begin{aligned}&\nu _l = \theta _l. \end{aligned}$$

(199)

Consequently (13) and (14) are also satisfied in this case, and the real part (193) simplifies to

$$\begin{aligned}&\left( a-bv\right) P_l'' - \left( \omega +\kappa \left( a \kappa +\alpha -b \omega \right) \right) P_l +c_1 \left( \xi -\kappa \left( \beta +\theta \right) \right) P_l^{2n+1} \nonumber \\&\quad \quad +c_2 \left( \xi -\kappa \left( \beta +\theta \right) \right) P_l^{4n+1} + c_1\eta P_l P_{{\bar{l}}}^{2n} \nonumber \\&\qquad + c_2 \eta P_l P_{{\bar{l}}}^{4n} -Q_l P_{{\bar{l}}} = 0. \end{aligned}$$

(200)

By virtue of the balancing principle, it is explored that

$$\begin{aligned} P_{{ {\bar{l}}}}=P_l. \end{aligned}$$

(201)

So, Eq. (200) changes to

$$\begin{aligned}&\left( a-bv\right) P_l'' - \left( \omega +\kappa \left( a \kappa +\alpha -b \omega \right) +Q_l \right) P_l +c_1 \left( \xi +\eta -\kappa \left( \beta +\theta \right) \right) P_l^{2n+1} \nonumber \\&\qquad +c_2 \left( \xi + \eta -\kappa \left( \beta +\theta \right) \right) P_l^{4n+1} = 0. \end{aligned}$$

(202)

In order for finding closed form solutions, the transform

$$\begin{aligned} P_l=U_l^\frac{1}{2n} \end{aligned}$$

(203)

is applied to (202) and so (202) turns into

$$\begin{aligned}&2n (a-b v) U_l U_l'' -(2n-1)(a-b v) \left( U_l' \right) ^2 -4n^2 \left( \omega + \kappa (a \kappa +\alpha -b \omega )+Q_l \right) U_l^2 \nonumber \\&\quad +4 c_1 n^2 \left( \xi _l+\eta _l -\kappa \left( \beta _l+\theta _l\right) \right) U_l^{3} +4 c_2 n^2 \left( \xi _l+\eta _l -\kappa \left( \beta _l+\theta _l\right) \right) U_l^{4} = 0. \end{aligned}$$

(204)

6.1 Extended $G^{\prime }/G-$expansion approach

Balancing $U_l U_l''$ or $\left( U_l' \right) ^2$ with $U_l^{4}$ in Eq. (84) leads to $M = 1$. Therefore, one reaches

$$\begin{aligned} U_l(\zeta )=&\alpha _0^{(l)}+\alpha _1^{(l)}\left( \frac{G'}{G}\right) + \beta _1^{(l)}\sqrt{\sigma \left( 1+\frac{1}{\mu }\left( \frac{G'}{G}\right) ^2 \right) }+ \gamma _1^{(l)}\left( \frac{G'}{G}\right) ^{-1} +\delta _1^{(l)}\frac{1}{\sqrt{\sigma \left( 1+\frac{1}{\mu }\left( \frac{G'}{G}\right) ^2 \right) }}. \end{aligned}$$

(205)

Proceeding as in previous sections, the following solutions are discovered:

When $\mu <0$ and $A_1^2>A_2^2$, one extracts optic solitons as the following:

$$\begin{aligned} q(x,t)=&\left\{ \displaystyle - \frac{ \mu (n+1)(a-bv)}{c_1 n^2 \chi _1} \left( 1 \mp \tanh \left[ \sqrt{- \mu } \left\{ x - \textstyle \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \right\} ^{\frac{1}{2n}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{\mu (a-b v)+\kappa n^2 (a \kappa +\alpha )+n^2 Q_l}{n^2 (b \kappa -1)}\right) t + \theta \right) \right] \end{aligned}$$

(206)

$$\begin{aligned} r(x,t)=&\left\{ \displaystyle - \frac{ \mu (n+1)(a-bv)}{c_1 n^2 \chi _2} \left( 1 \mp \tanh \left[ \sqrt{- \mu } \left\{ x - \textstyle \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \right\} ^{\frac{1}{2n}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{\mu (a-b v)+\kappa n^2 (a \kappa +\alpha )+n^2 Q_l}{n^2 (b \kappa -1)}\right) t + \theta \right) \right] \end{aligned}$$

(207)

for

$$\begin{aligned} c_2=&\displaystyle \frac{c_1^2 n^2 (2 n+1) \chi _l}{4 \mu (n+1)^2 (a-b v)} \end{aligned}$$

(208)

$$\begin{aligned} q(x,t)=&\left\{ \displaystyle - \frac{ \mu (n+1)(a-bv)}{c_1 n^2 \chi _1} \left( 1 \pm \coth \left[ \sqrt{- \mu } \left\{ x -\textstyle \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \right\} ^{\frac{1}{2n}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{\mu (a-b v)+\kappa n^2 (a \kappa +\alpha )+n^2 Q_l}{n^2 (b \kappa -1)}\right) t + \theta \right) \right] \end{aligned}$$

(209)

$$\begin{aligned} r(x,t)=&\left\{ \displaystyle - \frac{ \mu (n+1)(a-bv)}{c_1 n^2 \chi _2} \left( 1 \pm \coth \left[ \sqrt{- \mu } \left\{ x - \textstyle \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \right\} ^{\frac{1}{2n}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{\mu (a-b v)+\kappa n^2 (a \kappa +\alpha )+n^2 Q_l}{n^2 (b \kappa -1)}\right) t + \theta \right) \right] \end{aligned}$$

(210)

for

$$\begin{aligned} c_2=&\displaystyle \frac{c_1^2 n^2 (2 n+1) \chi _l}{4 \mu (n+1)^2 (a-b v)} \end{aligned}$$

(211)

$$\begin{aligned} q(x,t)=&\left\{ \displaystyle - \frac{ \mu (n+1)(a-bv)}{4 c_1 n^2 \chi _1} \left( 1 \mp \tanh \left[ \sqrt{- \mu } \left\{ x - \textstyle \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right. \right. \nonumber \\&\left. \left. - i {{\,{\mathrm{sech}}\,}}\left[ \sqrt{- \mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \right\} ^{\frac{1}{2n}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{\mu (a-b v)+4 \kappa n^2 (a \kappa +\alpha )+4 n^2 Q_l}{4 n^2 (b \kappa -1)}\right) t + \theta \right) \right] \end{aligned}$$

(212)

$$\begin{aligned} r(x,t)=&\left\{ \displaystyle - \frac{ \mu (n+1)(a-bv)}{4 c_1 n^2 \chi _2} \left( 1 \mp \tanh \left[ \sqrt{- \mu } \left\{ x - \textstyle \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right. \right. \nonumber \\&\left. \left. - i {{\,{\mathrm{sech}}\,}}\left[ \sqrt{- \mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \right\} ^{\frac{1}{2n}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{\mu (a-b v)+4 \kappa n^2 (a \kappa +\alpha )+4 n^2 Q_l}{4 n^2 (b \kappa -1)}\right) t + \theta \right) \right] \end{aligned}$$

(213)

for

$$\begin{aligned} c_2=&\displaystyle \frac{c_1^2 n^2 (2 n+1) \chi _l}{\mu (n+1)^2 (a-b v)} \end{aligned}$$

(214)

$$\begin{aligned} q(x,t)=&\left\{ \displaystyle - \frac{ 4 \mu (n+1)(a-bv)}{c_1 n^2 \chi _1} \left( 1 \pm {{\,{\mathrm{csch}}\,}}\left[ 2\sqrt{- \mu } \left\{ x - \textstyle \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + 2\zeta _0 \right] \right) \right\} ^{\frac{1}{2n}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{4 \mu (a-b v)+\kappa n^2 (a \kappa +\alpha )+n^2 Q_l}{n^2 (b \kappa -1)}\right) t + \theta \right) \right] \end{aligned}$$

(215)

$$\begin{aligned} r(x,t)=&\left\{ \displaystyle - \frac{ 4 \mu (n+1)(a-bv)}{c_1 n^2 \chi _2} \left( 1 \pm {{\,{\mathrm{csch}}\,}}\left[ 2\sqrt{- \mu } \left\{ x -\textstyle \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + 2\zeta _0 \right] \right) \right\} ^{\frac{1}{2n}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{4 \mu (a-b v)+\kappa n^2 (a \kappa +\alpha )+n^2 Q_l}{n^2 (b \kappa -1)}\right) t + \theta \right) \right] . \end{aligned}$$

(216)

for

$$\begin{aligned} c_2= \displaystyle -\frac{c_1^2 n^2 (2 n+1) \chi _l}{16 \mu (n+1)^2 (a-b v)} \end{aligned}$$

(217)

where $\zeta _0=\tanh ^{-1}(A_2/A_1)$ and

$$\begin{aligned} \chi _{l} = \eta _l+\xi _l -\kappa \left( \beta _l+\theta _l\right) . \end{aligned}$$

(218)

If, however, $\mu >0$, one achieves periodic waves as:

$$\begin{aligned} q(x,t)=&\left\{ \displaystyle - \frac{ \mu (n+1)(a-bv)}{c_1 n^2 \chi _1} \left( 1 \pm i \tan \left[ \sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \right\} ^{\frac{1}{2n}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{\mu (a-b v)+\kappa n^2 (a \kappa +\alpha )+n^2 Q_l}{n^2 (b \kappa -1)}\right) t + \theta \right) \right] \end{aligned}$$

(219)

$$\begin{aligned} r(x,t)=&\left\{ \displaystyle - \frac{ \mu (n+1)(a-bv)}{c_1 n^2 \chi _2} \left( 1 \pm i \tan \left[ \sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \right\} ^{\frac{1}{2n}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{\mu (a-b v)+\kappa n^2 (a \kappa +\alpha )+n^2 Q_l}{n^2 (b \kappa -1)}\right) t + \theta \right) \right] \end{aligned}$$

(220)

$$\begin{aligned} q(x,t)=&\left\{ \displaystyle - \frac{ \mu (n+1)(a-bv)}{c_1 n^2 \chi _1} \left( 1 \pm i \cot \left[ \sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \right\} ^{\frac{1}{2n}} \nonumber \\&\times \exp \left[ i \left( - \kappa x +\left( \displaystyle \frac{\mu (a-b v)+\kappa n^2 (a \kappa +\alpha )+n^2 Q_l}{n^2 (b \kappa -1)}\right) t + \theta \right) \right] \end{aligned}$$

(221)

$$\begin{aligned} r(x,t)=&\left\{ \displaystyle - \frac{ \mu (n+1)(a-bv)}{c_1 n^2 \chi _2} \left( 1 \pm i \cot \left[ \sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \right\} ^{\frac{1}{2n}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{\mu (a-b v)+\kappa n^2 (a \kappa +\alpha )+n^2 Q_l}{n^2 (b \kappa -1)}\right) t + \theta \right) \right] \end{aligned}$$

(222)

$$\begin{aligned} q(x,t)=&\left\{ \displaystyle - \frac{ \mu (n+1)(a-bv)}{4 c_1 n^2 \chi _1} \left( 1 \pm i \tan \left[ \sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right. \right. \nonumber \\&\left. \left. - i \sec \left[ \sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \right\} ^{\frac{1}{2n}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{\mu (a-b v)+4 \kappa n^2 (a \kappa +\alpha )+4 n^2 Q_l}{4 n^2 (b \kappa -1)}\right) t + \theta \right) \right] \end{aligned}$$

(223)

$$\begin{aligned} r(x,t)=&\left\{ \displaystyle - \frac{ \mu (n+1)(a-bv)}{4 c_1 n^2 \chi _2} \left( 1 \pm i \tan \left[ \sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right. \right. \nonumber \\&\left. \left. - i \sec \left[ \sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \right\} ^{\frac{1}{2n}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{\mu (a-b v)+4 \kappa n^2 (a \kappa +\alpha )+4 n^2 Q_l}{4 n^2 (b \kappa -1)}\right) t + \theta \right) \right] \end{aligned}$$

(224)

$$\begin{aligned} q(x,t)=&\left\{ \displaystyle - \frac{ 4 \mu (n+1)(a-bv)}{c_1 n^2 \chi _1} \left( 1 \pm i \csc \left[ 2 \sqrt{\mu } \left\{ x -\textstyle \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + 2 \zeta _0 \right] \right) \right\} ^{\frac{1}{2n}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{4 \mu (a-b v)+\kappa n^2 (a \kappa +\alpha )+n^2 Q_l}{n^2 (b \kappa -1)}\right) t + \theta \right) \right] \end{aligned}$$

(225)

$$\begin{aligned} r(x,t)=&\left\{ \displaystyle - \frac{ 4 \mu (n+1)(a-bv)}{c_1 n^2 \chi _2} \left( 1 \pm i \csc \left[ 2 \sqrt{\mu } \left\{ x - \textstyle \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + 2 \zeta _0 \right] \right) \right\} ^{\frac{1}{2n}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{4 \mu (a-b v)+\kappa n^2 (a \kappa +\alpha )+n^2 Q_l}{n^2 (b \kappa -1)}\right) t + \theta \right) \right] \end{aligned}$$

(226)

where $\zeta _0=\tan ^{-1}(A_1/A_2)$. These trigonometric function solution pairs exist as long as Eqs. (208), (211), (214) and (217) remain valid, respectively.

The profiles of dark solitons are represented by Fig. 8. The parameter values chosen are $n=1, \ \kappa = 1, \ a= 2, \ b= 3, \ \eta _1 = 1.6, \ \eta _2 = 1.5, \ \xi _1 = 1.4, \ \xi _2 = 1.2, \ \beta _1 = 0.6, \ \beta _2 = 0.4, \ \theta _1 = 0.5, \ \theta _2 = 0.1, \ \mu = -1 , \ \alpha = 1, \ \zeta _0 = 0, \ \omega = 1, \ c_1 = 1.$

Fig. 8

Numerical simulations of dark solitons (206) and (207)

7 Anti-cubic law

In this case, $F(s)=c_{1}s^{-2}+c_{2}s+c_{3}s^{2}$ where $c_{1}$, $c_{2}$ and $c_{3}$ are all constants. Therefore the system given by (1) and (2) collapses to

$$\begin{aligned}&iq_{t}+a_1q_{xx}+b_1q_{xt}+ \left\{ \xi _1 \left( \frac{c_{1}}{ \left| q\right| ^{4}} +c_{2}\left| q\right| ^{2}+c_{3}\left| q\right| ^{4}\right) + \eta _1 \left( \frac{c_{1}}{\left| r\right| ^{4}}+c_{2}\left| r\right| ^{2}+c_{3}\left| r\right| ^{4}\right) \right\} q \nonumber \\&\quad =Q_1 r + i \left\{ \alpha _1 q_{x} + \beta _1 \left( \left( \frac{c_{1}}{\left| q\right| ^{4}} +c_{2}\left| q\right| ^{2}+c_{3}\left| q\right| ^{4}\right) q\right) _{x}\right. \nonumber \\&\quad \quad \left. + \nu _1 \left( \frac{c_{1}}{\left| q\right| ^{4}} +c_{2}\left| q\right| ^{2}+c_{3}\left| q\right| ^{4} \right) _{x}q + \theta _1 \left( \frac{c_{1}}{\left| q\right| ^{4}} +c_{2}\left| q\right| ^{2}+c_{3}\left| q\right| ^{4}\right) q_{x} \right\} \end{aligned}$$

(227)

$$\begin{aligned}&ir_{t}+a_2r_{xx}+b_2r_{xt}+ \left\{ \xi _2 \left( \frac{c_{1}}{\left| r\right| ^{4}} +c_{2}\left| r\right| ^{2}+c_{3}\left| r\right| ^{4}\right) + \eta _2 \left( \frac{c_{1}}{\left| q\right| ^{4}} +c_{2}\left| q\right| ^{2}+c_{3}\left| q\right| ^{4}\right) \right\} r \ \nonumber \\&\quad =Q_2 q + i \left\{ \alpha _2 r_{x} + \beta _2 \left( \left( \frac{c_{1}}{\left| r\right| ^{4}}+c_{2}\left| r\right| ^{2}+c_{3}\left| r\right| ^{4}\right) r\right) _{x}\right. \nonumber \\&\quad \quad \left. + \nu _2 \left( \frac{c_{1}}{\left| r\right| ^{4}}+c_{2}\left| r\right| ^{2}+c_{3}\left| r\right| ^{4} \right) _{x}r + \theta _2 \left( \frac{c_{1}}{\left| r\right| ^{4}}+c_{2}\left| r\right| ^{2}+c_{3}\left| r\right| ^{4}\right) r_{x} \right\} . \end{aligned}$$

(228)

Substitute (5) and (6) into (227) and (228). The recovered real and imaginary parts are

$$\begin{aligned}&c_1\left( \xi _l-\kappa \left( \beta _l+\theta _l\right) \right) P_{{\bar{l}}}^4 +c_1\eta _l P_l{}^4-P_l{}^3 P_{{\bar{l}}}^4 \left( P_l \left( \kappa \left( a_l \kappa +\alpha _l-b_l \omega \right) \right. \right. \nonumber \\&\quad \quad \left. \left. -\eta _l P_{{\bar{l}}}^2 \left( c_3 P_{{\bar{l}}}^2+c_2\right) +\omega \right) +\left( b_l v-a_l\right) P_l''+c_3 P_l{}^5 \left( \kappa \left( \beta _l+\theta _l\right) -\xi _l\right) \right. \nonumber \\&\quad \quad \left. +c_2 P_l{}^3 \left( \kappa \left( \beta _l+\theta _l\right) -\xi _l\right) +Q_l P_{{\bar{l}}}\right) =0 \end{aligned}$$

(229)

and

$$\begin{aligned}&P_l' \left( P_l{}^4 \left( 2 a_l \kappa +\alpha _l-b_l (\kappa v+\omega )+v\right) +c_1 \left( -3 \beta _l+\theta _l-4 \nu _l\right) +c_3 P_l{}^8 \left( 5 \beta _l+\theta _l+4 \nu _l\right) \right. \nonumber \\&\qquad \left. +c_2 P_l{}^6 \left( 3 \beta _l+\theta _l+2 \nu _l\right) \right) =0 \end{aligned}$$

(230)

respectively. From (230) is possible to get the solitons speed (11) as long as the constraints

$$\begin{aligned}&-3 \beta _l+\theta _l-4 \nu _l=0 \end{aligned}$$

(231)

$$\begin{aligned}&5 \beta _l+\theta _l+4 \nu _l=0 \end{aligned}$$

(232)

$$\begin{aligned}&3 \beta _l+\theta _l+2 \nu _l=0 \end{aligned}$$

(233)

remain valid. From here

$$\begin{aligned}&\beta _l + \nu _l = 0 \end{aligned}$$

(234)

$$\begin{aligned}&\beta _l + \theta _l = 0 \end{aligned}$$

(235)

$$\begin{aligned}&\nu _l = \theta _l. \end{aligned}$$

(236)

Consequently (13) and (14) are also satisfied in this case, and the real part (229) becomes

$$\begin{aligned}&c_1\left( \xi _l-\kappa \left( \beta _l+\theta _l\right) \right) P_{{\bar{l}}}^4 +c_1\eta _l P_l{}^4-P_l{}^3 P_{{\bar{l}}}^4 \left( P_l \left( \kappa \left( a \kappa +\alpha -b \omega \right) \right. \right. \nonumber \\&\quad \quad \left. \left. -\eta _l P_{{\bar{l}}}^2 \left( c_3 P_{{\bar{l}}}^2+c_2\right) +\omega \right) +\left( b v-a\right) P_l''+c_3 P_l{}^5 \left( \kappa \left( \beta _l+\theta _l\right) -\xi _l\right) \right. \nonumber \\&\quad \quad \left. +c_2 P_l{}^3 \left( \kappa \left( \beta _l+\theta _l\right) -\xi _l\right) +Q_l P_{{\bar{l}}}\right) =0. \end{aligned}$$

(237)

Employing the balancing principle gives

$$\begin{aligned} P_{ {\bar{l}}}=P_l \end{aligned}$$

(238)

and then, Eq. (237) becomes to

$$\begin{aligned}&c_1 \left( -\beta _l\kappa +\eta _l-\theta _l \kappa +\xi _l\right) -P_l{}^3 \left( P_l\left( a \kappa ^2+\alpha \kappa -b \kappa \omega +Q_l+\omega \right) +(b v-a) P_l''\right. \nonumber \\&\quad \quad \left. +c_3 P_l{}^5 \left( \beta _l \kappa -\eta _l+\theta _l \kappa -\xi _l\right) +c_2 P_l{}^3 \left( \beta _l \kappa -\eta _l+\theta _l \kappa -\xi _l\right) \right) =0. \end{aligned}$$

(239)

To derive closed form solutions, one utilizes the transformation

$$\begin{aligned} P_l=U_l^\frac{1}{2}. \end{aligned}$$

(240)

Thus (239) turns into

$$\begin{aligned}&-4 U_l{}^2 \left( a \kappa ^2+\alpha \kappa -b \kappa \omega +Q_l+\omega \right) +2 (a-b v) U_l U_l'' -4 c_1 \left( \beta _l \kappa -\eta _l+\theta _l \kappa -\xi _l\right) \nonumber \\&\quad \quad -4 c_3 U_l{}^4 \left( \beta _l \kappa -\eta _l+\theta _l \kappa -\xi _l\right) -4 c_2 U_l{}^3 \left( \beta _l \kappa -\eta _l+\theta _l \kappa -\xi _l\right) +(b v-a) (U_l'){}^2=0. \end{aligned}$$

(241)

7.1 EXTENDED $G^{\prime }/G-$EXPANSION APPROACH

Balancing $U_l U_l''$ or $\left( U_l' \right) ^2$ with $U_l^{4}$ in Eq. (241) yields $M = 1$. Thus,

$$\begin{aligned} U_l(\zeta )=&\alpha _0^{(l)}+\alpha _1^{(l)}\left( \frac{G'}{G}\right) + \beta _1^{(l)}\sqrt{\sigma \left( 1+\frac{1}{\mu }\left( \frac{G'}{G}\right) ^2 \right) }+ \gamma _1^{(l)}\left( \frac{G'}{G}\right) ^{-1} +\delta _1^{(l)}\frac{1}{\sqrt{\sigma \left( 1+\frac{1}{\mu }\left( \frac{G'}{G}\right) ^2 \right) }} . \end{aligned}$$

(242)

Proceeding as in previous sections, the solutions acquired are:

For $\mu <0$ and $A_1^2>A_2^2$, one gets optic solitons as

$$\begin{aligned} q(x,t)=&\left\{ \displaystyle -\frac{3 c_2}{8 c_3} \pm \frac{\sqrt{3}}{2}\sqrt{\frac{\mu (a-bv)}{c_3\chi _1}} \tanh \left[ \sqrt{- \mu } \left\{ x - \left( \textstyle \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right\} ^{\frac{1}{2}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{16 c_3 \left( 2 a \kappa ^2+2 \alpha \kappa -a \mu +b \mu v+2 Q_l\right) +9 c_2^2 \chi _l}{32 c_3 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(243)

$$\begin{aligned} r(x,t)=&\left\{ \displaystyle -\frac{3 c_2}{8 c_3} \pm \frac{\sqrt{3}}{2}\sqrt{\frac{\mu (a-bv)}{c_3\chi _2}} \tanh \left[ \sqrt{- \mu } \left\{ x - \textstyle \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right\} ^{\frac{1}{2}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{16 c_3 \left( 2 a \kappa ^2+2 \alpha \kappa -a \mu +b \mu v+2 Q_l\right) +9 c_2^2 \chi _l}{32 c_3 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(244)

for

$$\begin{aligned} c_1=&\displaystyle -\frac{3 \left( 16 c_3 \mu (a-b v)-3 c_2^2 \chi _l\right) {}^2}{4096 c_3^3 \chi _l^2} \end{aligned}$$

(245)

$$\begin{aligned} q(x,t)=&\left\{ \displaystyle -\frac{3 c_2}{8 c_3} \pm \frac{\sqrt{3}}{2}\sqrt{\frac{\mu (a-bv)}{c_3\chi _1}} i{{\,{\mathrm{sech}}\,}}\left[ \sqrt{- \mu } \left\{ x - \textstyle \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right\} ^{\frac{1}{2}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{8 c_3 \left( 4 \kappa (a \kappa +\alpha )+\mu (a-b v)+4 Q_l\right) +9 c_2^2 \chi _l}{32 c_3 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(246)

$$\begin{aligned} r(x,t)=&\left\{ \displaystyle -\frac{3 c_2}{8 c_3} \pm \frac{\sqrt{3}}{2}\sqrt{\frac{\mu (a-bv)}{c_3\chi _2}} i{{\,{\mathrm{sech}}\,}}\left[ \sqrt{- \mu } \left\{ x - \textstyle \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right\} ^{\frac{1}{2}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{8 c_3 \left( 4 \kappa (a \kappa +\alpha )+\mu (a-b v)+4 Q_l\right) +9 c_2^2 \chi _l}{32 c_3 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(247)

for

$$\begin{aligned} c_1=&\displaystyle - {\frac{9 c_2^2}{4096 c_3^3} \left( 3 c_2^2 + \frac{16 c_3 \mu (a-b v)}{\chi _l}\right) } \end{aligned}$$

(248)

$$\begin{aligned} q(x,t)=&\left\{ \displaystyle -\frac{3 c_2}{8 c_3} \pm \frac{\sqrt{3}}{2}\sqrt{\frac{\mu (a-bv)}{c_3\chi _1}} \coth \left[ \sqrt{- \mu } \left\{ x -\textstyle \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right\} ^{\frac{1}{2}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{16 c_3 \left( 2 a \kappa ^2+2 \alpha \kappa -a \mu +b \mu v+2 Q_l\right) +9 c_2^2 \chi _l}{32 c_3 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(249)

$$\begin{aligned} r(x,t)=&\left\{ \displaystyle -\frac{3 c_2}{8 c_3} \pm \frac{\sqrt{3}}{2}\sqrt{\frac{\mu (a-bv)}{c_3\chi _2}} \coth \left[ \sqrt{- \mu } \left\{ x - \textstyle \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right\} ^{\frac{1}{2}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{16 c_3 \left( 2 a \kappa ^2+2 \alpha \kappa -a \mu +b \mu v+2 Q_l\right) +9 c_2^2 \chi _l}{32 c_3 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(250)

for

$$\begin{aligned} c_1=&\displaystyle -\frac{3 \left( 16 c_3 \mu (a-b v)-3 c_2^2 \chi _l\right) {}^2}{4096 c_3^3 \chi _l^2} \end{aligned}$$

(251)

$$\begin{aligned} q(x,t)=&\left\{ \displaystyle -\frac{3 c_2}{8 c_3} + \frac{\sqrt{3}}{4}\sqrt{ \frac{\mu (a-bv)}{c_3\chi _1}}\left( \tanh \left[ \sqrt{- \mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right. \right. \nonumber \\&\displaystyle \left. \left. \pm i{{\,{\mathrm{sech}}\,}}\left[ \sqrt{- \mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \right\} ^{\frac{1}{2}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{4 c_3 \left( 8 a \kappa ^2+8 \alpha \kappa -a \mu +b \mu v+8 Q_l\right) +9 c_2^2 \chi _l}{32 c_3 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(252)

$$\begin{aligned} r(x,t)=&\left\{ \displaystyle -\frac{3 c_2}{8 c_3} + \frac{\sqrt{3}}{4}\sqrt{ \frac{\mu (a-bv)}{c_3\chi _2}}\left( \tanh \left[ \sqrt{- \mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right. \right. \nonumber \\&\displaystyle \left. \left. \pm i{{\,{\mathrm{sech}}\,}}\left[ \sqrt{- \mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \right\} ^{\frac{1}{2}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{4 c_3 \left( 8 a \kappa ^2+8 \alpha \kappa -a \mu +b \mu v+8 Q_l\right) +9 c_2^2 \chi _l}{32 c_3 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(253)

for

$$\begin{aligned} c_1=&\displaystyle -\frac{3 \left( 4 c_3 \mu (a-b v)-3 c_2^2 \chi _l\right) {}^2}{4096 c_3^3 \chi _l^2} \end{aligned}$$

(254)

$$\begin{aligned} q(x,t)=&\left\{ \displaystyle -\frac{3 c_2}{8 c_3} \pm \sqrt{3}\sqrt{\frac{\mu (a-bv)}{c_3\chi _1}} \coth \left[ 2\sqrt{- \mu } \left\{ x -\textstyle \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + 2\zeta _0 \right] \right\} ^{\frac{1}{2}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{16 c_3 \left( 2 \kappa (a \kappa +\alpha )+2 Q_l+\ell _2\right) +9 c_2^2 \chi _l}{32 c_3 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(255)

$$\begin{aligned} r(x,t)=&\left\{ \displaystyle -\frac{3 c_2}{8 c_3} \pm \sqrt{3}\sqrt{\frac{\mu (a-bv)}{c_3\chi _2}} \coth \left[ 2\sqrt{- \mu } \left\{ x -\textstyle \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + 2\zeta _0 \right] \right\} ^{\frac{1}{2}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{16 c_3 \left( 2 \kappa (a \kappa +\alpha )+2 Q_l+\ell _2\right) +9 c_2^2 \chi _l}{32 c_3 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(256)

for

$$\begin{aligned} c_1= \displaystyle -\frac{3 \left( 2048 c_3^2 \mu \ell _1 (a-b v)+9 c_2^4 \chi _l^2+96 c_3 c_2^2 \ell _2 \chi _l\right) }{4096 c_3^3 \chi _l^2} \end{aligned}$$

(257)

where $\zeta _0=\tanh ^{-1}(A_2/A_1)$ and

$$\begin{aligned} \begin{array}{ll} \chi _{l} = \eta _l+\xi _l -\kappa \left( \beta _l+\theta _l\right) \\ \\ \ell _1 = \mu (a -b v) -\sqrt{a-b v} \sqrt{\mu ^2 (a-b v)} \\ \\ \ell _2= \mu (-a +b v) +3 \sqrt{a-b v} \sqrt{\mu ^2 (a-b v)}. \end{array} \end{aligned}$$

(258)

If, however, $\mu >0$, one has periodic waves as

$$\begin{aligned} q(x,t)=&\left\{ \displaystyle \displaystyle -\frac{3 c_2}{8 c_3} \mp \frac{\sqrt{3}}{2}\sqrt{-\frac{\mu (a-bv)}{c_3\chi _1}} \tan \left[ \sqrt{ \mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right\} ^{\frac{1}{2}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{16 c_3 \left( 2 a \kappa ^2+2 \alpha \kappa -a \mu +b \mu v+2 Q_l\right) +9 c_2^2 \chi _l}{32 c_3 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(259)

$$\begin{aligned} r(x,t)=&\left\{ \displaystyle \displaystyle -\frac{3 c_2}{8 c_3} \mp \frac{\sqrt{3}}{2}\sqrt{-\frac{\mu (a-bv)}{c_3\chi _2}} \tan \left[ \sqrt{ \mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right\} ^{\frac{1}{2}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{16 c_3 \left( 2 a \kappa ^2+2 \alpha \kappa -a \mu +b \mu v+2 Q_l\right) +9 c_2^2 \chi _l}{32 c_3 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(260)

$$\begin{aligned} q(x,t)=&\left\{ \displaystyle -\frac{3 c_2}{8 c_3} \pm \frac{\sqrt{3}}{2}\sqrt{-\frac{\mu (a-bv)}{c_3\chi _1}} \sec \left[ \sqrt{ \mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right\} ^{\frac{1}{2}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{8 c_3 \left( 4 \kappa (a \kappa +\alpha )+\mu (a-b v)+4 Q_l\right) +9 c_2^2 \chi _l}{32 c_3 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(261)

$$\begin{aligned} r(x,t)=&\left\{ \displaystyle -\frac{3 c_2}{8 c_3} \pm \frac{\sqrt{3}}{2}\sqrt{-\frac{\mu (a-bv)}{c_3\chi _2}} \sec \left[ \sqrt{ \mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right\} ^{\frac{1}{2}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{8 c_3 \left( 4 \kappa (a \kappa +\alpha )+\mu (a-b v)+4 Q_l\right) +9 c_2^2 \chi _l}{32 c_3 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(262)

$$\begin{aligned} q(x,t)=&\left\{ \displaystyle -\frac{3 c_2}{8 c_3} \mp \frac{\sqrt{3}}{2}\sqrt{-\frac{\mu (a-bv)}{c_3\chi _1}} \cot \left[ \sqrt{ \mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right\} ^{\frac{1}{2}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{16 c_3 \left( 2 a \kappa ^2+2 \alpha \kappa -a \mu +b \mu v+2 Q_l\right) +9 c_2^2 \chi _l}{32 c_3 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(263)

$$\begin{aligned} r(x,t)=&\left\{ \displaystyle -\frac{3 c_2}{8 c_3} \mp \frac{\sqrt{3}}{2}\sqrt{-\frac{\mu (a-bv)}{c_3\chi _2}} \cot \left[ \sqrt{ \mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right\} ^{\frac{1}{2}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{16 c_3 \left( 2 a \kappa ^2+2 \alpha \kappa -a \mu +b \mu v+2 Q_l\right) +9 c_2^2 \chi _l}{32 c_3 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(264)

$$\begin{aligned} q(x,t)=&\left\{ \displaystyle -\frac{3 c_2}{8 c_3} - \frac{\sqrt{3}}{4}\sqrt{-\frac{\mu (a-bv)}{c_3\chi _1}} \left( \tan \left[ \sqrt{ \mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right. \right. \nonumber \\&\left. \left. \mp \sec \left[ \sqrt{ \mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \right\} ^{\frac{1}{2}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{4 c_3 \left( 8 a \kappa ^2+8 \alpha \kappa -a \mu +b \mu v+8 Q_l\right) +9 c_2^2 \chi _l}{32 c_3 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(265)

$$\begin{aligned} r(x,t)=&\left\{ \displaystyle -\frac{3 c_2}{8 c_3} - \frac{\sqrt{3}}{4}\sqrt{-\frac{\mu (a-bv)}{c_3\chi _2}} \left( \tan \left[ \sqrt{ \mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right. \right. \nonumber \\&\left. \left. \mp \sec \left[ \sqrt{ \mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \right\} ^{\frac{1}{2}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{4 c_3 \left( 8 a \kappa ^2+8 \alpha \kappa -a \mu +b \mu v+8 Q_l\right) +9 c_2^2 \chi _l}{32 c_3 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(266)

$$\begin{aligned} q(x,t)=&\left\{ \displaystyle -\frac{3 c_2}{8 c_3} \mp \sqrt{3}\sqrt{-\frac{\mu (a-bv)}{c_3\chi _1}} \csc \left[ 2\sqrt{ \mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + 2 \zeta _0 \right] \right\} ^{\frac{1}{2}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{16 c_3 \left( 2 \kappa (a \kappa +\alpha )+2 Q_l+\ell _2\right) +9 c_2^2 \chi _l}{32 c_3 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(267)

$$\begin{aligned} r(x,t)=&\left\{ \displaystyle -\frac{3 c_2}{8 c_3} \mp \sqrt{3}\sqrt{-\frac{\mu (a-bv)}{c_3\chi _2}} \csc \left[ 2\sqrt{ \mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + 2 \zeta _0 \right] \right\} ^{\frac{1}{2}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{16 c_3 \left( 2 \kappa (a \kappa +\alpha )+2 Q_l+\ell _2\right) +9 c_2^2 \chi _l}{32 c_3 (b \kappa -1)} \right) t + \theta \right) \right] \end{aligned}$$

(268)

where $\zeta _0=\tan ^{-1}(A_1/A_2)$. These trigonometric function solution pairs exist as long as Eqs. (245), (248), (251), (254) and (257) remain valid, respectively.

The profiles of dark solitons are stood for by Fig. 9. The parameter values chosen are $\kappa = 1, \ a= 1, \ b= 2, \ \xi _1 = 1.5, \ \xi _2 = 1.9, \ \beta _1 = 1.6, \ \beta _2 = 0.4, \ \theta _1 = 1.4, \ \theta _2 = 0.7, \ \eta _1 = 1.7, \ \eta _2 = 2.1, \ \mu = -1 , \ \alpha = 1, \ \zeta _0 = 0, \ \omega = 1, \ c_2 = 2, \ c_3 = 3 .$

Fig. 9

Numerical simulations of dark solitons (243) and (244)

8 Polynomial law

In this case, $F(s)=c_{1}s+c_{2}s^{2}+c_{3}s^{3}$ where $c_{1}$, $c_{2}$ and $c_{3}$ are all constants. Therefore the system (1)–(2) is given by

$$\begin{aligned}&iq_{t}+a_1q_{xx}+b_1q_{xt}+ \left\{ \xi _1 \left( c_{1}\left| q\right| ^{2} +c_{2} \left| q\right| ^{4}+c_{3} \left| q\right| ^{6}\right) + \eta _1 \left( c_{1} \left| r\right| ^{2}+c_{2} \left| r\right| ^{4}+c_{3} \left| r\right| ^{6}\right) \right\} q \nonumber \\&\quad =Q_1 r + i \left\{ \alpha _1 q_{x} + \beta _1 \left( \left( c_{1} \left| q\right| ^{2}+c_{2} \left| q\right| ^{4}+c_{3} \left| q\right| ^{6}\right) q\right) _{x}\right. \nonumber \\&\quad \quad \left. + \nu _1 \left( c_{1} \left| q\right| ^{2} +c_{2} \left| q\right| ^{4}+c_{3} \left| q\right| ^{6} \right) _{x}q + \theta _1 \left( c_{1} \left| q\right| ^{2} +c_{2} \left| q\right| ^{4}+c_{3}\left| q\right| ^{6}\right) q_{x}\right\} \end{aligned}$$

(269)

$$\begin{aligned}&ir_{t}+a_2r_{xx}+b_2r_{xt}+ \left\{ \xi _2 \left( c_{1}\left| r\right| ^{2} +c_{2}\left| r\right| ^{4}+c_{3}\left| r\right| ^{6}\right) + \eta _2 \left( c_{1}\left| q\right| ^{2}+c_{2}\left| q\right| ^{4}+c_{3}\left| q\right| ^{6}\right) \right\} r \nonumber \\&\quad =Q_2 q + i \left\{ \alpha _2 r_{x} + \beta _2 \left( \left( c_{1}\left| r\right| ^{2}+c_{2}\left| r\right| ^{4}+c_{3}\left| r\right| ^{6}\right) r\right) _{x}\right. \nonumber \\&\qquad \left. + \nu _2 \left( c_{1} \left| r\right| ^{2} +c_{2}\left| r\right| ^{4}+c_{3}\left| r\right| ^{6} \right) _{x}r + \theta _2 \left( c_{1}\left| r\right| ^{2} +c_{2}\left| r\right| ^{4}+c_{3}\left| r\right| ^{6}\right) r_{x}\right\} . \end{aligned}$$

(270)

Put (5) and (6) into (269) and (270). The real part gives

$$\begin{aligned}&\left( a_l -b_l v \right) P_l''+\left( b_l\kappa \omega - \omega -a_l \kappa ^2 - \alpha _l \kappa \right) P_l +\left( \xi _l -\kappa (\beta _l + \theta _l)\right) \left( c_1 P_l^{3} + c_2 P_l^{5} + c_3 P_l^{7}\right) \nonumber \\&\qquad + (c_1 +c_2 P_{{\bar{l}}}^2 +c_3 P_{{\bar{l}}}^4)\eta _lP_lP_{{\bar{l}}}^2 -Q_lP_{{\bar{l}}} = 0 \end{aligned}$$

(271)

while the imaginary part implies

$$\begin{aligned}&\left( v+2a_l \kappa +\alpha _l - b_l (v \kappa +\omega ) \right) P_l' +c_1 \left( 3 \beta _l+\theta _l+2 \nu _l\right) P_l^2 P_l' +c_2 \left( 5 \beta _l+\theta _l+4 \nu _l\right) P_l^4 P_l' \nonumber \\&\qquad +c_3 \left( 7 \beta _l+\theta _l+6 \nu _l\right) P_l^6 P_l' = 0. \end{aligned}$$

(272)

From (272) is possible in order to procure the solitons speed (11) as long as the constraints

$$\begin{aligned}&3 \beta _l+\theta _l+2 \nu _l =0 \end{aligned}$$

(273)

$$\begin{aligned}&5 \beta _l+\theta _l+4 \nu _l= 0 \end{aligned}$$

(274)

$$\begin{aligned}&7 \beta _l+\theta _l+6 \nu _l=0 \end{aligned}$$

(275)

remain valid. Then form (273) and (275)

$$\begin{aligned}&\beta _l + \nu _l = 0 \end{aligned}$$

(276)

$$\begin{aligned}&\beta _l + \theta _l = 0 \end{aligned}$$

(277)

$$\begin{aligned}&\theta _l = \nu _l. \end{aligned}$$

(278)

As a results (13) and (14) are also satisfied in this case, and the real part (271) becomes

$$\begin{aligned}&\left( a -b v \right) P_l''+\left( b\kappa \omega - \omega -a \kappa ^2 - \alpha \kappa \right) P_l +\left( \xi _l -\kappa (\beta _l + \theta _l)\right) \left( c_1 P_l^{3} + c_2 P_l^{5} + c_3 P_l^{7}\right) \ \nonumber \\&\quad \quad +(c_1 +c_2 P_{{\bar{l}}}^2 +c_3 P_{{\bar{l}}}^4)\eta _lP_lP_{{\bar{l}}}^2 -Q_lP_{{\bar{l}}} = 0. \end{aligned}$$

(279)

From the balancing principle, one gets

$$\begin{aligned} P_{{ {\bar{l}}}}=P_l. \end{aligned}$$

(280)

Thus, Eq. (279) reduces to

$$\begin{aligned}&\left( a -b v \right) P_l''+\left( b\kappa \omega - \omega -a \kappa ^2 - \alpha \kappa \right) P_l \nonumber \\&\qquad +\left( \xi _l + \eta _l -\kappa (\beta _l + \theta _l)\right) \left( c_1 P_l^{3} + c_2 P_l^{5} + c_3 P_l^{7}\right) -Q_lP_l = 0. \end{aligned}$$

(281)

Set

$$\begin{aligned} P_l=U_l^\frac{1}{3} \end{aligned}$$

(282)

so that (281) transform to

$$\begin{aligned}&3 (a-b v) U_l U_l'' - 2 (a-b v) (U_l')^2 - 9 \left( \omega +\kappa (a \kappa +\alpha -b \omega ) +Q_l\right) U_l^2 \nonumber \\&\qquad +9 c_3 \left( \eta _l +\xi _l -\kappa (\beta _l +\theta _l)\right) U_l^4=0. \end{aligned}$$

(283)

8.1 Extended $G^{\prime }/G-$expansion approach

According to the homogeneous balance method, one reveals

$$\begin{aligned} U_l(\zeta )=&\alpha _0^{(l)}+\alpha _1^{(l)}\left( \frac{G'}{G}\right) + \beta _1^{(l)}\sqrt{\sigma \left( 1+\frac{1}{\mu }\left( \frac{G'}{G}\right) ^2 \right) }+ \gamma _1^{(l)}\left( \frac{G'}{G}\right) ^{-1} +\delta _1^{(l)}\frac{1}{\sqrt{\sigma \left( 1+\frac{1}{\mu }\left( \frac{G'}{G}\right) ^2 \right) }}. \end{aligned}$$

(284)

Proceeding as in previous sections, the secured solutions are:

If $\mu <0$ and $A_1^2>A_2^2$, one attains complexiton solutions

$$\begin{aligned} q(x,t)=&\left\{ \displaystyle \pm \frac{2}{3} \sqrt{-\frac{ \mu (a-b v)}{c_3 \chi _1}} {{\,{\mathrm{sech}}\,}}\left[ \sqrt{-\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right\} ^{\frac{1}{3}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{9 \kappa (a \kappa +\alpha )+\mu (a-b v)+9 Q_l}{9 b \kappa -9}\right) t + \theta \right) \right] \end{aligned}$$

(285)

$$\begin{aligned} r(x,t)=&\left\{ \displaystyle \pm \frac{2}{3} \sqrt{-\frac{ \mu (a-b v)}{c_3 \chi _2}} {{\,{\mathrm{sech}}\,}}\left[ \sqrt{-\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right\} ^{\frac{1}{3}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{9 \kappa (a \kappa +\alpha )+\mu (a-b v)+9 Q_l}{9 b \kappa -9}\right) t + \theta \right) \right] \end{aligned}$$

(286)

and singular solitons

$$\begin{aligned} q(x,t)=&\left\{ \displaystyle \mp \frac{4i}{3} \sqrt{-\frac{ \mu (a-b v)}{c_3 \chi _1}} {{\,{\mathrm{csch}}\,}}\left[ 2\sqrt{-\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + 2\zeta _0 \right] \right\} ^{\frac{1}{3}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{9 \kappa (a \kappa +\alpha )+4 \mu (a-b v)+9 Q_l}{9 b \kappa -9}\right) t + \theta \right) \right] \end{aligned}$$

(287)

$$\begin{aligned} r(x,t)=&\left\{ \displaystyle \mp \frac{4i}{3} \sqrt{-\frac{ \mu (a-b v)}{c_3 \chi _2}} {{\,{\mathrm{csch}}\,}}\left[ 2\sqrt{-\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + 2\zeta _0 \right] \right\} ^{\frac{1}{3}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{9 \kappa (a \kappa +\alpha )+4 \mu (a-b v)+9 Q_l}{9 b \kappa -9}\right) t + \theta \right) \right] \end{aligned}$$

(288)

where $\zeta _0=\tanh ^{-1}(A_2/A_1)$ and

$$\begin{aligned} \chi _{l} = \eta _l +\xi _l - \kappa \left( \beta _l+\theta _l\right) . \end{aligned}$$

(289)

If, however, $\mu >0$, one obtains periodic waves as

$$\begin{aligned} q(x,t)=&\left\{ \displaystyle \pm \frac{2i}{3} \sqrt{\frac{ \mu (a-b v)}{c_3 \chi _1}} \sec \left[ \sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right\} ^{\frac{1}{3}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{9 \kappa (a \kappa +\alpha )+\mu (a-b v)+9 Q_l}{9 b \kappa -9}\right) t + \theta \right) \right] \end{aligned}$$

(290)

$$\begin{aligned} r(x,t)=&\left\{ \displaystyle \pm \frac{2i}{3} \sqrt{\frac{ \mu (a-b v)}{c_3 \chi _2}} \sec \left[ \sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right\} ^{\frac{1}{3}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{9 \kappa (a \kappa +\alpha )+\mu (a-b v)+9 Q_l}{9 b \kappa -9}\right) t + \theta \right) \right] \end{aligned}$$

(291)

$$\begin{aligned} q(x,t)=&\left\{ \displaystyle \mp \frac{4i}{3} \sqrt{\frac{ \mu (a-b v)}{c_3 \chi _1}}\csc \left[ 2\sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + 2\zeta _0 \right] \right\} ^{\frac{1}{3}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{9 \kappa (a \kappa +\alpha )+4 \mu (a-b v)+9 Q_l}{9 b \kappa -9}\right) t + \theta \right) \right] \end{aligned}$$

(292)

$$\begin{aligned} r(x,t)=&\left\{ \displaystyle \mp \frac{4i}{3} \sqrt{\frac{ \mu (a-b v)}{c_3 \chi _2}}\csc \left[ 2\sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + 2\zeta _0 \right] \right\} ^{\frac{1}{3}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{9 \kappa (a \kappa +\alpha )+4 \mu (a-b v)+9 Q_l}{9 b \kappa -9}\right) t + \theta \right) \right] \end{aligned}$$

(293)

where $\zeta _0=\tan ^{-1}(A_1/A_2)$.

The profiles of bright solitons are given by Fig. 10. The parameter values picked are $\kappa = 1, \ a= 2, \ b= 3, \ \xi _1 = 1.6, \ \xi _2 = 1.8, \ \beta _1 = 0.4, \ \beta _2 = 0.6, \ \theta _1 = 0.1, \ \theta _2 = 0.4, \ \eta _1 = 0.7, \ \eta _2 = 1.9, \ \mu = -1 , \ \alpha = 1, \ \zeta _0 = 0, \ \omega = 1, \ c_3=-3.$

Fig. 10

Numerical simulations of bright solitons (285) and (286)

9 Triple-power law

In this case, $F(s)=c_{1}s^{n}+c_{2}s^{2n}+c_{3}s^{3n}$ where $c_{1}$, $c_{2}$ and $c_{3}$ are all constants. Therefore Eqs. (1) and (2) collapse to

$$\begin{aligned}&iq_{t}+a_1q_{xx}+b_1q_{xt}+ \left\{ \xi _1 \left( c_{1}\left| q \right| ^{2n} +c_{2} \left| q \right| ^{4n}+c_{3} \left| q \right| ^{6n} \right) \right. \nonumber \\&\qquad \left. + \eta _1 \left( c_{1} \left| r \right| ^{2n}+c_{2} \left| r \right| ^{4n}+c_{3} \left| r \right| ^{6n} \right) \right\} q \nonumber \\&\quad =Q_1 r + i \left\{ \alpha _1 q_{x} \right. \left. + \beta _1 \left( \left( c_{1} \left| q\right| ^{2n}+c_{2} \left| q\right| ^{4n}+c_{3}\left| q\right| ^{6n}\right) q\right) _{x} + \nu _1 \left( c_{1}\left| q\right| ^{2n} +c_{2}\left| q\right| ^{4n}+c_{3}\left| q\right| ^{6n} \right) _{x}q \right. \nonumber \\&\qquad \left. + \theta _1 \left( c_{1}\left| q\right| ^{2n} +c_{2}\left| q\right| ^{4n}+c_{3}\left| q\right| ^{6n}\right) q_{x}\right\} \end{aligned}$$

(294)

$$\begin{aligned}&ir_{t}+a_2r_{xx}+b_2r_{xt}+ \left\{ \xi _2 \left( c_{1}\left| r\right| ^{2n} +c_{2}\left| r\right| ^{4n}+c_{3}\left| r\right| ^{6n}\right) \right. \nonumber \\&\qquad \left. + \eta _2 (c_{1}\left| q\right| ^{2n}+c_{2}\left| q\right| ^{4n}+c_{3}\left| q\right| ^{6n}) \right\} r \nonumber \\&\quad =Q_2 q + i \left\{ \alpha _2 r_{x} \right. \left. + \beta _2 \left( \left( c_{1}\left| r\right| ^{2n}+c_{2}\left| r\right| ^{4n}+c_{3}\left| r\right| ^{6n}\right) r\right) _{x} + \nu _2 \left( c_{1}\left| r\right| ^{2n} +c_{2}\left| r\right| ^{4n}+c_{3}\left| r\right| ^{6n} \right) _{x}r \right. \nonumber \\&\qquad \left. + \theta _2 \left( c_{1}\left| r\right| ^{2n} +c_{2}\left| r\right| ^{4n}+c_{3}\left| r\right| ^{6n}\right) r_{x}\right\} . \end{aligned}$$

(295)

Upon inserting (5) and (6) into (294) and (295) the resulting real part recovered is

$$\begin{aligned}&\left( a_l -b_l v \right) P_l''+\left( b_l\kappa \omega - \omega -a_l \kappa ^2 - \alpha _l \kappa \right) P_l +\left( \xi _l -\kappa (\beta _l + \theta _l)\right) \nonumber \\&\qquad \times \left( c_1 P_l^{2n+1} + c_2 P_l^{4n+1} + c_3 P_l^{6n+1}\right) +(c_1 +c_2 P_{{\bar{l}}}^{2n} +c_3 P_{{\bar{l}}}^{4n})\eta _l P_l P_{{\bar{l}}}^{2n} -Q_l P_{{\bar{l}}} = 0 \end{aligned}$$

(296)

and for the imaginary part

$$\begin{aligned}&\left( v+2a_l \kappa +\alpha _l - b_l (v \kappa +\omega ) \right) P_l' +c_1 \left( (2n+1)\beta _l+\theta _l+2n \nu _l\right) P_l^{2n} P_l' \nonumber \\&\qquad +c_2 \left( (4n+1)\beta _l + \theta _l + 4n \nu _l\right) P_l^{4n} P_l' +c_3 \left( (6n+1) \beta _l + \theta _l + 6n \nu _l\right) P_l^{6n} P_l' = 0. \end{aligned}$$

(297)

From (297) is possible to derive the solitons speed (11) as long as the constraints

$$\begin{aligned}&(2n+1)\beta _l+\theta _l+2n \nu _l = 0 \end{aligned}$$

(298)

$$\begin{aligned}&(4n+1)\beta _l + \theta _l + 4n \nu _l = 0 \end{aligned}$$

(299)

$$\begin{aligned}&(6n+1) \beta _l + \theta _l + 6n \nu _l = 0 \end{aligned}$$

(300)

remain valid. Then, from (298) and (300)

$$\begin{aligned}&\beta _l + \nu _l = 0 \end{aligned}$$

(301)

$$\begin{aligned}&\beta _l + \theta _l = 0 \end{aligned}$$

(302)

$$\begin{aligned}&\theta _l = \nu _l. \end{aligned}$$

(303)

Consequently (13) and (14) are also satisfied in this case, and the real part (296) becomes

$$\begin{aligned}&\left( a -b v \right) P_l''+\left( b\kappa \omega - \omega -a \kappa ^2 - \alpha \kappa \right) P_l +\left( \xi _l -\kappa (\beta _l + \theta _l)\right) \nonumber \\&\qquad \times \left( c_1 P_l^{2n+1} + c_2 P_l^{4n+1} + c_3 P_l^{6n+1}\right) + (c_1 +c_2 P_{{\bar{l}}}^{2n} +c_3 P_{{\bar{l}}}^{4n})\eta _l P_l P_{{\bar{l}}}^{2n} -Q_l P_{{\bar{l}}} = 0. \end{aligned}$$

(304)

The balancing principle implies that

$$\begin{aligned} P_{{ {\bar{l}}}}=P_l. \end{aligned}$$

(305)

Therefore, Eq. (304) modifies to

$$\begin{aligned}&\left( a -b v \right) P_l''+\left( b\kappa \omega - \omega -a \kappa ^2 - \alpha \kappa \right) P_l +\left( \xi _l + \eta _l -\kappa (\beta _l + \theta _l)\right) \left( c_1 P_l^{2n+1} + c_2 P_l^{4n+1} + c_3 P_l^{6n+1}\right) -Q_lP_l = 0. \end{aligned}$$

(306)

To investigate closed form solutions to the adopted model, the transformation

$$\begin{aligned} P_l=U_l^\frac{1}{3n} \end{aligned}$$

(307)

is done and in this case, (306) turns into

$$\begin{aligned}&3n (a-b v) U_l U_l'' - (3n-1) (a-b v) (U_l')^2 - 9n^2 \left( \omega +\kappa (a \kappa +\alpha -b \omega ) +Q_l\right) U_l^2 \nonumber \\&\qquad +9 c_3 n^2\left( \eta _l +\xi _l -\kappa (\beta _l +\theta _l)\right) U_l^4=0. \end{aligned}$$

(308)

9.1 Extended $G^{\prime }/G-$expansion approach

From the principle of homogeneous balance

$$\begin{aligned} U_l(\zeta )=&\alpha _0^{(l)}+\alpha _1^{(l)}\left( \frac{G'}{G}\right) + \beta _1^{(l)}\sqrt{\sigma \left( 1+\frac{1}{\mu }\left( \frac{G'}{G}\right) ^2 \right) }+ \gamma _1^{(l)}\left( \frac{G'}{G}\right) ^{-1} +\delta _1^{(l)}\frac{1}{\sqrt{\sigma \left( 1+\frac{1}{\mu }\left( \frac{G'}{G}\right) ^2 \right) }}. \end{aligned}$$

(309)

Proceeding as in previous sections, the solutions retrieved are:

When $\mu <0$ and $A_1^2>A_2^2$, one secures bright solitons

$$\begin{aligned} q(x,t)=&\left\{ \displaystyle \pm \frac{1}{3n} \sqrt{ - \frac{ \mu (3 n+1) (a-b v)}{ c_3 \chi _1}} {{\,{\mathrm{sech}}\,}}\left[ \sqrt{-\mu } \left\{ x - \textstyle \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right\} ^{\frac{1}{3n}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{\mu (a-b v)+9 \kappa n^2 (a \kappa +\alpha )+9 n^2 Q_l}{9 n^2 (b \kappa -1)}\right) t + \theta \right) \right] \end{aligned}$$

(310)

$$\begin{aligned} r(x,t)=&\left\{ \displaystyle \pm \frac{1}{3n} \sqrt{ - \frac{ \mu (3 n+1) (a-b v)}{ c_3 \chi _2}} {{\,{\mathrm{sech}}\,}}\left[ \sqrt{-\mu } \left\{ x - \textstyle \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right\} ^{\frac{1}{3n}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{\mu (a-b v)+9 \kappa n^2 (a \kappa +\alpha )+9 n^2 Q_l}{9 n^2 (b \kappa -1)}\right) t + \theta \right) \right] \end{aligned}$$

(311)

and complexiton solutions

$$\begin{aligned} q(x,t)=&\left\{ \displaystyle \mp \frac{2i}{3n} \sqrt{ - \frac{ \mu (3 n+1) (a-b v)}{ c_3 \chi _1}} {{\,{\mathrm{csch}}\,}}\left[ 2\sqrt{-\mu } \left\{ x - \textstyle \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + 2\zeta _0 \right] \right\} ^{\frac{1}{3n}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{4 \mu (a-b v)+9 \kappa n^2 (a \kappa +\alpha )+9 n^2 Q_l}{9 n^2 (b \kappa -1)}\right) t + \theta \right) \right] \end{aligned}$$

(312)

$$\begin{aligned} r(x,t)=&\left\{ \displaystyle \mp \frac{2i}{3n} \sqrt{ - \frac{ \mu (3 n+1) (a-b v)}{ c_3 \chi _2}} {{\,{\mathrm{csch}}\,}}\left[ 2\sqrt{-\mu } \left\{ x - \textstyle \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + 2\zeta _0 \right] \right\} ^{\frac{1}{3n}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{4 \mu (a-b v)+9 \kappa n^2 (a \kappa +\alpha )+9 n^2 Q_l}{9 n^2 (b \kappa -1)}\right) t + \theta \right) \right] \end{aligned}$$

(313)

where $\zeta _0=\tanh ^{-1}(A_2/A_1)$ and

$$\begin{aligned} \chi _{l} = \eta _l+\xi _l -\kappa \left( \beta _l+\theta _l\right) . \end{aligned}$$

(314)

If, however, $\mu >0$, one acquires periodic waves as

$$\begin{aligned} q(x,t)=&\left\{ \displaystyle \mp \frac{i}{3n} \sqrt{ \frac{ \mu (3 n+1) (a-b v)}{ c_3 \chi _1}}\sec \left[ \sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right\} ^{\frac{1}{3n}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{\mu (a-b v)+9 \kappa n^2 (a \kappa +\alpha )+9 n^2 Q_l}{9 n^2 (b \kappa -1)}\right) t + \theta \right) \right] \end{aligned}$$

(315)

$$\begin{aligned} r(x,t)=&\left\{ \displaystyle \mp \frac{i}{3n} \sqrt{ \frac{ \mu (3 n+1) (a-b v)}{ c_3 \chi _2}}\sec \left[ \sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right\} ^{\frac{1}{3n}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{\mu (a-b v)+9 \kappa n^2 (a \kappa +\alpha )+9 n^2 Q_l}{9 n^2 (b \kappa -1)}\right) t + \theta \right) \right] \end{aligned}$$

(316)

$$\begin{aligned} q(x,t)=&\left\{ \displaystyle \pm \frac{2i}{3n} \sqrt{ \frac{ \mu (3 n+1) (a-b v)}{ c_3 \chi _1}}\csc \left[ 2\sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + 2\zeta _0 \right] \right\} ^{\frac{1}{3n}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{4 \mu (a-b v)+9 \kappa n^2 (a \kappa +\alpha )+9 n^2 Q_l}{9 n^2 (b \kappa -1)}\right) t + \theta \right) \right] \end{aligned}$$

(317)

$$\begin{aligned} r(x,t)=&\left\{ \displaystyle \pm \frac{2i}{3n} \sqrt{ \frac{ \mu (3 n+1) (a-b v)}{ c_3 \chi _2}}\csc \left[ 2\sqrt{\mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + 2\zeta _0 \right] \right\} ^{\frac{1}{3n}} \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \displaystyle \frac{4 \mu (a-b v)+9 \kappa n^2 (a \kappa +\alpha )+9 n^2 Q_l}{9 n^2 (b \kappa -1)}\right) t + \theta \right) \right] \end{aligned}$$

(318)

where $\zeta _0=\tan ^{-1}(A_1/A_2)$.

The profiles of bright solitons are represented by Fig.11. The chosen parameter values are $n=2, \ \kappa = 1, \ a= 1, \ b= 2, \ \xi _1 = 1.6, \ \xi _2 = 1.7, \ \beta _1 = 0.8, \ \beta _2 = 0.2, \ \theta _1 = 0.7, \ \theta _2 = 0.4, \ \eta _1 = 2, \ \eta _2 = 2.2, \ \mu = -1 , \ \alpha = 1, \ \zeta _0 = 0, \ \omega = 1, \ c_3 = -2 .$

Fig. 11

Numerical simulations of bright solitons (310) and (311)

10 Parabolic law with weak non-local nonlinearity

In this case, $F(s)=c_{1}s+c_{2}s^2+c_{3}s_{xx}$ where $c_{1}$, $c_{2}$ and $c_{3}$ are constants. Therefore the system (1)–(2) is given by

$$\begin{aligned}&iq_{t}+a_1q_{xx}+b_1q_{xt}+ \left\{ \xi _1 \left( c_{1}\left| q\right| ^{2} +c_{2}\left| q\right| ^{4}+c_{3}\left( \left| q\right| ^{2}\right) _{xx}\right) + \eta _1 \left( c_{1}\left| r\right| ^{2}+c_{2}\left| r\right| ^{4}+c_{3}\left( \left| r\right| ^{2}\right) _{xx}\right) \right\} q \nonumber \\&\quad =Q_1 r + i \left\{ \alpha _1 q_{x} + \beta _1 \left( \left( c_{1}\left| q\right| ^{2} +c_{2}\left| q\right| ^{4}+c_{3}\left( \left| q\right| ^{2}\right) _{xx}\right) q\right) _{x}\nonumber \right. \\&\qquad + \nu _1 \left( c_{1}\left| q\right| ^{2} +c_{2}\left| q\right| ^{4}+c_{3}\left( \left| q\right| ^{2}\right) _{xx} \right) _{x}q \nonumber \\&\qquad \left. + \theta _1 \left( c_{1}\left| q\right| ^{2} +c_{2}\left| q\right| ^{4}+c_{3}\left( \left| q\right| ^{2}\right) _{xx}\right) q_{x} \right\} \end{aligned}$$

(319)

$$\begin{aligned}&ir_{t}+a_2r_{xx}+b_2r_{xt}+ \left\{ \xi _2 \left( c_{1}\left| r\right| ^{2}+c_{2}\left| r\right| ^{4}+c_{3}\left( \left| r\right| ^{2}\right) _{xx}\right) + \eta _2 \left( c_{1}\left| q\right| ^{2} +c_{2}\left| q\right| ^{4}+c_{3}\left( \left| q\right| ^{2}\right) _{xx}\right) \right\} r \nonumber \\&\quad =Q_2 q + i \left\{ \alpha _2 r_{x} + \beta _2 \left( \left( c_{1}\left| r\right| ^{2}+c_{2}\left| r\right| ^{4}+c_{3}\left( \left| r\right| ^{2}\right) _{xx}\right) r\right) _{x}\nonumber \right. \\&\qquad + \nu _2 \left( c_{1}\left| r\right| ^{2}+c_{2}\left| r\right| ^{4}+c_{3}\left( \left| r\right| ^{2}\right) _{xx} \right) _{x}r \nonumber \\&\qquad \left. + \theta _2 \left( c_{1}\left| r\right| ^{2}+c_{2}\left| r\right| ^{4}+c_{3}\left( \left| r\right| ^{2}\right) _{xx}\right) r_{x} \right\} . \end{aligned}$$

(320)

Upon putting (5) and (6) into (319) and (320) the recovered real part is

$$\begin{aligned}&P_l \left( a_l \kappa ^2+\alpha _l \kappa -b_l \kappa \omega +2 \beta _l c_3 \kappa P_l'{}^2-2 c_3 \eta _l P_{{\bar{l}}}'{}^2-2 c_3 \eta _l P_{{\bar{l}}} P_{{\bar{l}}}''-\eta _l P_{{\bar{l}}}{}^2 \left( c_2 P_{{\bar{l}}}{}^2+c_1\right) \right. \nonumber \\&\quad \quad \left. +2 c_3 \theta _l \kappa P_l'{}^2-2 c_3 \xi _l P_l'{}^2+\omega \right) +\left( b_l v-a_l\right) P_l''+2 c_3 P_l{}^2 P_l'' \left( \kappa \left( \beta _l+\theta _l\right) -\xi _l\right) \nonumber \\&\quad \quad +c_2 P_l{}^5 \left( \kappa \left( \beta _l+\theta _l\right) -\xi _l\right) +c_1 P_l{}^3 \left( \kappa \left( \beta _l+\theta _l\right) -\xi _l\right) +Q_l P_{{\bar{l}}}=0 \end{aligned}$$

(321)

while the imaginary part reads as

$$\begin{aligned}&P_l' \left( 2 a_l \kappa +\alpha _l-b_l (\kappa v+\omega )+P_l \left( 2 c_3 \left( 4 \beta _l+\theta _l+3 \nu _l\right) P_l''+c_2 P_l{}^3 \left( 5 \beta _l+\theta _l+4 \nu _l\right) \right. \right. \nonumber \\&\qquad \left. \left. +c_1 P_l \left( 3 \beta _l+\theta _l+2 \nu _l\right) \right) +v\right) +2 c_3 \left( \beta _l+\theta _l\right) P_l'{}^3+2 c_3 \left( \beta _l+\nu _l\right) P_l{}^2 P_l{}^{(3)}=0. \end{aligned}$$

(322)

From (322) is possible to get the solitons speed (11) as long as the constraints

$$\begin{aligned}&4 \beta _l+\theta _l+3 \nu _l=0 \end{aligned}$$

(323)

$$\begin{aligned}&5 \beta _l+\theta _l+4 \nu _l=0 \end{aligned}$$

(324)

$$\begin{aligned}&3 \beta _l+\theta _l+2 \nu _l=0 \end{aligned}$$

(325)

remain valid. From here

$$\begin{aligned}&\beta _l + \theta _l =0 \end{aligned}$$

(326)

$$\begin{aligned}&\beta _l + \nu _l=0 \end{aligned}$$

(327)

$$\begin{aligned}&\nu _l =\theta _l. \end{aligned}$$

(328)

As a consequence (13) and (14) are also satisfied in this case, and the real part (321) changes to

$$\begin{aligned}&P_l \left( a \kappa ^2+\alpha \kappa -b \kappa \omega +2 \beta _l c_3 \kappa P_l'{}^2-2 c_3 \eta _l P_{{\bar{l}}}'{}^2-2 c_3 \eta _l P_{{\bar{l}}} P_{{\bar{l}}}''-\eta _l P_{{\bar{l}}}{}^2 \left( c_2 P_{{\bar{l}}}{}^2+c_1\right) \right. \nonumber \\&\qquad \left. +2 c_3 \theta _l \kappa P_l'{}^2-2 c_3 \xi _l P_l'{}^2+\omega \right) +\left( b v-a\right) P_l''+2 c_3 P_l{}^2 P_l'' \left( \kappa \left( \beta _l+\theta _l\right) -\xi _l\right) \nonumber \\&\qquad +c_2 P_l{}^5 \left( \kappa \left( \beta _l+\theta _l\right) -\xi _l\right) +c_1 P_l{}^3 \left( \kappa \left( \beta _l+\theta _l\right) -\xi _l\right) +Q_l P_{{\bar{l}}}=0. \end{aligned}$$

(329)

The balancing principle brings about

$$\begin{aligned} P_{ {\bar{l}}}=P_l. \end{aligned}$$

(330)

Then, Eq. (329) becomes

$$\begin{aligned}&P_l \left( a \kappa ^2+\alpha \kappa -b \kappa \omega -2 c_3 \left( \eta _l+\xi _l\right) P_l'{}^2+Q_l+\omega \right) +(b v-a) P_l'' \nonumber \\&\qquad -2 c_3 \left( \eta _l+\xi _l\right) P_l{}^2 P_l''-c_2 \left( \eta _l+\xi _l\right) P_l{}^5-c_1 \left( \eta _l+\xi _l\right) P_l{}^3=0. \end{aligned}$$

(331)

10.1 Extended $G^{\prime }/G-$expansion approach

Balancing $P_l^2 P_l''$ with $P_l^5$ in Eq. (331) yields $M = 1$. Thus, one gets

$$\begin{aligned} P_l(\zeta )=&\alpha _0^{(l)}+\alpha _1^{(l)}\left( \frac{G'}{G}\right) + \beta _1^{(l)}\sqrt{\sigma \left( 1+\frac{1}{\mu }\left( \frac{G'}{G}\right) ^2 \right) }+ \gamma _1^{(l)}\left( \frac{G'}{G}\right) ^{-1} +\delta _1^{(l)}\frac{1}{\sqrt{\sigma \left( 1+\frac{1}{\mu }\left( \frac{G'}{G}\right) ^2 \right) }}. \end{aligned}$$

(332)

Proceeding as in previous sections, the solutions obtained are:

When $\mu <0$ and $A_1^2>A_2^2$, one secures optic solitons as

$$\begin{aligned} q(x,t)=&\displaystyle \pm \sqrt{\frac{2\mu (a-bv)}{\left( 8 c_3 \mu +c_1\right) \left( \eta _1+\xi _1\right) }} \tanh \left[ \sqrt{- \mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \nonumber \\&\times \exp \left[ i \left( - \kappa x + \textstyle \left( \frac{c_1 \left( a \kappa ^2+\alpha \kappa -2 a \mu +2 b \mu v+Q_l\right) +4 c_3 \mu \left( 2 \kappa (a \kappa +\alpha )-3 \mu (a-b v)+2 Q_l\right) }{(b \kappa -1) \left( 8 c_3 \mu +c_1\right) }\right) t + \theta \right) \right] \end{aligned}$$

(333)

$$\begin{aligned} r(x,t)=&\displaystyle \pm \sqrt{\frac{2\mu (a-bv)}{\left( 8 c_3 \mu +c_1\right) \left( \eta _2+\xi _2\right) }} \tanh \left[ \sqrt{- \mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \nonumber \\&\times \exp \left[ i \left( - \kappa x + \textstyle \left( \frac{c_1 \left( a \kappa ^2+\alpha \kappa -2 a \mu +2 b \mu v+Q_l\right) +4 c_3 \mu \left( 2 \kappa (a \kappa +\alpha )-3 \mu (a-b v)+2 Q_l\right) }{(b \kappa -1) \left( 8 c_3 \mu +c_1\right) }\right) t + \theta \right) \right] \end{aligned}$$

(334)

for

$$\begin{aligned} c_2=&\displaystyle \frac{3 c_3 \left( 8 c_3 \mu +c_1\right) \left( \eta _l+\xi _l\right) }{a-b v} \end{aligned}$$

(335)

$$\begin{aligned} q(x,t)=&\displaystyle \pm \sqrt{\frac{2 \mu (a-b v)}{ \left( 4 c_3 \mu -c_1 \right) \left( \eta _1+\xi _1\right) }} {{\,{\mathrm{sech}}\,}}\left[ \sqrt{- \mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \frac{\kappa (a \kappa +\alpha )+\mu (a-b v)+Q_l}{b \kappa -1}\right) t + \theta \right) \right] \end{aligned}$$

(336)

$$\begin{aligned} r(x,t)=&\displaystyle \pm \sqrt{\frac{2 \mu (a-b v)}{ \left( 4 c_3 \mu -c_1 \right) \left( \eta _2+\xi _2\right) }} {{\,{\mathrm{sech}}\,}}\left[ \sqrt{- \mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \frac{\kappa (a \kappa +\alpha )+\mu (a-b v)+Q_l}{b \kappa -1}\right) t + \theta \right) \right] \end{aligned}$$

(337)

for

$$\begin{aligned} c_2=&\displaystyle \frac{3 c_3 \left( c_1-4 c_3 \mu \right) \left( \eta _l+\xi _l\right) }{a-b v} \end{aligned}$$

(338)

$$\begin{aligned} q(x,t)=&\displaystyle \pm \sqrt{\frac{2\mu (a-bv)}{\left( 8 c_3 \mu +c_1\right) \left( \eta _1+\xi _1\right) }} \coth \left[ \sqrt{- \mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \nonumber \\&\times \exp \left[ i \left( - \kappa x + \textstyle \left( \frac{c_1 \left( a \kappa ^2+\alpha \kappa -2 a \mu +2 b \mu v+Q_l\right) +4 c_3 \mu \left( 2 \kappa (a \kappa +\alpha )-3 \mu (a-b v)+2 Q_l\right) }{(b \kappa -1) \left( 8 c_3 \mu +c_1\right) }\right) t + \theta \right) \right] \end{aligned}$$

(339)

$$\begin{aligned} r(x,t)=&\displaystyle \pm \sqrt{\frac{2\mu (a-bv)}{\left( 8 c_3 \mu +c_1\right) \left( \eta _2+\xi _2\right) }} \coth \left[ \sqrt{- \mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \nonumber \\&\times \exp \left[ i \left( - \kappa x + \textstyle \left( \frac{c_1 \left( a \kappa ^2+\alpha \kappa -2 a \mu +2 b \mu v+Q_l\right) +4 c_3 \mu \left( 2 \kappa (a \kappa +\alpha )-3 \mu (a-b v)+2 Q_l\right) }{(b \kappa -1) \left( 8 c_3 \mu +c_1\right) }\right) t + \theta \right) \right] \end{aligned}$$

(340)

for

$$\begin{aligned} c_2=&\displaystyle \frac{3 c_3 \left( 8 c_3 \mu +c_1\right) \left( \eta _l+\xi _l\right) }{a-b v} \end{aligned}$$

(341)

$$\begin{aligned} q(x,t)=&\displaystyle \sqrt{\frac{\mu (a-bv)}{ \left( 4 c_3 \mu +2 c_1\right) \left( \eta _1+\xi _1\right) }}\left( \pm \tanh \left[ \sqrt{- \mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right. \nonumber \\&\left. + i {{\,{\mathrm{sech}}\,}}\left[ \sqrt{- \mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \nonumber \\&\times \exp \left[ i \left( - \kappa x + \textstyle \left( \frac{2 c_1 \left( 2 a \kappa ^2+2 \alpha \kappa -a \mu +b \mu v+2 Q_l\right) +c_3 \mu \left( 8 \kappa (a \kappa +\alpha )-3 \mu (a-b v)+8 Q_l\right) }{4 (b \kappa -1) \left( 2 c_3 \mu +c_1\right) }\right) t + \theta \right) \right] \end{aligned}$$

(342)

$$\begin{aligned} r(x,t)=&\displaystyle \sqrt{\frac{\mu (a-bv)}{ \left( 4 c_3 \mu +2 c_1\right) \left( \eta _2+\xi _2\right) }}\left( \pm \tanh \left[ \sqrt{- \mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right. \nonumber \\&\left. + i {{\,{\mathrm{sech}}\,}}\left[ \sqrt{- \mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \nonumber \\&\times \exp \left[ i \left( - \kappa x + \textstyle \left( \frac{2 c_1 \left( 2 a \kappa ^2+2 \alpha \kappa -a \mu +b \mu v+2 Q_l\right) +c_3 \mu \left( 8 \kappa (a \kappa +\alpha )-3 \mu (a-b v)+8 Q_l\right) }{4 (b \kappa -1) \left( 2 c_3 \mu +c_1\right) }\right) t + \theta \right) \right] \end{aligned}$$

(343)

for

$$\begin{aligned} c_2=&\displaystyle \frac{3 c_3 \left( 2 c_3 \mu +c_1\right) \left( \eta _l+\xi _l\right) }{a-b v} \end{aligned}$$

(344)

$$\begin{aligned} q(x,t)=&\displaystyle \mp \sqrt{-\frac{ 8\mu (a-b v)}{\left( 16 c_3 \mu -c_1\right) \left( \eta _1+\xi _1\right) }} {{\,{\mathrm{csch}}\,}}\left[ 2\sqrt{- \mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + 2\zeta _0 \right] \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \frac{\kappa (a \kappa +\alpha )+4 \mu (a-b v)+Q_l}{b \kappa -1}\right) t + \theta \right) \right] \end{aligned}$$

(345)

$$\begin{aligned} r(x,t)=&\displaystyle \mp \sqrt{-\frac{ 8\mu (a-b v)}{\left( 16 c_3 \mu -c_1\right) \left( \eta _2+\xi _2\right) }} {{\,{\mathrm{csch}}\,}}\left[ 2\sqrt{- \mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + 2\zeta _0 \right] \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \frac{\kappa (a \kappa +\alpha )+4 \mu (a-b v)+Q_l}{b \kappa -1}\right) t + \theta \right) \right] \end{aligned}$$

(346)

for

$$\begin{aligned} c_2= \displaystyle \frac{3 c_3 \left( c_1-16 c_3 \mu \right) \left( \eta _l+\xi _l\right) }{a-b v} \end{aligned}$$

(347)

where $\zeta _0=\tanh ^{-1}(A_2/A_1).$ If, however, $\mu >0$, one constructs periodic waves as

$$\begin{aligned} q(x,t)=&\displaystyle \mp \sqrt{\frac{2\mu (b v- a)}{\left( 8 c_3 \mu +c_1\right) \left( \eta _1+\xi _1\right) }} \tan \left[ \sqrt{ \mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \nonumber \\&\times \exp \left[ i \left( - \kappa x + \textstyle \left( \frac{c_1 \left( a \kappa ^2+\alpha \kappa -2 a \mu +2 b \mu v+Q_l\right) +4 c_3 \mu \left( 2 \kappa (a \kappa +\alpha )-3 \mu (a-b v)+2 Q_l\right) }{(b \kappa -1) \left( 8 c_3 \mu +c_1\right) }\right) t + \theta \right) \right] \end{aligned}$$

(348)

$$\begin{aligned} r(x,t)=&\displaystyle \mp \sqrt{\frac{2\mu (b v- a)}{\left( 8 c_3 \mu +c_1\right) \left( \eta _2+\xi _2\right) }} \tan \left[ \sqrt{ \mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \nonumber \\&\times \exp \left[ i \left( - \kappa x + \textstyle \left( \frac{c_1 \left( a \kappa ^2+\alpha \kappa -2 a \mu +2 b \mu v+Q_l\right) +4 c_3 \mu \left( 2 \kappa (a \kappa +\alpha )-3 \mu (a-b v)+2 Q_l\right) }{(b \kappa -1) \left( 8 c_3 \mu +c_1\right) }\right) t + \theta \right) \right] \end{aligned}$$

(349)

$$\begin{aligned} q(x,t)=&\displaystyle \pm \sqrt{ \frac{2 \mu (a-b v)}{{ \left( 4 c_3\mu - c_1 \right) \left( \eta _1+\xi _1\right) }}} \sec \left[ \sqrt{ \mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \frac{\kappa (a \kappa +\alpha )+\mu (a-b v)+Q_l}{b \kappa -1}\right) t + \theta \right) \right] \end{aligned}$$

(350)

$$\begin{aligned} r(x,t)=&\displaystyle \pm \sqrt{ \frac{2 \mu (a-b v)}{{ \left( 4 c_3\mu - c_1 \right) \left( \eta _2+\xi _2\right) }}} \sec \left[ \sqrt{ \mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \frac{\kappa (a \kappa +\alpha )+\mu (a-b v)+Q_l}{b \kappa -1}\right) t + \theta \right) \right] \end{aligned}$$

(351)

$$\begin{aligned} q(x,t)=&\displaystyle \mp \sqrt{\frac{2 \mu (b v-a)}{\left( 8 c_3 \mu +c_1\right) \left( \eta _1+\xi _1\right) }} \cot \left[ \sqrt{ \mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \nonumber \\&\times \exp \left[ i \left( - \kappa x + \textstyle \left( \frac{c_1 \left( a \kappa ^2+\alpha \kappa -2 a \mu +2 b \mu v+Q_l\right) +4 c_3 \mu \left( 2 \kappa (a \kappa +\alpha )-3 \mu (a-b v)+2 Q_l\right) }{(b \kappa -1) \left( 8 c_3 \mu +c_1\right) }\right) t + \theta \right) \right] \end{aligned}$$

(352)

$$\begin{aligned} r(x,t)=&\displaystyle \mp \sqrt{\frac{2 \mu (b v-a)}{\left( 8 c_3 \mu +c_1\right) \left( \eta _2+\xi _2\right) }} \cot \left[ \sqrt{ \mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \nonumber \\&\times \exp \left[ i \left( - \kappa x + \textstyle \left( \frac{c_1 \left( a \kappa ^2+\alpha \kappa -2 a \mu +2 b \mu v+Q_l\right) +4 c_3 \mu \left( 2 \kappa (a \kappa +\alpha )-3 \mu (a-b v)+2 Q_l\right) }{(b \kappa -1) \left( 8 c_3 \mu +c_1\right) }\right) t + \theta \right) \right] \end{aligned}$$

(353)

$$\begin{aligned} q(x,t)=&\displaystyle \sqrt{\frac{\mu (b v-a)}{\left( 4 c_3 \mu +2c_1\right) \left( \eta _1+\xi _1\right) }} \left( \mp \tan \left[ \sqrt{ \mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right. \nonumber \\&\left. + \sec \left[ \sqrt{ \mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \nonumber \\&\times \exp \left[ i \left( - \kappa x + \textstyle \left( \frac{2 c_1 \left( 2 a \kappa ^2+2 \alpha \kappa -a \mu +b \mu v+2 Q_l\right) +c_3 \mu \left( 8 \kappa (a \kappa +\alpha )-3 \mu (a-b v)+8 Q_l\right) }{4 (b \kappa -1) \left( 2 c_3 \mu +c_1\right) }\right) t + \theta \right) \right] \end{aligned}$$

(354)

$$\begin{aligned} r(x,t)=&\displaystyle \sqrt{\frac{\mu (b v-a)}{\left( 4 c_3 \mu +2c_1\right) \left( \eta _2+\xi _2\right) }} \left( \mp \tan \left[ \sqrt{ \mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right. \nonumber \\&\left. + \sec \left[ \sqrt{ \mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + \zeta _0 \right] \right) \nonumber \\&\times \exp \left[ i \left( - \kappa x + \textstyle \left( \frac{2 c_1 \left( 2 a \kappa ^2+2 \alpha \kappa -a \mu +b \mu v+2 Q_l\right) +c_3 \mu \left( 8 \kappa (a \kappa +\alpha )-3 \mu (a-b v)+8 Q_l\right) }{4 (b \kappa -1) \left( 2 c_3 \mu +c_1\right) }\right) t + \theta \right) \right] \end{aligned}$$

(355)

$$\begin{aligned} q(x,t)=&\displaystyle \mp \sqrt{ \frac{8 \mu (a-b v)}{\left( 16 c_3 \mu -c_1\right) \left( \eta _1+\xi _1\right) }} \csc \left[ 2\sqrt{ \mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + 2 \zeta _0 \right] \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \frac{\kappa (a \kappa +\alpha )+4 \mu (a-b v)+Q_l}{b \kappa -1}\right) t + \theta \right) \right] \end{aligned}$$

(356)

$$\begin{aligned} r(x,t)=&\displaystyle \mp \sqrt{ \frac{8 \mu (a-b v)}{\left( 16 c_3 \mu -c_1\right) \left( \eta _2+\xi _2\right) }} \csc \left[ 2\sqrt{ \mu } \left\{ x - \left( \frac{b\omega - 2a\kappa -\alpha }{1-b \kappa } \right) t \right\} + 2 \zeta _0 \right] \nonumber \\&\times \exp \left[ i \left( - \kappa x + \left( \frac{\kappa (a \kappa +\alpha )+4 \mu (a-b v)+Q_l}{b \kappa -1}\right) t + \theta \right) \right] \end{aligned}$$

(357)

where $\zeta _0=\tan ^{-1}(A_1/A_2)$. These trigonometric function solution pairs exist as long as Eqs. (335), (338), (341), (344) and (347) remain valid, respectively.

The profiles of dark solitons are stood for by Fig. 12. The parameter values chosen are $\kappa = 2, \ a= 1, \ b= 2, \ \xi _1 = 3, \ \xi _2 = 4.2, \ \eta _1 = 2, \ \eta _2 = 3.5, \ \mu = -1 , \ \alpha = 1, \ \zeta _0 = 0, \ \omega = 1, \ c_1 = 0.1, \ c_3 =0.3 .$

That of bright solitons are represented by Fig. 13. In this case, the parameter values picked are $\kappa = 2, \ a= 1, \ b= 2, \ \xi _1 = 3.3, \ \xi _2 = 5.3, \ \eta _1 = 4.6, \ \eta _2 = 3.6, \ \mu = -1 , \ \alpha = 1, \ \zeta _0 = 0, \ \omega = 1, \ c_1 = 0.2, \ c_3 =0.1 .$

Fig. 12

Numerical simulations of dark solitons (333) and (334)

Fig. 13

Numerical simulations of bright solitons (336) and (337)

11 Conclusions

This paper is an application of the extended $G^{\prime }/G$–expansion scheme to retrieve solitons and complexitons to magneto-optic waveguides that are considered with a wide range of nonlinear refractive index structures. The soliton solutions that emerged appear with parameter restructures that fall out naturally from its mathematical structure. These soliton solutions in magneto-optic waveguides open up a floodgate of opportunities to venture further along in this avenue (Dötsch et al. 2005; Haider 2017; Hasegawa and Miyazaki 1992; Kudryashov 2019; Shoji and Mizumoto 2018; Kudryashov 2019a, b, 2020, 2019; Kudryashov and Antonova 2020; Kudryashov 2020a, b, c, 2021a, b, c; Kudryashov and Safanova 2021; Biswas 2020; Zayed et al. 2020, Zayed et al. 2021; Vega–Guzman et al. 2021; Gonzalez–Gaxiola et al. 2021; Yildirim et al. 2021, 2021, xxxx, yyyy, Zayed et al. 2020a, b, c, d, 2021a, b; Daoui et al. 2021; Biswas et al. 2020; Liu et al. 2020a, b; Yu et al. 2020; Yan et al. 2020; Biswas et al. 2020; Meradji et al. 2020; Merabti et al. 2020; Sassaroli et al. 2001; Višňovskỳ et al. 2006). One now needs to locate the conservation laws for these solitons and consequently address the quasi-monochromatic dynamics of such solitons with weak perturbation effects. Subsequently, the quasi-stationary soliton solutions will be recovered for such models. With the aid of quasimonochromatic dynamics, quasi-particle theory shall be established to suppress the intra-channel collision of solitons while propagating through such magneto-optic waveguides. The results of these research activities shall be disseminated. Those hungry readers are suggested to stay tuned with patience!!!