New optical soliton solutions for nonlinear complex fractional Schrödinger equation via new auxiliary equation method and novel $({G'}/{G})$-expansion method

Abstract

In this research, we apply two different techniques on nonlinear complex fractional nonlinear Schrödinger equation which is a very important model in fractional quantum mechanics. Nonlinear Schrödinger equation is one of the basic models in fibre optics and many other branches of science. We use the conformable fractional derivative to transfer the nonlinear real integer-order nonlinear Schrödinger equation to nonlinear complex fractional nonlinear Schrödinger equation. We apply new auxiliary equation method and novel $\left( {G'}/{G}\right)$-expansion method on nonlinear complex fractional Schrödinger equation to obtain new optical forms of solitary travelling wave solutions. We find many new optical solitary travelling wave solutions for this model. These solutions are obtained precisely and efficiency of the method can be demonstrated.

1 Introduction

At the beginning of the twentieth century, experimental guides proposed that atomic particles were also wave-like in nature. For example, electrons were established to give diffraction patterns when passed through a double slit just like light waves. Therefore, it was logical to suppose that a wave equation could demonstrate the attitude of atomic particles. Nonlinear complex fractional Schrödinger equation is one of the most fundamental equation in fractional quantum mechanics which was formulated by Nick Laskin in 1999. The properties of the nonlinear complex fractional Schrödinger equation are the same as linearity, real energy eigenstates, space and time derivatives, local conservation of probability, positive energy, analytic continuation to diffusion and regularity. Many properties of it discover new one by getting new form of solitary travelling wave solutions. However, physically interpreting the wave is one of the main philosophical problems of quantum mechanics. This great model appeared to all the world as one of the results obtained by extending the Feynman path integral, form the Brownian-like to Hévy-like quantum mechanical paths. Nick Laskin in 1999 put the formula of nonlinear complex fractional Schrödinger equation as follows:

$$\begin{aligned} ih\frac{\partial \psi }{\partial t}=D_{\alpha }(-h^{2}\Delta )^{{\alpha }/{2}} \psi +V\psi , \end{aligned}$$

(1.1)

where $\psi$ is the Schrödinger wave function which represents the quantum mechanical probability amplitude for the particle, V is the potential energy which is a function of (p, t) such that p is 3D positive vector, t is the time, h is the Planck constant, $\Delta$ is the Laplace operator, $D_{\alpha }$ is a scale constant and $\alpha$ belongs to an open interval (0, 1). Nonlinear complex fractional Schrödinger equation is considered as one of the most powerful models because of its many applications. We just mention a few applications of this model to show how this model has a great influence on science.

• The nuclear charge of the atom when V(p) refers to the energy of the hydrogen-like atom can be defined as the negative value of the mathematical calculation of the product of atomic number (Z) and the square of the electron charge (e) divided by the absolute value of position vector (p).

  $$\begin{aligned} V(p)=-Z \, \frac{e^{2}}{|p|}. \end{aligned}$$

• Infinite potential in one dimension can be considered as the evidence of the discrete energy spectrum and has the following value:

  $$\begin{aligned} V(x)=\left\{ \begin{array}{l} \, \infty \Rightarrow x\,<\,-\alpha , \\ 0 \Rightarrow -\alpha \,<\,x\,<\,\alpha , \\ \infty \Rightarrow x\,>\,-\alpha . \end{array}\right. \end{aligned}$$

• Fractional quantum oscillator which is fractional quantum mechanical with Hamiltonian operator

  $$\begin{aligned} H_{\alpha ,\, \beta }=D_{\alpha } \left( -h^{2}\,\Delta \right) ^{{\alpha }/{2}}+Q^{2}\, |p|^{\beta }, \end{aligned}$$

  where $1 < \alpha \le 2$, $1 < \beta \le 2$ and Q is the interaction constant.

• Fractional quantum mechanics in solid state system.

Many researchers applied several techniques on nonlinear complex fractional Schrödinger equation [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]. In ref. [1], McLachlan applied a variational method,  in ref. [2], Kaup and Newell applied inverse scattering techniques,  in ref. [3], Whitehead proved the existence of a class of exact eigenvalues and eigenfunctions,  in ref. [4], Ray applied the methods of Burgan et al,  in ref. [5], Gendenshtein applied type of hidden symmetry,  in ref. [6], Ziolkowski applied the homogeneous method,  in ref. [7], Gagnon and Winternitz applied Lie symmetry groups,  in ref. [8], Malfliet and Hereman applied the tanh method,  in ref. [9], Kanna and Lakshmanan obtained explicit multisoliton solutions (up to four-soliton solutions),  in ref. [10], Yan applied the generalised method,  in ref. [11], Kruglov et al applied ansatz method,  in ref. [12], Biazar and Ghazvini applied He’s homotopy perturbation method,  in ref. [13], Ikhdair and Sever applied the Nikiforov–Uvarov method,  in ref. [14], Wazwaz applied the variational iteration method,  in ref. [15], Ma and Chen applied symmetry algebra,  in ref. [16], Tezcan and Sever applied an appropriate coordinate transformation,  in ref. [17], Taghizadeh et al applied the first integral method,  in ref. [18], Arda and Sever applied Laplace transform approach,  in ref. [19], Seadawy applied the function transformation method,  in ref. [20], Seadawy applied variational method,  in ref. [21], Seadawy applied the amplitude ansatz method,  in ref. [22], Yue et al applied a class of ordinary differential equations. All of these great researchers applied several methods as we mentioned [23,24,25,26,27,28,29,30,31,32,33,34]. Over the years, we believe in the importance of this equation and that there are many properties that have not been discovered so far. In this research, we used two methods to find new forms of solutions to this amazing model.

The rest of this paper is organised as follows: In §2, we use new auxiliary equation method [35] and novel $\left( {G'}/{G}\right)$-expansion method [36, 37] to get the exact and solitary travelling wave solutions of nonlinear complex fractional nonlinear Schrödinger equation. In §3, conclusions are given.

2 Formulation for nonlinear complex fractional Schrödinger equation

Consider the nonlinear complex fractional Schrödinger equation in the form [37, 39]

$$\begin{aligned} \frac{\partial ^{\alpha }Q}{\partial t^{\alpha }}+ i \frac{\partial ^{2} Q}{\partial x^{2}}+ \frac{\partial }{\partial x}(\left| Q\right| ^{2} Q)=0, \end{aligned}$$

(2.1)

such that $0<\alpha <1.$

Using the travelling wave transformation $Q(x,t)=v(\xi )\mathrm {e}^{i\,\eta }$  and conformable fractional derivative

$$\begin{aligned} \xi= & {} i\,k\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) ,\\ \eta= & {} \left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) \end{aligned}$$

to transfer the nonlinear complex fractional Schrödinger equation to nonlinear integer order Schrödinger equation and for further properties about conformable fractional derivative you can see [35, 40,41,42], we obtain

$$\begin{aligned} \left\{ {\begin{array}{l} \, \dfrac{\partial ^{\alpha }\,Q}{\partial \, t^{\alpha }}=i\,\left( \epsilon \,v+2\,\omega \,k\,v'\right) \mathrm {e}^{i\,\eta },\\ \dfrac{\partial ^{2} Q}{\partial \, x^{2}}= -\,\left( \omega ^{2}\,v+2\,\omega \,k\,v'+k^{2}\,v''\right) \mathrm {e}^{i\,\eta },\\ \dfrac{\partial }{\partial \, x}\, (\left| Q\right| ^{2} Q)=i\, \left( \omega \,v^{3}+3\,k\,v^{2}\,v''\right) \mathrm {e}^{i\,\eta }. \end{array}}\right. \end{aligned}$$

(2.2)

Substituting (2.2) into eq. (2.1), we get that the nonlinear complex fractional Schrödinger equation transform into nonlinear ordinary differential equation as follows [43,44,45,46,47,48,49]:

$$\begin{aligned} \left( \epsilon -\omega ^{2}\right) v-k^{2}\,v''+\omega \,v^{3}+3\,k\,v^{2}\,v'=0. \end{aligned}$$

(2.3)

Balancing between the highest derivative term and nonlinear term in eq. (2.3), $(v'' \,\mathrm {and}\, v^{2}\,v')$ $\Rightarrow$ $(N+2=2\,N+N+1)$ $\Rightarrow$ $\left( N=\frac{1}{2}\right)$. So, we use another transformation $v(\xi )=u^{{1}/{2}} (\xi )$ in eq. (2.3). We obtain

$$\begin{aligned}&4\,\omega \,u^{3}+4\,(\epsilon -\omega ^{2})\,u^{2}+6\,k\,u^{2}\,u'+k^{2}\,u'^{2}\nonumber \\&\quad -\,2\,k\,u\,u''=0. \end{aligned}$$

(2.4)

Balancing between the highest derivative term and nonlinear term in eq. (2.3), $(u\,u'' \, \mathrm {and} \, u^{2}\,u')$ $\Rightarrow$ $(N+N+2=2\,N+N+1)$ $\Rightarrow$ $\left( N=1\right)$.

2.1 Exact and solitary wave solution of nonlinear complex fractional Schrödinger equation by using new auxiliary equation method

Using the default for precision solution using new auxiliary equation method on nonlinear complex fractional Schrödinger equation, we obtain

$$\begin{aligned} u(\xi )=a_{0}+a_{1}\, a^{f(\xi )}. \end{aligned}$$

(2.5)

Substituting eq. (2.6) and its derivatives into eq. (2.4) and equating the coefficient of different power of $a^{i\,f(\xi )}$ to zero, we obtain a system of algebraic equations by solving it with any computer program like Maple, Mathematica, Matlab and so on. We get

Case I

$$\begin{aligned}&\alpha =0,\quad \beta ={\frac{-2\,\omega }{3\,k}},\quad \sigma =0,\quad \epsilon ={\frac{10\,{\omega }^{2}}{9}},\nonumber \\&a_{{0}}=0,\quad a_{{1}}=a_{{1}}, \end{aligned}$$

so that, the exact travelling wave solution of nonlinear complex fractional Schrödinger equation (2.4) is in the form

$$\begin{aligned} u(\xi )=a_{1}\, a^{f(\xi )}. \end{aligned}$$

(2.6)

Therefore, the solitary wave solutions:

$$\begin{aligned}&u(\xi )=\frac{a_{1}\left( -\left( 1+\mathrm {e}^{2\,\beta \, \xi }\right) \pm \sqrt{2\left( \mathrm {e}^{4\,\beta \, \xi }+1\right) }\right) }{\mathrm {e}^{2\,\beta \,\xi }-1}, \end{aligned}$$

(2.7)

$$\begin{aligned}&Q(x,t)=\left[ \frac{a_{1}\left( -\left( 1+\mathrm {e}^{2i\,\beta \, k\,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) }\right) \pm \sqrt{2\left( \mathrm {e}^{4\,i\,\beta \, k\,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) }+1\right) }\right) }{\mathrm {e}^{2\,i\,\beta \,k\,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) }-1} \right] ^{{1}/{2}}\, \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) } \end{aligned}$$

(2.8)

or

$$\begin{aligned}&u(\xi )=\frac{a_{1}\left( -\left( 1+\mathrm {e}^{2\,\beta \, \xi }\right) \pm \sqrt{\mathrm {e}^{4\,\beta \, \xi }+6\,\mathrm {e}^{2\,\beta \, \xi }+1}\right) }{2\,\mathrm {e}^{2\,\beta \,\xi }}, \end{aligned}$$

(2.9)

$$\begin{aligned}&Q(x,t)=\left[ \frac{a_{1}\left( -\left( 1+\mathrm {e}^{2\,i\,\beta \,k\, \left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) }\right) \pm \sqrt{\mathrm {e}^{4\,i\,\beta \, k\,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) }+6\,\mathrm {e}^{2\, i\,\beta \,k\,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) }+1}\right) }{2\,\mathrm {e}^{2\,i\,\beta \,k\,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) }} \right] ^{{1}/{2}}\, \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) }.\nonumber \\ \end{aligned}$$

(2.10)

Case II

$$\begin{aligned} \alpha= & {} 0,\beta =2\,{\frac{\omega }{k}},\quad \sigma =2\,{\frac{a_{{1}}}{k}},\nonumber \\ \epsilon= & {} 2\,{\omega }^{2},\quad a_{{0}}=0,\quad a_{{1}}=a_{{1}}, \end{aligned}$$

so that, the exact travelling wave solution of nonlinear complex fractional Schrödinger equation (2.4) is in the form

$$\begin{aligned} u(\xi )=a_{1}\, a^{f(\xi )}. \end{aligned}$$

(2.11)

Therefore, the solitary wave solutions when $\beta ^{2}-\,\alpha \,\sigma <0 \,\mathrm {and} \, \sigma \ne 0$ are

$$\begin{aligned}&u(\xi )=a_{1}\left[ \frac{-\beta }{\sigma }+\frac{\sqrt{-\beta ^{2}}}{\sigma } \tan \left( \frac{\sqrt{-\beta ^{2}}}{2} \, \xi \right) \right] , \end{aligned}$$

(2.12)

$$\begin{aligned}&Q(x,t)=\left[ a_{1}\left[ \frac{-\beta }{\sigma }+\frac{\sqrt{-\beta ^{2}}}{\sigma } \tan \left( \frac{\sqrt{\beta ^{2}}}{2} \, k\,\right. \right. \right. \nonumber \\&~~~~~~~~~~~~~~\left. \left. \left. \times \left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) \right) \right] \right] ^{{1}/{2}}\, \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) } \end{aligned}$$

(2.13)

or

$$\begin{aligned}&u(\xi )=a_{1}\left[ \frac{-\beta }{\sigma }+\frac{\sqrt{-\beta ^{2}}}{\sigma }\, \cot \left( \frac{\sqrt{-\beta ^{2}}}{2} \, \xi \right) \right] , \end{aligned}$$

(2.14)

$$\begin{aligned} \\&Q(x,t)=\left[ a_{1}\,\left[ \frac{-\beta }{\sigma }+\frac{\sqrt{-\beta ^{2}}}{\sigma }\, \cot \left( \frac{\sqrt{\beta ^{2}}}{2} \, k\,\right. \right. \right. \nonumber \\&~~~~~~~~~~~~~~\left. \left. \left. \times \left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) \right) \right] \right] ^{{1}/{2}}\, \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) }.\nonumber \\ \end{aligned}$$

(2.15)

When $\beta ^{2}-\,\alpha \,\sigma >0 \,\mathrm {and} \, \sigma \ne 0$

$$\begin{aligned}&u(\xi )=a_{1}\,\left[ \frac{-\beta }{\sigma }-\frac{\beta }{ \sigma } \tanh \left( \frac{\beta }{2} \, \xi \right) \right] , \end{aligned}$$

(2.16)

$$\begin{aligned}&Q(x,t)=\left[ a_{1}\,\left[ \frac{-\beta }{\sigma }-\frac{\beta }{ \sigma }\, \right. \right. \nonumber \\&~~~~~~~~~~~~~~~\left. \left. \times \tanh \left( \frac{i\,k\,\beta }{2} \,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) \right) \right] \right] ^{{1}/{2}}\,\nonumber \\&~~~~~~~~~~~~~~~\times \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) } \end{aligned}$$

(2.17)

or

$$\begin{aligned}&u(\xi )=a_{1}\,\left[ \frac{-\beta }{\sigma }-\frac{\beta }{ \sigma }\, \coth \left( \frac{\beta }{2} \, \xi \right) \right] , \end{aligned}$$

(2.18)

$$\begin{aligned}&Q(x,t)=\left[ a_{1}\,\left[ \frac{-\beta }{\sigma }-\frac{\beta }{ \sigma }\, \right. \right. \nonumber \\&\left. \left. ~~~~~~~~~~~~~~~\times \coth \left( \frac{i\,k\,\beta }{2} \,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) \right) \right] \right] ^{{1}/{2}}\,\nonumber \\&~~~~~~~~~~~~~~~~\times \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) }. \end{aligned}$$

(2.19)

When $\beta =k,\, \sigma =2\,k,\,\alpha =0$

$$\begin{aligned}&u(\xi )=\frac{a_{1}\,\mathrm {e}^{k\,\xi }}{1-\mathrm {e}^{k\,\xi }}, \end{aligned}$$

(2.20)

$$\begin{aligned}&Q(x,t)=\left[ \frac{a_{1}\,\mathrm {e}^{i\,k^{2}\,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) }}{1-\mathrm {e}^{i\,k^{2}\,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) }} \right] ^{{1}/{2}}\nonumber \\&\qquad \qquad \quad \times \, \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) }. \end{aligned}$$

(2.21)

When $\alpha =0$

$$\begin{aligned}&u(\xi )=\frac{a_{1}\,\beta \,\mathrm {e}^{\beta \,\xi }}{1+\frac{\sigma }{2}\, \mathrm {e}^{\beta \,\xi } }, \end{aligned}$$

(2.22)

$$\begin{aligned}&Q(x,t)=\left[ \frac{a_{1}\,\beta \,\mathrm {e}^{i\,\beta \,k\,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) }}{1+\frac{\sigma }{2}\, \mathrm {e}^{i\,\beta \,k\,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) } } \right] ^{{1}/{2}}\,\nonumber \\&\quad ~~~~~~~~~~~~~\times \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) }. \end{aligned}$$

(2.23)

Case III

$$\begin{aligned}&\omega =\frac{-3}{2}\,\beta \,k,\quad \sigma =0,\quad \epsilon =\frac{5}{2}\,{\beta }^{2}{k}^{2},\nonumber \\&a_{{0}}=a_{{0}},\quad a_{{1}}=a_{{1}} \end{aligned}$$

so that, the exact travelling wave solution of nonlinear complex fractional Schrödinger equation (2.4) is in the form

$$\begin{aligned} u(\xi )=a_{0}+a_{1}\, a^{f(\xi )}. \end{aligned}$$

(2.24)

Therefore, the solitary wave solutions, when $\beta =k,\, \alpha =2\,k,\,\sigma =0$ are

$$\begin{aligned}&u(\xi )=a_{0}+a_{1}\big [\mathrm {e}^{k\,\xi }-1\big ], \end{aligned}$$

(2.25)

$$\begin{aligned}&Q(x,t)=\left[ a_{0}+a_{1}\left[ \mathrm {e}^{i\,k^{2}\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) }\right. \right. -1\Bigg ] \Bigg ]^{{1}/{2}}\,\nonumber \\&~~~~~~~~~~~~~~~\times \, \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) }. \end{aligned}$$

(2.26)

When $2\,\beta =\alpha +\sigma$

$$\begin{aligned}&u(\xi )=a_{0}+a_{1}\,\big [{1-\alpha \, \mathrm{e}^{\frac{1}{2}(\alpha -\sigma )\xi }}\big ], \end{aligned}$$

(2.27)

$$\begin{aligned}&Q(x,t)=\left[ a_{0}+a_{1}\,\left[ {1}-\alpha \, \mathrm{e}^{\frac{i\,k}{2}(\alpha -\sigma )\,\,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) }\right] \right] ^{{1}/{2}}\,\nonumber \\&~~~~~~~~~~~~~~\times \,\mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) } \end{aligned}$$

(2.28)

or

$$\begin{aligned}&u(\xi )=a_{0}-a_{1}\,\big [{\alpha \, \mathrm{e}^{\frac{1}{2}(\alpha -\sigma )\xi }+1}\big ], \end{aligned}$$

(2.29)

$$\begin{aligned}&Q(x,t)=\left[ a_{0}-a_{1}\,\left[ {\alpha \, \mathrm{e}^{\frac{i\,k\,}{2}(\alpha -\sigma )\,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) }+1}\right] \right] ^{{1}/{2}}\,\nonumber \\&~~~~~~~~~~~~~~\times \,\mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) }. \end{aligned}$$

(2.30)

When $\alpha =0$

$$\begin{aligned}&u(\xi )=a_{0}+\frac{a_{1}\,\beta \,\mathrm {e}^{\beta \,\xi }}{1+\frac{\sigma }{2}\, \mathrm {e}^{\beta \,\xi } }, \end{aligned}$$

(2.31)

$$\begin{aligned}&Q(x,t)=\left[ a_{0}+\frac{a_{1}\,\beta \,\mathrm {e}^{i\,\beta \,k\,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) }}{1+\frac{\sigma }{2}\, \mathrm {e}^{i\,\beta \,k\,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) } } \right] ^{{1}/{2}}\,\nonumber \\&~~~~~~~~~~~~~~\times \,\mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) }. \end{aligned}$$

(2.32)

Case IV

$$\begin{aligned} \alpha= & {} {\frac{a_{{0}} \left( \beta \,k-2\,a_{{0}} \right) }{k\,a_{{1}}}},\\ \omega= & {} \frac{1}{2}\,\beta \,k-2\,a_{{0}},\quad a_{0}\,\left( \frac{\beta \,k}{2}-\omega \right) ,\quad a_{1}=\frac{k\,\sigma }{2} \end{aligned}$$

so that, the exact travelling wave solution of nonlinear complex fractional Schrödinger equation (2.4) is in the form

$$\begin{aligned} u(\xi )=\frac{1}{2}\,\left[ \left( \frac{\beta \,k}{2}-\omega \right) +k\,\sigma \, a^{f(\xi )}\right] . \end{aligned}$$

(2.33)

Therefore, the solitary wave solutions, when $\beta ^{2}-\,\alpha \,\sigma <0 \,\mathrm {and} \, \sigma \ne 0$ are

$$\begin{aligned}&u(\xi )=\frac{1}{2}\,\left[ -\left( \omega +\frac{\beta \,k}{2}\right) +k{\sqrt{-\left( \beta ^{2}-\,\alpha \,\sigma \right) }}\,\right. \nonumber \\&~~~~~~~~~~~~\times \,\left. \tan \left( \frac{\sqrt{-\left( \beta ^{2}-\,\alpha \,\sigma \right) }}{2} \, \xi \right) \right] , \end{aligned}$$

(2.34)

$$\begin{aligned}&Q(x,t)=\left[ \frac{1}{2}\left[ -\left( \omega +\frac{\beta \,k}{2}\right) +k{\sqrt{\left( \beta ^{2}-\,\alpha \,\sigma \right) }}\right. \right. \nonumber \\&\quad \times \tan \left( \frac{k\,\sqrt{\left( \beta ^{2}-\,\alpha \,\sigma \right) }}{2}\left. \left. \times \,\,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) \right) \right] \right] ^{{1}/{2}}\nonumber \\&\quad \times \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) } \end{aligned}$$

(2.35)

or

$$\begin{aligned}&u(\xi )=\frac{1}{2}\,\left[ -\left( \omega +\frac{\beta \,k}{2}\right) +k\,{\sqrt{-\left( \beta ^{2}-\,\alpha \,\sigma \right) }}\,\right. \nonumber \\&\quad \qquad \left. \times \cot \left( \frac{\sqrt{-\left( \beta ^{2}-\,\alpha \,\sigma \right) }}{2} \, \xi \right) \right] , \end{aligned}$$

(2.36)

$$\begin{aligned}&Q(x,t)=\left[ \frac{1}{2}\,\left[ -\left( \omega +\frac{\beta \,k}{2}\right) +k\,{\sqrt{-\left( \beta ^{2}-\,\alpha \,\sigma \right) }}\,\right. \right. \nonumber \\&~~~\left. \left. \times \cot \left( \frac{k\,\sqrt{\left( \beta ^{2}-\,\alpha \,\sigma \right) }}{2} \,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) \right) \right] \right] ^{{1}/{2}}\,\nonumber \\&~~~~\times \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) }. \end{aligned}$$

(2.37)

When $\beta ^{2}-\,\alpha \,\sigma >0 \,\hbox {and} \, \sigma \ne 0$

$$\begin{aligned}&u(\xi )=\frac{-1}{2}\,\left[ \left( \omega +\frac{\beta \,k}{2}\right) +k\,{\sqrt{\left( \beta ^{2}-\,\alpha \,\sigma \right) }}\,\right. \nonumber \\&~~~~~~~~~\left. \times \tanh \left( \frac{\sqrt{\left( \beta ^{2}-\,\alpha \,\sigma \right) }}{2} \, \xi \right) \right] , \end{aligned}$$

(2.38)

$$\begin{aligned}&Q(x,t)=\left[ \frac{-1}{2}\,\left[ \left( \omega +\frac{\beta \,k}{2}\right) +k\,{\sqrt{\left( \beta ^{2}-\,\alpha \,\sigma \right) }}\,\right. \right. \nonumber \\&~~~~~~~~~~~~~~\times \tanh \left( \frac{k\,\sqrt{-\left( \beta ^{2}-\,\alpha \,\sigma \right) }}{2}\right. \nonumber \\&~~~~~~~~~~~~~\left. \left. \left. \times \left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \right] \right] ^{{1}/{2}}\mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) } \end{aligned}$$

(2.39)

or

$$\begin{aligned}&u(\xi )=\frac{-1}{2}\,\left[ \left( \omega +\frac{\beta \,k}{2}\right) +k\,{\sqrt{\left( \beta ^{2}-\,\alpha \,\sigma \right) }}\right. \nonumber \\&\quad ~~~~~~~~\left. \times \coth \left( \frac{\sqrt{\left( \beta ^{2}-\,\alpha \,\sigma \right) }}{2} \, \xi \right) \right] , \end{aligned}$$

(2.40)

$$\begin{aligned}&Q(x,t)=\left[ \frac{-1}{2}\,\left[ \left( \omega +\frac{\beta \,k}{2}\right) +k\,{\sqrt{\left( \beta ^{2}-\,\alpha \,\sigma \right) }}\,\right. \right. \nonumber \\&\qquad ~~~~~~~~\times \coth \left( \frac{k\,\sqrt{-\left( \beta ^{2}-\,\alpha \,\sigma \right) }}{2} \right. \nonumber \\&\qquad ~~~~~~\,\left. \left. \left. \times \left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) \right) \right] \right] ^{{1}/{2}}\mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) }. \end{aligned}$$

(2.41)

When $\beta ^{2}+\,\alpha ^{2} >0 , \sigma \ne 0 \,\mathrm {and} \, \sigma =-\,\alpha$

$$\begin{aligned}&u(\xi )=\frac{-1}{2}\,\left[ \left( \omega +\frac{\beta \,k}{2}\right) +k\,{\,\sqrt{\left( \beta ^{2}+\alpha ^{2}\right) }}\,\right. \nonumber \\&~~~~~~~~~~\left. \times \tanh \left( \frac{\sqrt{\left( \beta ^{2}+\alpha ^{2}\right) }}{2}\, \xi \right) \right] , \end{aligned}$$

(2.42)

$$\begin{aligned}&Q(x,t)=\left[ \frac{-1}{2}\,\left[ \left( \omega +\frac{\beta \,k}{2}\right) +k\,{\sqrt{\left( \beta ^{2}+\alpha ^{2}\right) }}\,\right. \right. \nonumber \\&~~~~\left. \left. \times \tanh \left( \frac{k\,\sqrt{-\left( \beta ^{2}+\alpha ^{2}\right) }}{2}\,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) \right) \right] \right] ^{{1}/{2}}\,\nonumber \\&~~~~\times \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) } \end{aligned}$$

(2.43)

or

$$\begin{aligned}&u(\xi )=\frac{-1}{2}\,\left[ \left( \omega +\frac{\beta \,k}{2}\right) +k\,{\sqrt{\left( \beta ^{2}+\alpha ^{2}\right) }}\,\right. \nonumber \\&\times \left. \coth \left( \frac{\sqrt{\left( \beta ^{2}+\alpha ^{2}\right) }}{2}\, \xi \right) \right] , \end{aligned}$$

(2.44)

$$\begin{aligned} Q(x,t)= & {} \left[ \frac{1}{2}\,\left[ \left( \frac{\beta \,k}{2}-\omega \right) -k\,\left[ {\beta }+{\sqrt{\beta ^{2}+\alpha ^{2}}}\,\right. \right. \right. \nonumber \\&\times \left. \left. \left. \coth \left( \frac{k\,\sqrt{-\left( \beta ^{2}+\alpha ^{2}\right) }}{2}\, \left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) \right) \right] \right] \right] ^{{1}/{2}}\,\nonumber \\&\times \,\mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) }. \end{aligned}$$

(2.45)

When $\beta ^{2}+\,\alpha ^{2} <0, \, \sigma \ne 0 \,\mathrm {and} \, \sigma =-\,\alpha$

$$\begin{aligned}&u(\xi )=\frac{-1}{2}\,\left[ \left( \omega +\frac{\beta \,k}{2}\right) +k\,\sqrt{- \left( \beta ^{2}+\alpha ^{2}\right) }\, \right. \nonumber \\&\quad \quad \quad \left. \times \tan \left( \frac{\sqrt{- \left( \beta ^{2}+\alpha ^{2}\right) }}{2}\, \xi \right) \right] , \end{aligned}$$

(2.46)

$$\begin{aligned}&Q(x,t)=\left[ \frac{-1}{2}\,\left[ \left( \omega +\frac{\beta \,k}{2}\right) +k\,{\sqrt{- \left( \beta ^{2}+\alpha ^{2}\right) }}\, \right. \right. \nonumber \\&\quad \quad \quad \quad \left. \left. \times \tan \left( \frac{k\,\sqrt{ \left( \beta ^{2}+\alpha ^{2}\right) }}{2}\,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) \right) \right] \right] ^{{1}/{2}}\,\nonumber \\&\quad \quad \quad \quad ~\!\times \,\mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) } \end{aligned}$$

(2.47)

or

$$\begin{aligned}&u(\xi )=\frac{-1}{2}\,\left[ \left( \omega +\frac{\beta \,k}{2}\right) +k\,{\sqrt{- \left( \beta ^{2}+\alpha ^{2}\right) }}\, \right. \nonumber \\&\quad \times \left. \cot \left( \frac{\sqrt{- \left( \beta ^{2}+\alpha ^{2}\right) }}{2}\, \xi \right) \right] , \end{aligned}$$

(2.48)

$$\begin{aligned} Q(x,t)= & {} \left[ \frac{-1}{2}\,\left[ \left( \omega +\frac{\beta \,k}{2}\right) +k\,{\sqrt{- \left( \beta ^{2}+\alpha ^{2}\right) }}\, \right. \right. \nonumber \\&\quad \quad \times \left. \left. \cot \left( \frac{k\,\sqrt{ \left( \beta ^{2}+\alpha ^{2}\right) }}{2}\, \left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) \right) \right] \right] ^{{1}/{2}}\,\nonumber \\&\quad \quad \times \,\mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) }. \end{aligned}$$

(2.49)

When $\beta ^{2}-\,\alpha ^{2}<0 \,\mathrm {and} \, \sigma =\alpha$

$$\begin{aligned}&u(\xi )=\frac{1}{2}\,\left[ -\left( \omega +\frac{\beta \,k}{2}\right) +k{\sqrt{-\left( \beta ^{2}-\,\alpha ^{2}\right) }}\,\right. \nonumber \\&\left. \times \tan \left( \frac{\sqrt{-\left( \beta ^{2}-\,\alpha ^{2}\right) }}{2}\, \xi \right) \right] , \end{aligned}$$

(2.50)

$$\begin{aligned}&Q(x,t)=\left[ \frac{1}{2}\,\left[ -\left( \omega +\frac{\beta \,k}{2}\right) +k{\sqrt{-\left( \beta ^{2}-\,\alpha ^{2}\right) }}\,\right. \right. \nonumber \\&\quad \quad \quad \quad \left. \left. \times \tan \left( \frac{k\,\sqrt{\left( \beta ^{2}-\,\alpha ^{2}\right) }}{2}\,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) \right) \right] \right] ^{{1}/{2}}\nonumber \\&\quad \quad \quad \quad \ \times \,\mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) } \end{aligned}$$

(2.51)

or

$$\begin{aligned}&u(\xi )=\frac{1}{2}\,\left[ -\left( \omega +\frac{\beta \,k}{2}\right) +k{\sqrt{-\left( \beta ^{2}-\,\alpha ^{2}\right) }}\,\right. \nonumber \\&\times \left. \cot \left( \frac{\sqrt{-\left( \beta ^{2}-\,\alpha ^{2}\right) }}{2}\, \xi \right) \right] , \end{aligned}$$

(2.52)

$$\begin{aligned}&Q(x,t)=\left[ \frac{1}{2}\,\left[ -\left( \omega +\frac{\beta \,k}{2}\right) +k{\sqrt{-\left( \beta ^{2}-\,\alpha ^{2}\right) }}\,\right. \right. \nonumber \\&\quad \quad \quad \quad \left. \left. \times \cot \left( \frac{k\,\sqrt{\left( \beta ^{2}-\,\alpha ^{2}\right) }}{2}\, \left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) \right) \right] \right] ^{{1}/{2}}\nonumber \\&\quad \quad \quad \ \quad \times \,\mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) }. \end{aligned}$$

(2.53)

When $\beta ^{2}-\,\alpha ^{2}>0 ~\hbox {and} ~ \sigma =\alpha$

$$\begin{aligned}&u(\xi )=\frac{1}{2}\,\left[ -\left( \omega +\frac{\beta \,k}{2}\right) +k{\sqrt{\beta ^{2}-\,\alpha ^{2}}}\,\right. \nonumber \\&\quad \quad \quad \times \left. \tanh \left( \frac{\sqrt{\beta ^{2}-\alpha ^{2}}}{2}\, \xi \right) \right] , \end{aligned}$$

(2.54)

$$\begin{aligned}&Q(x,t)=\left[ \frac{1}{2}\,\left[ -\left( \omega +\frac{\beta \,k}{2}\right) +k{\sqrt{\beta ^{2}-\,\alpha ^{2}}}\,\right. \right. \nonumber \\&\quad \quad \left. \left. \times \tanh \left( \frac{k\sqrt{-\left( \beta ^{2}-\alpha ^{2}\right) }}{2}\,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) \right) \right] \right] ^{{1}/{2}}\,\nonumber \\&\quad \quad \times \,\mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) } \end{aligned}$$

(2.55)

or

$$\begin{aligned}&u(\xi )=\frac{1}{2}\,\left[ -\left( \omega +\frac{\beta \,k}{2}\right) +k{\sqrt{\beta ^{2}-\,\alpha ^{2}}}\,\right. \nonumber \\&\quad \quad \quad \times \left. \coth \left( \frac{\sqrt{\beta ^{2}-\alpha ^{2}}}{2}\, \xi \right) \right] , \end{aligned}$$

(2.56)

$$\begin{aligned}&Q(x,t)=\left[ \frac{1}{2}\,\left[ -\left( \omega +\frac{\beta \,k}{2}\right) +k{\sqrt{\beta ^{2}-\,\alpha ^{2}}}\,\right. \right. \nonumber \\&\qquad \left. \left. \times \coth \left( \frac{k\,\sqrt{-\left( \beta ^{2}-\alpha ^{2}\right) }}{2}\, \left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) \right) \right] \right] ^{{1}/{2}}\, \nonumber \\&\qquad \ \times \,\mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) }. \end{aligned}$$

(2.57)

When $\alpha \,\sigma <0, \, \sigma \ne 0 \,\mathrm {and} \, \beta = 0$

$$\begin{aligned}&u(\xi )=\frac{1}{2}\,\left[ -\omega +k\,\sqrt{{-\alpha \,\sigma }}\, \tanh \left( \frac{\sqrt{-\alpha \,\sigma }}{2} \, \xi \right) \right] ,\nonumber \\ \end{aligned}$$

(2.58)

$$\begin{aligned}&Q(x,t)=\left[ \frac{1}{2}\,\left[ -\omega +\,k\,\sqrt{{-\alpha \,\sigma }}\, \right. \right. \nonumber \\&\quad \quad \quad \quad \quad \times \left. \left. \tanh \left( \frac{k\,\sqrt{\alpha \,\sigma }}{2} \,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) \right) \right] \right] ^{{1}/{2}}\,\nonumber \\&\quad \quad \quad \quad \quad \times \,\mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) } \end{aligned}$$

(2.59)

or

$$\begin{aligned}&u(\xi )=\frac{1}{2}\,\left[ -\omega +\,k\,\sqrt{{-\alpha \,\sigma }}\, \coth \left( \frac{\sqrt{-\alpha \,\sigma }}{2} \, \xi \right) \right] ,\nonumber \\ \end{aligned}$$

(2.60)

$$\begin{aligned}&Q(x,t)=\left[ \frac{1}{2}\,\left[ -\omega +k\,\sqrt{{-\alpha \,\sigma }}\,\right. \right. \nonumber \\&\quad \quad \quad \quad \left. \left. \times \coth \left( \frac{k\,\sqrt{\alpha \,\sigma }}{2} \,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) \right) \right] \right] ^{{1}/{2}}\,\nonumber \\&\quad \quad \quad \quad \ \times \,\mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) }. \end{aligned}$$

(2.61)

When $\beta =0\ \mathrm {and} \ \alpha =-\sigma$

$$\begin{aligned}&u(\xi )=\frac{1}{2}\,\left[ -\omega +k\,\sigma \left[ \frac{-\left( 1+\mathrm {e}^{2\,\alpha \, \xi }\right) \pm \sqrt{2\left( \mathrm {e}^{4\,\alpha \, \xi }+1\right) }}{\mathrm {e}^{2\,\alpha \,\xi }-1}\right] \right] , \end{aligned}$$

(2.62)

$$\begin{aligned}&Q(x,t)=\left[ \frac{1}{2}\left[ -\omega +k\,\sigma \left[ \frac{-\left( 1+\mathrm {e}^{2\,i\,\alpha \, k\,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) }\right) \pm \sqrt{2\left( \mathrm {e}^{4\,i\,\alpha \, k\,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) }+1\right) }}{\mathrm {e}^{2\,i\,\alpha \,k\,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) }-1}\right] \right] \right] ^{{1}/{2}}\,\mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) }\nonumber \\ \end{aligned}$$

(2.63)

or

$$\begin{aligned}&u(\xi )=\frac{1}{2}\,\left[ -\omega +k\,\sigma \left[ \frac{-\left( 1+\mathrm {e}^{2\,\alpha \, \xi }\right) \pm \sqrt{\mathrm {e}^{4\,\alpha \, \xi }+6\,\mathrm {e}^{2\,\alpha \, \xi }+1}}{2\,\mathrm {e}^{2\,\alpha \,\xi }}\right] \right] , \end{aligned}$$

(2.64)

$$\begin{aligned}&Q(x,t)=\left[ \frac{1}{2}\,\left[ -\omega +k\,\sigma \left[ \frac{-\left( 1+\mathrm {e}^{2\,i\,\alpha \,k\,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) }\right) \pm \sqrt{\mathrm {e}^{4\,i\,\alpha \,k\,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) }+6\,\mathrm {e}^{2\,i\,\alpha \, k\,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) }+1}}{2\,\mathrm {e}^{2\,i\,\alpha \,k\,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) }}\right] \right] \right] ^{{1}/{2}}\nonumber \\&\quad \quad \quad \quad \quad \times \mathrm{e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) }. \end{aligned}$$

(2.65)

When $\beta ^{2}=\alpha \,\sigma$

$$\begin{aligned}&u(\xi )=\frac{1}{2}\,\left[ \left( \frac{\beta \,k}{2}-\omega \right) -\frac{k\,\left( \beta \,\xi +2\right) }{\xi }\right] , \end{aligned}$$

(2.66)

$$\begin{aligned}&Q(x,t)=\left[ \frac{1}{2}\,\left[ \left( \frac{\beta \,k}{2}-\omega \right) \right. \right. \nonumber \\&\quad \quad \quad \quad \quad \quad \left. \left. +\frac{i\,\left( i\,\beta \,k\,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) +2\right) }{\,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) }\right] \right] ^{{1}/{2}}\,\nonumber \\&\quad \quad \quad \quad \quad \quad \times \,\mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) }. \end{aligned}$$

(2.67)

When $\beta =k,\, \sigma =2\,k\, \text {and } \alpha =0$

$$\begin{aligned}&u(\xi )=\frac{1}{2}\,\left[ \left( \frac{\beta \,k}{2}-\omega \right) +\frac{k\,\sigma \,\mathrm {e}^{k\,\xi }}{1-\mathrm {e}^{k\,\xi }}\right] , \end{aligned}$$

(2.68)

$$\begin{aligned}&Q(x,t)=\left[ \frac{1}{2}\,\left[ \left( \frac{\beta \,k}{2}-\omega \right) \right. \right. \nonumber \\&\quad \quad \quad \quad \quad \quad \left. \left. +\frac{k\,\sigma \,\mathrm {e}^{i\,k^{2}\,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) }}{1-\mathrm {e}^{i\,k^{2}\,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) }}\right] \right] ^{{1}/{2}}\,\nonumber \\&\quad \quad \quad \quad \quad \quad \times \,\mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) }. \end{aligned}$$

(2.69)

When $2\,\beta =\alpha +\sigma$

$$\begin{aligned} u(\xi )= & {} \frac{1}{2}\,\left[ \left( \frac{\beta \,k}{2}-\omega \right) +k\,\sigma \left[ \frac{1-\alpha \, \mathrm {e}^{\frac{i\,k}{2}(\alpha -\sigma )\xi }}{1-\sigma \, \mathrm {e}^{\frac{1}{2}(\alpha -\sigma )\xi }}\right] \right] ,\nonumber \\ \end{aligned}$$

(2.70)

$$\begin{aligned} Q(x,t)= & {} \left[ \frac{1}{2}\,\left[ \left( \frac{\beta \,k}{2}-\omega \right) \right. \right. \nonumber \\&\left. \left. +k\,\sigma \left[ \frac{1-\alpha \, \mathrm {e}^{\frac{i\,k}{2}(\alpha -\sigma )\,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) }}{1-\sigma \, \mathrm {e}^{\frac{i\,k}{2}(\alpha -\sigma )\,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) }}\right] \right] \right] ^{{1}/{2}}\,\nonumber \\&\times \,\mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) } \end{aligned}$$

(2.71)

or

$$\begin{aligned} u(\xi )= & {} \frac{1}{2}\,\left[ \left( \frac{\beta \,k}{2}-\omega \right) +k\,\sigma \left[ \frac{\alpha \, \mathrm {e}^{\frac{1}{2}(\alpha -\sigma )\xi }+1}{-\sigma \, \mathrm {e}^{\frac{1}{2}(\alpha -\sigma )\xi }-1}\right] \right] ,\nonumber \\ \end{aligned}$$

(2.72)

$$\begin{aligned} Q(x,t)= & {} \left[ \frac{1}{2}\,\left[ \left( \frac{\beta \,k}{2}-\omega \right) \right. \right. \nonumber \\&\left. \left. +k\,\sigma \left[ \frac{\alpha \, \mathrm {e}^{\frac{i\,k}{2}(\alpha -\sigma )\,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) }+1}{-\sigma \, \mathrm {e}^{\frac{i\,k}{2}(\alpha -\sigma )\,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) }-1}\right] \right] \right] ^{{1}/{2}}\,\nonumber \\&\times \,\mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) }. \end{aligned}$$

(2.73)

When $-2\,\beta =\alpha +\sigma$

$$\begin{aligned}&u(\xi )=\frac{1}{2}\,\left[ \left( \frac{\beta \,k}{2}-\omega \right) +k\,\sigma \, \left[ \frac{\mathrm {e}^{\frac{1}{2}\left( \alpha -\sigma \right) \,\xi }+\alpha }{\mathrm {e}^{\frac{1}{2}\left( \alpha -\sigma \right) \,\xi }+\sigma }\right] \right] ,\nonumber \\\end{aligned}$$

(2.74)

$$\begin{aligned} Q(x,t)= & {} \left[ \frac{1}{2}\,\left[ \left( \frac{\beta \,k}{2}-\omega \right) \right. \right. \nonumber \\&\left. \left. +\,k\,\sigma \left[ \frac{\mathrm {e}^{\frac{i\,k}{2}\left( \alpha -\sigma \right) \,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) }+\alpha }{\mathrm {e}^{\frac{i\,k}{2}\left( \alpha -\sigma \right) \,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) }+\sigma }\right] \right] \right] ^{{1}/{2}}\, \nonumber \\&\times \,\mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) }. \end{aligned}$$

(2.75)

When $\alpha =0$

$$\begin{aligned}&u(\xi )=\frac{1}{2}\,\left[ \left( \frac{\beta \,k}{2}-\omega \right) +k\,\sigma \left[ \frac{\beta \,\mathrm{e}^{\beta \,\xi }}{1+\frac{\sigma }{2}\, \mathrm{e}^{\beta \,\xi } }\right] \right] ,\nonumber \\ \end{aligned}$$

(2.76)

$$\begin{aligned} Q(x,t)= & {} \left[ \frac{1}{2}\,\left[ \left( \frac{\beta \,k}{2}-\omega \right) \right. \right. \nonumber \\&\left. \left. +\,k\,\sigma \left[ \frac{\beta \,\mathrm {e}^{i\,\beta \,k\,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) }}{1+\frac{\sigma }{2}\, \mathrm {e}^{i\,\beta \,k\,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) } }\right] \right] \right] ^{{1}/{2}}\,\nonumber \\&\times \,\mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) }. \end{aligned}$$

(2.77)

When $\beta =\alpha =\sigma \ne 0$

$$\begin{aligned}&u(\xi )=\frac{1}{2}\,\left[ \left( \frac{\beta \,k}{2}-\omega \right) -\frac{k\,\left( \alpha \,\xi +2\right) }{\xi }\right] , \end{aligned}$$

(2.78)

$$\begin{aligned} Q(x,t)= & {} \left[ \frac{1}{2}\,\left[ \left( \frac{\beta \,k}{2}-\omega \right) \right. \right. \nonumber \\&\left. \left. +\,\frac{i\,k\,\sigma \left( i\,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) +2\right) }{\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) }\right] \right] ^{{1}/{2}}\,\nonumber \\&\times \,\mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) }. \end{aligned}$$

(2.79)

When $\beta =\alpha =0$

$$\begin{aligned}&u(\xi )=\frac{-1}{2}\,\left[ \omega +\frac{2\,k\,}{ \xi }\right] ,\end{aligned}$$

(2.80)

$$\begin{aligned} Q(x,t)= & {} \left[ \frac{-1}{2}\,\left[ \omega -\frac{2\,i\,}{\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) }\right] \right] ^{{1}/{2}}\,\nonumber \\&\times \,\mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) }. \end{aligned}$$

(2.81)

When $\beta =0 \hbox { and } \alpha =\sigma$

$$\begin{aligned}&u(\xi )=\frac{1}{2}\,\left[ -\omega +k\,\sigma \left[ \tan \left( \frac{\alpha \,\xi +C}{2}\right) \right] \right] , \end{aligned}$$

(2.82)

$$\begin{aligned} Q(x,t)= & {} \left[ \frac{1}{2}\,\left[ -\omega +k\,\sigma \left[ \right. \right. \right. \nonumber \\&\left. \left. \left. \times \tan \left( \frac{i\,\alpha \,k\,\left( x+\frac{2\,\omega \, t^{\alpha }}{\alpha }\right) +C}{2}\right) \right] \right] \right] ^{{1}/{2}}\, \nonumber \\&\ \times \,\mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) }, \end{aligned}$$

(2.83)

where C is an arbitrary constant.

2.2 Exact and solitary wave solutions of nonlinear complex fractional Schrödinger equation by using novel $\left( {G'}/{G}\right)$-expansion method

Using the default for precision solution using novel $\left( {G'}/{G}\right)$-expansion method on nonlinear complex fractional Schrödinger equation, we obtain

$$\begin{aligned} u(\xi )=\frac{a_{-1}}{\left( d+\frac{G'}{G}\right) }+a_{0}+a_{1}\left( d+\frac{G'}{G}\right) . \end{aligned}$$

(2.84)

Substituting eq. (2.84) and its derivatives into eq. (2.4) and equating the coefficient of different power of $\left( d+\frac{G'}{G}\right) ^{i}$ to zero, we obtain a system of algebraic equations by solving it with any computer program like Maple, Mathematica, Matlab and so on. We get

Case I

$$\begin{aligned} \lambda= & {} {\frac{\mu \, \left( 2\,d\,a_{{0}}+a_{{-1}} \right) }{d \left( d\,a_{{0}}+a_{{-1}} \right) }},\quad \omega ={\frac{3\,\mu \,a_{{-1}}k}{2\,d \left( d\,a_{{0}}+a_{{-1}}\right) }},\\ v= & {} {\frac{{d}^{2}a_{{0}}+d\,a_{{-1}}+\mu \,a_{{0}}}{d \left( d\,a_{{0}}+a_{{-1}} \right) }},\\ \epsilon= & {} {\frac{5\,{k}^{2}{\mu }^{2}{a_{{-1}}}^{2}}{2\,{d}^{2}\left( ,a_{{0}}+a_{{-1}} \right) ^{2}}},\quad a_{{-1}}=a_{{-1}},\\&a_{{0}}=a_{{0}},\quad a_{{1}}=0. \end{aligned}$$

From the coefficients in Case I, we obtained the exact travelling wave solutions of nonlinear complex fractional Schrödinger equation in the following form:

$$\begin{aligned} u(\xi )=a_{0}+\frac{a_{-1}}{\left( d+\frac{G'}{G}\right) }. \end{aligned}$$

(2.85)

Therefore, the solitary travelling wave solutions will be in the following forms: When $\Omega =\lambda ^{2}-4\, \lambda \, \mu +4\, \mu >0$ and $\lambda (v-1)\ne 0$ or $\mu (v-1)\ne 0$

$$\begin{aligned} u(\xi )= & {} a_{0}+\frac{a_{-1}}{ \left( d-\frac{1}{2(v-1)}\left( \lambda +\sqrt{\Omega } \, \tanh \left( \frac{\sqrt{\Omega }}{2}\,\xi \right) \right) \right) },\nonumber \\ \end{aligned}$$

(2.86)

$$\begin{aligned} Q(x,t)= & {} \left[ a_{0}+\frac{a_{-1}}{ \left( d-\frac{1}{2(v-1)}\left( \lambda +\sqrt{\Omega } \, \tanh \left( \frac{i\,k\,\sqrt{\Omega }}{2}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \right) \right) } \right] ^{{1}/{2}}\, \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) },\end{aligned}$$

(2.87)

$$\begin{aligned}&u(\xi )=a_{0}+\frac{a_{-1}}{ \left( d-\frac{1}{2(v-1)}\left( \lambda +\sqrt{\Omega } \, \coth \left( \frac{\sqrt{\Omega }}{2}\,\xi \right) \right) \right) },\end{aligned}$$

(2.88)

$$\begin{aligned} Q(x,t)= & {} \left[ a_{0}+\frac{a_{-1}}{ \left( d-\frac{1}{2(v-1)}\left( \lambda +\sqrt{\Omega } \, \coth \left( \frac{i\,k\,\sqrt{\Omega }}{2}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \right) \right) } \right] ^{{1}/{2}}\, \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) },\end{aligned}$$

(2.89)

$$\begin{aligned}&u(\xi )=a_{0}+\frac{a_{-1}}{ \left( d-\frac{1}{2(v-1)}(\lambda +\sqrt{\Omega } \,( \tanh (\sqrt{\Omega }\,\xi )\pm i\, \mathrm {sech}(\sqrt{\Omega }\,\xi )))\right) }, \end{aligned}$$

(2.90)

$$\begin{aligned} Q(x,t)= & {} \left[ a_{0}+\frac{a_{-1}}{ \left( d-\frac{1}{2(v-1)}\left( \lambda +\sqrt{\Omega }\left( \tanh \left( i\,k\,\sqrt{\Omega }\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \pm i\, \mathrm {sech}\left( i\,k\,\sqrt{\Omega }\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \right) \right) \right) } \right] ^{{1}/{2}}\, \nonumber \\&\quad \times \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) },\end{aligned}$$

(2.91)

$$\begin{aligned}&u(\xi )=a_{0}+\frac{a_{-1}}{ \left( d-\frac{1}{2(v-1)}\left( \lambda +\sqrt{\Omega } \,\left( \coth \left( \sqrt{\Omega }\,\xi \right) \pm \, \mathrm {csch}\left( \sqrt{\Omega }\,\xi \right) \right) \right) \right) }, \end{aligned}$$

(2.92)

$$\begin{aligned} Q(x,t)= & {} \left[ a_{0}+\frac{a_{-1}}{ \left( d-\frac{1}{2(v-1)}\left( \lambda +\sqrt{\Omega } \,\left( \coth \left( i\,k\,\sqrt{\Omega }\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \pm \, \mathrm {csch}\left( i\,k\,\sqrt{\Omega }\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \right) \right) \right) } \right] ^{{1}/{2}}\,\nonumber \\&\quad \times \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) }, \end{aligned}$$

(2.93)

$$\begin{aligned}&u(\xi )=a_{0}+\frac{a_{-1}}{\left( d-\frac{1}{4(v-1)}\left( 2\,\lambda +\sqrt{\Omega }\left( \tanh \left( \frac{\sqrt{\Omega }}{4}\,\xi \right) \pm \, \coth \left( \frac{\sqrt{\Omega }}{4}\,\xi \right) \right) \right) \right) }, \end{aligned}$$

(2.94)

$$\begin{aligned} Q(x,t)= & {} \left[ a_{0}+\frac{a_{-1}}{\left( d-\frac{1}{4(v-1)}\left( 2\,\lambda +\sqrt{\Omega }\left( \tanh \left( \frac{i\,k\,\sqrt{\Omega }}{4}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \pm \, \coth \left( \frac{i\,k\,\sqrt{\Omega }}{4}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \right) \right) \right) } \right] ^{{1}/{2}}\,\nonumber \\&\quad \times \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) },\end{aligned}$$

(2.95)

$$\begin{aligned}&u(\xi )=a_{0}+\frac{a_{-1}}{ \left( d+\frac{1}{2(v-1)}\left( -\lambda +\frac{\pm \sqrt{\Omega \,\left( A^{2}+B^{2}\right) }-A\,\sqrt{\Omega }\, \cosh \left( \sqrt{\Omega }\,\xi \right) }{A\, \sinh \left( \sqrt{\Omega }\, \xi \right) +B}\right) \right) },\end{aligned}$$

(2.96)

$$\begin{aligned} Q(x,t)= & {} \left[ a_{0}+\frac{a_{-1}}{ \left( d+\frac{1}{2(v-1)}\left( -\lambda +\frac{\pm \sqrt{\Omega \,\left( A^{2}+B^{2}\right) }-A\,\sqrt{\Omega }\, \cosh \left( i\,k\,\sqrt{\Omega }\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) }{A\, \sinh \left( i\,k\,\sqrt{\Omega }\, \left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) +B}\right) \right) } \right] ^{{1}/{2}}\, \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) }, \end{aligned}$$

(2.97)

$$\begin{aligned}&u(\xi )=a_{0}+\frac{a_{-1}}{ \left( d+\frac{1}{2(v-1)}\left( -\lambda +\frac{\pm \sqrt{\Omega \,\left( A^{2}+B^{2}\right) }+A\,\sqrt{\Omega }\, \cosh \left( \sqrt{\Omega }\,\xi \right) }{A\, \sinh \left( \sqrt{\Omega }\, \xi \right) +B}\right) \right) }, \end{aligned}$$

(2.98)

$$\begin{aligned} Q(x,t)= & {} \left[ a_{0}+\frac{a_{-1}}{ \left( d+\frac{1}{2(v-1)}\left( -\lambda +\frac{\pm \sqrt{\Omega \,\left( A^{2}+B^{2}\right) }+A\,\sqrt{\Omega }\, \cosh \left( i\,k\,\sqrt{\Omega }\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) }{A\, \sinh \left( i\,k\,\sqrt{\Omega }\, \left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) +B}\right) \right) } \right] ^{{1}/{2}}\, \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) },\end{aligned}$$

(2.99)

$$\begin{aligned}&u(\xi )=a_{0}+\frac{a_{-1}}{\left( d+\frac{2\,\mu \, \cosh \left( \frac{\sqrt{\Omega }}{2}\,\xi \right) }{\sqrt{\Omega }\, \sinh \left( \frac{\sqrt{\Omega }}{2}\,\xi \right) -\lambda \,\cosh \left( \frac{\sqrt{\Omega }}{2}\,\xi \right) }\right) },\end{aligned}$$

(2.100)

$$\begin{aligned} Q(x,t)= & {} \left[ a_{0}+\frac{a_{-1}}{\left( d+\frac{2\,\mu \, \cosh \left( \frac{\sqrt{\Omega }}{2}\,\xi \right) }{\sqrt{\Omega }\, \sinh \left( \frac{i\,k\,\sqrt{\Omega }}{2}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) -\lambda \,\cosh \left( \frac{i\,k\,\sqrt{\Omega }}{2}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) }\right) } \right] ^{{1}/{2}}\, \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) }, \end{aligned}$$

(2.101)

$$\begin{aligned}&u(\xi )=a_{0}+\frac{a_{-1}}{ \left( d+\frac{2\,\mu \, \sinh \left( \frac{\sqrt{\Omega }}{2}\,i\,k\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) }{\sqrt{\Omega }\, \cosh \left( \frac{\sqrt{\Omega }}{2}\,i\,k\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) -\lambda \,\sinh \left( \frac{\sqrt{\Omega }}{2}\,i\,k\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) }\right) }, \end{aligned}$$

(2.102)

$$\begin{aligned} Q(x,t)= & {} \left[ a_{0}+\frac{a_{-1}}{ \left( d+\frac{2\,\mu \, \sinh \left( \frac{i\,k\,\sqrt{\Omega }}{2}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) }{\sqrt{\Omega }\, \cosh \left( \frac{i\,k\,\sqrt{\Omega }}{2}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) -\lambda \,\sinh \left( \frac{i\,k\,\sqrt{\Omega }}{2}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) }\right) } \right] ^{{1}/{2}}\, \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) },\end{aligned}$$

(2.103)

$$\begin{aligned}&u(\xi )=a_{0}+\frac{a_{-1}}{\left( d+\frac{2\,\mu \, \cosh \left( {\sqrt{\Omega }}\,\xi \right) }{\sqrt{\Omega }\, \sinh \left( {\sqrt{\Omega }}\,\xi \right) -\lambda \,\cosh \left( {\sqrt{\Omega }}\,\xi \right) \pm i\,\sqrt{\Omega }}\right) },\end{aligned}$$

(2.104)

$$\begin{aligned} Q(x,t)= & {} \left[ a_{0}+\frac{a_{-1}}{\left( d+\frac{2\,\mu \, \cosh \left( {i\,k\,\sqrt{\Omega }}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) }{\sqrt{\Omega }\, \sinh \left( {i\,k\,\sqrt{\Omega }}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) -\lambda \,\cosh \left( {i\,k\,\sqrt{\Omega }}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \pm i\,\sqrt{\Omega }}\right) } \right] ^{{1}/{2}}\, \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) },\end{aligned}$$

(2.105)

$$\begin{aligned}&u(\xi )=a_{0}+\frac{a_{-1}}{\left( d+\frac{2\,\mu \, \sinh \left( {\sqrt{\Omega }}\,\xi \right) }{\sqrt{\Omega }\, \cosh \left( {\sqrt{\Omega }}\,\xi \right) -\lambda \sinh \left( {\sqrt{\Omega }}\,\xi \right) \pm i\,\sqrt{\Omega }}\right) }, \end{aligned}$$

(2.106)

$$\begin{aligned}&Q(x,t)=\left[ a_{0}+\frac{a_{-1}}{\left( d+\frac{2\,\mu \, \sinh \left( {i\,k\,\sqrt{\Omega }}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) }{\sqrt{\Omega }\, \cosh \left( {i\,k\,\sqrt{\Omega }}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) -\lambda \,\sinh \left( {i\,k\,\sqrt{\Omega }}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \pm i\,\sqrt{\Omega }}\right) } \right] ^{{1}/{2}}\, \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) }, \end{aligned}$$

(2.107)

where A, B are arbitrary real constants and $A^{2}+B^{2}>0$.

When $\Omega =\lambda ^{2}-4\, \lambda \, \mu +4\, \mu <0$ and $\lambda (v-1)\ne 0$ or $\mu (v-1)\ne 0$

$$\begin{aligned}&u(\xi )=a_{0}+\frac{a_{-1}}{ \left( d+\frac{1}{2(v-1)}\left( -\lambda +\sqrt{-\Omega } \, \tanh \left( \frac{\sqrt{-\Omega }}{2}\,\xi \right) \right) \right) }, \end{aligned}$$

(2.108)

$$\begin{aligned} Q(x,t)= & {} \left[ a_{0}+\frac{a_{-1}}{ \left( d+\frac{1}{2(v-1)}\left( -\lambda +\sqrt{-\Omega } \, \tanh \left( \frac{i\,k\,\sqrt{-\Omega }}{2}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \right) \right) } \right] ^{{1}/{2}}\, \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) },\end{aligned}$$

(2.109)

$$\begin{aligned}&u(\xi )=a_{0}+\frac{a_{-1}}{ \left( d-\frac{1}{2(v-1)}\left( \lambda +\sqrt{-\Omega } \, \coth \left( \frac{\sqrt{-\Omega }}{2}\,\xi \right) \right) \right) },\end{aligned}$$

(2.110)

$$\begin{aligned} Q(x,t)= & {} \left[ a_{0}+\frac{a_{-1}}{ \left( d-\frac{1}{2(v-1)}\left( \lambda +\sqrt{-\Omega } \, \coth \left( \frac{i\,k\,\sqrt{-\Omega }}{2}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \right) \right) } \right] ^{{1}/{2}}\, \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) },\end{aligned}$$

(2.111)

$$\begin{aligned}&u(\xi )=a_{0}+\frac{a_{-1}}{ \left( d+\frac{1}{2(v-1)}\left( -\lambda +\sqrt{-\Omega } \,\left( \tan \left( \sqrt{-\Omega }\,\xi \right) \pm \, \sec \left( \sqrt{-\Omega }\,\xi \right) \right) \right) \right) }, \end{aligned}$$

(2.112)

$$\begin{aligned} Q(x,t)= & {} \left[ a_{0}+\frac{a_{-1}}{ \left( d+\frac{1}{2(v-1)}\left( -\lambda +\sqrt{-\Omega } \,\left( \tan \left( i\,k\,\sqrt{-\Omega }\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \pm \, \sec \left( \sqrt{-\Omega }\,\xi \right) \right) \right) \right) } \right] ^{{1}/{2}}\,\nonumber \\&\quad \times \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) },\end{aligned}$$

(2.113)

$$\begin{aligned}&u(\xi )=a_{0}+\frac{a_{-1}}{ \left( d-\frac{1}{2(v-1)}\left( \lambda +\sqrt{-\Omega } \,\left( \cot \left( \sqrt{-\Omega }\,\xi \right) \pm \, \mathrm {csc}\left( \sqrt{-\Omega }\,\xi \right) \right) \right) \right) },\end{aligned}$$

(2.114)

$$\begin{aligned} Q(x,t)= & {} \left[ a_{0}+\frac{a_{-1}}{ \left( d-\frac{1}{2(v-1)}\left( \lambda +\sqrt{-\Omega } \,\left( \cot \left( i\,k\,\sqrt{-\Omega }\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \pm \, \mathrm {csc}\left( i\,k\,\sqrt{-\Omega }\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \right) \right) \right) } \right] ^{{1}/{2}}\, \nonumber \\&\quad \times \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) },\end{aligned}$$

(2.115)

$$\begin{aligned}&u(\xi )=a_{0}+\frac{a_{-1}}{\left( d+\frac{1}{4(v-1)}\left( -2\,\lambda +\sqrt{-\Omega } \,\left( \tan \left( \frac{\sqrt{-\Omega }}{4}\,\xi \right) -\, \cot \left( \frac{\sqrt{-\Omega }}{4}\,\xi \right) \right) \right) \right) },\end{aligned}$$

(2.116)

$$\begin{aligned} Q(x,t)= & {} \left[ \! a_{0}+\frac{a_{-1}}{\left( d+\frac{1}{4(v-1)}\left( -2\,\lambda +\sqrt{-\Omega } \,\left( \tan \left( \frac{i\,k\,\sqrt{-\Omega }}{4}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) -\, \cot \left( \frac{i\,k\,\sqrt{-\Omega }}{4}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \right) \right) \right) } \!\right] ^{{1}/{2}}\,\nonumber \\&\quad \times \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) },\end{aligned}$$

(2.117)

$$\begin{aligned}&u(\xi )=a_{0}+\frac{a_{-1}}{\left( d+\frac{1}{2(v-1)}\left( -\lambda +\frac{\pm \sqrt{-\Omega \,\left( A^{2}-B^{2}\right) }-A\,\sqrt{-\Omega }\, \cos \left( \sqrt{-\Omega }\,\xi \right) }{A\, \sin \left( \sqrt{-\Omega }\, \xi \right) +B}\right) \right) },\end{aligned}$$

(2.118)

$$\begin{aligned} Q(x,t)= & {} \left[ a_{0}+\frac{a_{-1}}{\left( d+\frac{1}{2(v-1)}\left( -\lambda +\frac{\pm \sqrt{-\Omega \,\left( A^{2}-B^{2}\right) }-A\,\sqrt{-\Omega }\, \cos \left( i\,k\,\sqrt{-\Omega }\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) }{A\, \sin \left( i\,k\,\sqrt{-\Omega }\, \left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) +B}\right) \right) } \right] ^{{1}/{2}}\, \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) },\nonumber \\\end{aligned}$$

(2.119)

$$\begin{aligned}&u(\xi )=a_{0}+\frac{a_{-1}}{\left( d+\frac{1}{2(v-1)}\left( -\lambda +\frac{\pm \sqrt{-\Omega \,\left( A^{2}-B^{2}\right) }+A\,\sqrt{-\Omega }\, \cos \left( \sqrt{-\Omega }\,\xi \right) }{A\, \sin \left( \sqrt{-\Omega }\, \xi \right) +B}\right) \right) },\end{aligned}$$

(2.120)

$$\begin{aligned} Q(x,t)= & {} \left[ a_{0}+\frac{a_{-1}}{\left( d+\frac{1}{2(v-1)}\left( -\lambda +\frac{\pm \sqrt{-\Omega \,\left( A^{2}-B^{2}\right) }+A\,\sqrt{-\Omega }\, \cos \left( i\,k\,\sqrt{-\Omega }\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) }{A\, \sin \left( i\,k\,\sqrt{-\Omega }\, \left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) +B}\right) \right) } \right] ^{{1}/{2}}\, \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) },\nonumber \\\end{aligned}$$

(2.121)

$$\begin{aligned}&u(\xi )=a_{0}+\frac{a_{-1}}{\left( d-\frac{2\,\mu \, \cos \left( \frac{\sqrt{-\Omega }}{2}\,\xi \right) }{\sqrt{-\Omega }\, \sin \left( \frac{\sqrt{-\Omega }}{2}\,\xi \right) +\lambda \,\cos \left( \frac{\sqrt{-\Omega }}{2}\,\xi \right) }\right) },\end{aligned}$$

(2.122)

$$\begin{aligned} Q(x,t)= & {} \left[ a_{0}+\frac{a_{-1}}{\left( d-\frac{2\,\mu \, \cos \left( \frac{i\,k\,\sqrt{-\Omega }}{2}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) }{\sqrt{-\Omega }\, \sin \left( \frac{i\,k\,\sqrt{-\Omega }}{2}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) +\lambda \,\cos \left( \frac{i\,k\,\sqrt{-\Omega }}{2}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) }\right) } \right] ^{{1}/{2}}\, \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) }, \end{aligned}$$

(2.123)

$$\begin{aligned}&u(\xi )=a_{0}+\frac{a_{-1}}{\left( d+\frac{2\,\mu \, \sin \left( \frac{\sqrt{-\Omega }}{2}\,\xi \right) }{\sqrt{-\Omega }\, \cos \left( \frac{\sqrt{-\Omega }}{2}\,\xi \right) -\lambda \,\sin \left( \frac{\sqrt{-\Omega }}{2}\,\xi \right) }\right) },\end{aligned}$$

(2.124)

$$\begin{aligned} Q(x,t)= & {} \left[ a_{0}+\frac{a_{-1}}{\left( d+\frac{2\,\mu \, \sin \left( \frac{i\,k\,\sqrt{-\Omega }}{2}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) }{\sqrt{-\Omega }\, \cos \left( \frac{i\,k\,\sqrt{-\Omega }}{2}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) -\lambda \,\sin \left( \frac{i\,k\,\sqrt{-\Omega }}{2}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) }\right) } \right] ^{{1}/{2}}\, \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) },\end{aligned}$$

(2.125)

$$\begin{aligned}&u(\xi )=a_{0}+\frac{a_{-1}}{\left( d-\frac{2\,\mu \, \cos \left( {\sqrt{-\Omega }}\,\xi \right) }{\sqrt{-\Omega }\, \sin \left( {\sqrt{-\Omega }}\,\xi \right) +\lambda \,\cos \left( {\sqrt{-\Omega }}\,\xi \right) \pm \,\sqrt{-\Omega }}\right) },\end{aligned}$$

(2.126)

$$\begin{aligned} Q(x,t)= & {} \left[ a_{0}+\frac{a_{-1}}{\left( d-\frac{2\,\mu \, \cos \left( {i\,k\,\sqrt{-\Omega }}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) }{\sqrt{-\Omega }\, \sin \left( {i\,k\,\sqrt{-\Omega }}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) +\lambda \,\cos \left( {i\,k\,\sqrt{-\Omega }}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \pm \,\sqrt{-\Omega }}\right) } \right] ^{{1}/{2}}\, \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) },\end{aligned}$$

(2.127)

$$\begin{aligned}&u(\xi )=a_{0}+\frac{a_{-1}}{\left( d+\frac{2\,\mu \, \sin \left( {\sqrt{-\Omega }}\,\xi \right) }{\sqrt{-\Omega }\, \cos \left( {\sqrt{-\Omega }}\,\xi \right) -\lambda \,\sin \left( {\sqrt{-\Omega }}\,\xi \right) \pm \,\sqrt{-\Omega }}\right) },\end{aligned}$$

(2.128)

$$\begin{aligned} Q(x,t)= & {} \left[ a_{0}+\frac{a_{-1}}{\left( d+\frac{2\,\mu \, \sin \left( {i\,k\,\sqrt{-\Omega }}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) }{\sqrt{-\Omega }\, \cos \left( {i\,k\,\sqrt{-\Omega }}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) -\lambda \,\sin \left( {\sqrt{-\Omega }}\,i\,k\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \pm \,\sqrt{-\Omega }}\right) } \right] ^{{1}/{2}}\, \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) }, \end{aligned}$$

(2.129)

where A, B are arbitrary real constants and $A^{2}-B^{2}>0$.

When $\mu =0$ and $\lambda (v-1)\ne 0$, we have

$$\begin{aligned} u(\xi )=a_{0}+\frac{a_{-1}}{ \left( d-\frac{\lambda \,k}{(v-1)\,\left( k+\cosh (\lambda \,\xi )-\sinh (\lambda \,\xi )\right) }\right) }, \end{aligned}$$

(2.130)

$$\begin{aligned} Q(x,t)= & {} \left[ a_{0}+\frac{a_{-1}}{ \left( d-\frac{\lambda \,k}{(v-1)\,\left( k+\cosh \left( i\,\lambda \,k\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) -\sinh \left( i\,\lambda \,k\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \right) }\right) } \right] ^{{1}/{2}}\, \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) },\end{aligned}$$

(2.131)

$$\begin{aligned}&u(\xi )=a_{0}+\frac{a_{-1}}{ \left( d-\frac{\lambda \,\left( \cosh (\lambda \,\xi )+\sinh (\lambda \,\xi )\right) }{(v-1)\,\left( k+\cosh (\lambda \,\xi )+\sinh (\lambda \,\xi )\right) }\right) },\end{aligned}$$

(2.132)

$$\begin{aligned} Q(x,t)= & {} \left[ a_{0}+\frac{a_{-1}}{ \left( d-\frac{\lambda \,\left( \cosh \left( i\,\lambda \,k\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) +\sinh \left( i\,\lambda \,k\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \right) }{(v-1)\,\left( k+\cosh \left( i\,\lambda \,k\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) +\sinh \left( i\,\lambda \,k\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \right) }\right) } \right] ^{{1}/{2}}\, \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) }, \end{aligned}$$

(2.133)

$$\begin{aligned}&u(\xi )=a_{0}+\frac{a_{-1}}{ \left( d-\frac{1}{(v-1)\,\xi + C}\right) },\end{aligned}$$

(2.134)

$$\begin{aligned} Q(x,t)= & {} \left[ a_{0}+\frac{a_{-1}}{ \left( d+\frac{i}{k\,(v-1)\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) + C}\right) } \right] ^{{1}/{2}}\,\nonumber \\&\quad \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) }, \end{aligned}$$

(2.135)

where C, k are arbitrary constants.

Case II

$$\begin{aligned}&\lambda =\frac{1}{k}\,\sqrt{\frac{2\,\epsilon }{5}},\quad \omega =-\frac{3}{2}\,\sqrt{\frac{2\,\epsilon }{5}},\quad v=1,\quad a_{-1}=0,\nonumber \\&a_{0}=a_{0},\quad a_{1}=\frac{-a_{0}\,\sqrt{\frac{2\,\epsilon }{5}} }{d\, \sqrt{\frac{2\,\epsilon }{5}}-k\,\mu }\nonumber \\ \end{aligned}$$

so that, the exact travelling wave solutions of nonlinear complex fractional Schrödinger equation will be in the following form:

$$\begin{aligned} u(\xi )=a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{{(2\,\epsilon }/{5})}-k\,\mu }\,\left( d+\frac{G'}{G}\right) \right] .\nonumber \\ \end{aligned}$$

(2.136)

Therefore, the solitary travelling wave solutions will be in these forms:

When $\Omega =\lambda ^{2}-4\, \lambda \, \mu +4\, \mu >0$ and $\lambda (v-1)\ne 0$ or $\mu (v-1)\ne 0$

$$\begin{aligned}&u(\xi )=a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{({2\,\epsilon }/{5})}-k\,\mu }\,\left( d-\frac{1}{2(v-1)}\right. \right. \nonumber \\&\quad \quad \quad \left. \left. \times \left( \lambda +\sqrt{\Omega } \tanh \left( \frac{\sqrt{\Omega }}{2}\,\xi \right) \right) \right) \right] ,\end{aligned}$$

(2.137)

$$\begin{aligned}&Q(x,t)=\left[ a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{({2\,\epsilon }/{5}})-k\,\mu }\,\left( d-\frac{1}{2(v-1)}\right. \right. \right. \nonumber \\&\quad \left. \left. \left. \times \left( \lambda +\sqrt{\Omega }\, \tanh \left( \frac{i\,k\,\sqrt{\Omega }}{2}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \right) \right) \right] \right] ^{{1}/{2}}\, \nonumber \\&\quad \,\times \,\mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) }, \end{aligned}$$

(2.138)

$$\begin{aligned}&u(\xi )=a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{({2\,\epsilon }/{5}})-k\,\mu }\, \left( d-\frac{1}{2(v-1)}\right. \right. \nonumber \\&\quad \quad \quad \quad \left. \left. \times \left( \lambda +\sqrt{\Omega } \, \coth \left( \frac{\sqrt{\Omega }}{2}\,\xi \right) \right) \right) \right] , \end{aligned}$$

(2.139)

$$\begin{aligned}&Q(x,t)=\left[ a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{({2\,\epsilon }/{5}})-k\,\mu }\, \left( d-\frac{1}{2(v-1)}\!\!\!\!\right. \right. \right. \nonumber \\&\quad \left. \left. \left. \times \left( \lambda +\sqrt{\Omega } \, \coth \left( \frac{i\,k\,\sqrt{\Omega }}{2}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \right) \right) \right] \right] ^{{1}/{2}}\,\nonumber \\&\quad \times \,\mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) }, \end{aligned}$$

(2.140)

$$\begin{aligned}&u(\xi )=a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{({2\,\epsilon }/{5}})-k\,\mu }\,\left( d-\frac{1}{2(v-1)}\right. \right. \nonumber \\&\quad \quad \quad \quad \times \left( \lambda +\sqrt{\Omega } \,\left( \tanh \left( \sqrt{\Omega }\,\xi \right) \right. \right. \nonumber \\&\quad \quad \quad \left. \left. \left. \left. \pm \, i\, \mathrm {sech}\left( \sqrt{\Omega }\,\xi \right) \right) \right) \right) \right] , \end{aligned}$$

(2.141)

$$\begin{aligned}&Q(x,t)=\left[ a_{0}\, \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{({2\,\epsilon }/{5})}-k\,\mu }\,\left( d-\frac{1}{2(v-1)}\right. \right. \right. \nonumber \\&\quad \times \left( \lambda +\sqrt{\Omega } \left( \tanh \left( i\,k\,\sqrt{\Omega }\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \right. \right. \nonumber \\&\quad \pm i\, \mathrm {sech}\left( i\,k\,\sqrt{\Omega }\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \bigg )\bigg )\bigg )\bigg ] \bigg ]^{{1}/{2}}\,\nonumber \\&\quad \times \,\mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) }, \end{aligned}$$

(2.142)

$$\begin{aligned}&u(\xi )=a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{({2\,\epsilon }/{5}})-k\,\mu }\, \left( d-\frac{1}{2(v-1)}\right. \right. \nonumber \\&\quad \quad \,\,\,\times \left( \lambda +\sqrt{\Omega } \,\left( \coth \left( \sqrt{\Omega }\,\xi \right) \right. \right. \nonumber \\&\quad \quad \,\left. \left. \left. \left. \pm \, \mathrm {csch}\left( \sqrt{\Omega }\,\xi \right) \right) \right) \right) \right] ,\end{aligned}$$

(2.143)

$$\begin{aligned}&Q(x,t)=\left[ a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{({2\,\epsilon }/{5})}-k\,\mu }\, \left( d-\frac{1}{2(v-1)}\right. \right. \right. \nonumber \\&\left. \left. \left. \quad \times \left( \lambda +\sqrt{\Omega } \,\left( \coth \left( i\,k\,\sqrt{\Omega }\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \right. \right. \right. \right. \right. \nonumber \\&\left. \left. \left. \left. \left. \quad \pm \, \mathrm {csch}\left( i\,k\,\sqrt{\Omega }\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \right) \right) \right) \right] \right] ^{{1}/{2}}\, \nonumber \\&\quad \times \,\mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) },\end{aligned}$$

(2.144)

$$\begin{aligned}&u(\xi )=a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{({2\,\epsilon }/{5})}-k\,\mu }\,\left( d-\frac{1}{4(v-1)}\right. \right. \nonumber \\&\quad \quad \quad \quad \times \left( 2\,\lambda +\sqrt{\Omega } \left( \tanh \left( \frac{\sqrt{\Omega }}{4}\,\xi \right) \right. \right. \nonumber \\&\left. \left. \left. \left. \quad \quad \quad \quad \!\pm \,\coth \left( \frac{\sqrt{\Omega }}{4}\,\xi \right) \right) \right) \right) \right] ,\end{aligned}$$

(2.145)

$$\begin{aligned}&Q(x,t)=\left[ a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{{(2\,\epsilon /{5})}}-k\,\mu }\,\left( d-\frac{1}{4(v-1)}\right. \right. \right. \nonumber \\&\left. \left. \left. \times \left( 2\,\lambda +\sqrt{\Omega } \left( \tanh \left( \frac{i\,k\,\sqrt{\Omega }}{4}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \right. \right. \right. \right. \right. \nonumber \\&\left. \left. \left. \left. \left. \pm \coth \left( \frac{i\,k\,\sqrt{\Omega }}{4}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \right) \right) \right) \right] \right] ^{{1}/{2}}\, \nonumber \\&\quad \times \,\mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) }, \end{aligned}$$

(2.146)

$$\begin{aligned}&u(\xi )=a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{({2\,\epsilon }/{5}})-k\,\mu }\, \left( d+\frac{1}{2(v-1)}\left( -\lambda +\frac{\pm \sqrt{\Omega \,\left( A^{2}+B^{2}\right) }-A\,\sqrt{\Omega }\, \cosh \left( \sqrt{\Omega }\,\xi \right) }{A\, \sinh \left( \sqrt{\Omega }\, \xi \right) +B}\right) \right) \right] ,\nonumber \\\end{aligned}$$

(2.147)

$$\begin{aligned} Q(x,t)= & {} \left[ a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{({2\,\epsilon }/{5}})-k\,\mu }\,\right. \right. \nonumber \\&\left. \left. \quad \times \left( d+\frac{1}{2(v-1)}\left( -\lambda +\frac{\pm \sqrt{\Omega \,\left( A^{2}+B^{2}\right) }-A\,\sqrt{\Omega }\, \cosh \left( i\,k\,\sqrt{\Omega }\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) }{A\, \sinh \left( i\,k\,\sqrt{\Omega }\, \left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) +B}\right) \right) \right] \right] ^{{1}/{2}}\,\nonumber \\&\quad \times \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) },\end{aligned}$$

(2.148)

$$\begin{aligned}&u(\xi )=a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{({2\,\epsilon }/{5}})-k\,\mu }\, \left( d+\frac{1}{2(v-1)}\left( -\lambda +\frac{\pm \sqrt{\Omega \,\left( A^{2}+B^{2}\right) }+A\,\sqrt{\Omega }\, \cosh \left( \sqrt{\Omega }\,\xi \right) }{A\, \sinh \big (\sqrt{\Omega }\, \xi \big )+B}\right) \right) \right] ,\nonumber \\ \end{aligned}$$

(2.149)

$$\begin{aligned} Q(x,t)= & {} \left[ a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{({2\,\epsilon }/{5}})-k\,\mu }\, \right. \right. \nonumber \\&\times \left. \left. \left( d+\frac{1}{2(v-1)}\left( -\lambda +\frac{\pm \sqrt{\Omega \,\left( A^{2}+B^{2}\right) }+A\,\sqrt{\Omega }\, \cosh \left( i\,k\,\sqrt{\Omega }\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) }{A\, \sinh \left( i\,k\sqrt{\Omega }\, \left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) +B}\right) \right) \right] \right] ^{{1}/{2}}\,\!\!\!\! \mathrm {e}^{i(\omega \,x+\frac{\epsilon \,t^{\alpha }}{\alpha })},\nonumber \\ \end{aligned}$$

(2.150)

$$\begin{aligned}&u(\xi )=a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{({2\,\epsilon }/{5}})-k\,\mu }\,\left( d+\frac{2\,\mu \, \cosh \left( \frac{\sqrt{\Omega }}{2}\,\xi \right) }{\sqrt{\Omega }\, \sinh \left( \frac{\sqrt{\Omega }}{2}\,\xi \right) -\lambda \,\cosh \left( \frac{\sqrt{\Omega }}{2}\,\xi \right) }\right) \right] ,\end{aligned}$$

(2.151)

$$\begin{aligned} Q(x,t)= & {} \left[ a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{({2\,\epsilon }/{5})}-k\,\mu }\right. \right. \nonumber \\&\left. \left. \times \left( d+\frac{2\,\mu \, \cosh \left( \frac{i\,k\,\sqrt{\Omega }}{2}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) }{\sqrt{\Omega }\, \sinh \left( \frac{\sqrt{\Omega }}{2}\,i\,k\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) -\lambda \,\cosh \left( \frac{i\,k\,\sqrt{\Omega }}{2}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) }\right) \right] \right] ^{{1}/{2}}\, \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) },\nonumber \\\end{aligned}$$

(2.152)

$$\begin{aligned}&u(\xi )=a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{({2\,\epsilon }/{5}})-k\,\mu }\, \left( d+\frac{2\,\mu \, \sinh \left( \frac{\sqrt{\Omega }}{2}\,\xi \right) }{\sqrt{\Omega }\, \cosh \left( \frac{\sqrt{\Omega }}{2}\,\xi \right) -\lambda \,\sinh \left( \frac{\sqrt{\Omega }}{2}\,\xi \right) }\right) \right] ,\end{aligned}$$

(2.153)

$$\begin{aligned} Q(x,t)= & {} \left[ a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{({2\,\epsilon }/{5}})-k\,\mu }\,\right. \right. \nonumber \\&\left. \left. \times \left( d+\frac{2\,\mu \, \sinh \left( \frac{i\,k\,\sqrt{\Omega }}{2}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) }{\sqrt{\Omega }\, \cosh \left( \frac{i\,k\,\sqrt{\Omega }}{2}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) -\lambda \,\sinh \left( \frac{i\,k\,\sqrt{\Omega }}{2}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) }\right) \right] \right] ^{{1}/{2}}\, \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) },\nonumber \\\end{aligned}$$

(2.154)

$$\begin{aligned}&u(\xi )=a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{({2\,\epsilon }/{5}})-k\,\mu }\,\left( d +\frac{2\,\mu \, \cosh \left( {\sqrt{\Omega }}\,\xi \right) }{\sqrt{\Omega }\, \sinh \left( {\sqrt{\Omega }}\,\xi \right) -\lambda \,\cosh \left( {\sqrt{\Omega }}\,\xi \right) \pm i\,\sqrt{\Omega }}\right) \right] ,\end{aligned}$$

(2.155)

$$\begin{aligned} Q(x,t)= & {} \left[ a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{({2\,\epsilon }/{5}})-k\,\mu }\,\right. \right. \nonumber \\&\left. \left. \times \left( d+\frac{2\,\mu \, \cosh \left( {\sqrt{\Omega }}\,\xi \right) }{\sqrt{\Omega }\, \sinh \left( {i\,k\,\sqrt{\Omega }}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) -\lambda \,\cosh \left( {i\,k\,\sqrt{\Omega }}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \pm i\,\sqrt{\Omega }}\right) \right] \right] ^{{1}/{2}}\nonumber \\&\times \,\mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) },\end{aligned}$$

(2.156)

$$\begin{aligned}&u(\xi )=a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{({2\,\epsilon }/{5}})-k\,\mu }\,\left( d+\frac{2\,\mu \, \sinh \big ({\sqrt{\Omega }}\,\xi \big )}{\sqrt{\Omega }\, \cosh \big ({\sqrt{\Omega }}\,\xi \big )-\lambda \,\sinh \big ({\sqrt{\Omega }}\,\xi \big )\pm i\,\sqrt{\Omega }}\right) \right] , \end{aligned}$$

(2.157)

$$\begin{aligned} Q(x,t)= & {} \left[ a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{({2\,\epsilon }/{5})}-k\,\mu }\,\right. \right. \nonumber \\&\left. \left. \times \left( d+\frac{2\,\mu \, \sinh \left( {\sqrt{\Omega }}\,i\,k\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) }{\sqrt{\Omega }\, \cosh \left( {\sqrt{\Omega }}\,i\,k\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) -\lambda \,\sinh \left( {\sqrt{\Omega }}\,i\,k\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \pm i\,\sqrt{\Omega }}\right) \right] \right] ^{{1}/{2}}\, \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) },\nonumber \\ \end{aligned}$$

(2.158)

where A, B are arbitrary real constants and $A^{2}+B^{2}>0$.

When $\Omega =\lambda ^{2}-4\, \lambda \, \mu +4\, \mu <0$ and $\lambda (v-1)\ne 0$ or $\mu (v-1)\ne 0$

$$\begin{aligned}&u(\xi )=a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{({2\,\epsilon }/{5}})-k\,\mu }\, \left( d+\frac{1}{2(v-1)}\left( -\lambda +\sqrt{-\Omega } \, \tanh \left( \frac{\sqrt{-\Omega }}{2}\,i\,k \left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \right) \right) \right] ,\nonumber \\\end{aligned}$$

(2.159)

$$\begin{aligned} Q(x,t)= & {} \left[ a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{({2\,\epsilon }/{5}})-k\,\mu }\, \left( d+\frac{1}{2(v-1)}\left( -\lambda +\sqrt{-\Omega } \, \tanh \left( \frac{k\,\sqrt{\Omega }}{2} \left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \right) \right) \right] \right] ^{{1}/{2}}\nonumber \\&\times \,\mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) },\end{aligned}$$

(2.160)

$$\begin{aligned}&u(\xi )=a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{({2\,\epsilon }/{5}})-k\,\mu }\, \left( d-\frac{1}{2(v-1)}\left( \lambda +\sqrt{-\Omega } \, \coth \left( \frac{\sqrt{-\Omega }}{2}\,\xi \right) \right) \right) \right] ,\end{aligned}$$

(2.161)

$$\begin{aligned} Q(x,t)= & {} \left[ a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{({2\,\epsilon }/{5}})-k\,\mu }\, \left( d-\frac{1}{2(v-1)}\left( \lambda +\sqrt{-\Omega } \, \coth \left( \frac{k\,\sqrt{\Omega }}{2}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \right) \right) \right] \right] ^{{1}/{2}}\nonumber \\&\times \,\mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) },\end{aligned}$$

(2.162)

$$\begin{aligned}&u(\xi )=a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{({2\,\epsilon }/{5}})-k\,\mu }\,\left( d+\frac{1}{2(v-1)} \left( -\lambda +\sqrt{-\Omega } \,\left( \tan \left( \sqrt{-\Omega }\,\xi \right) \pm \, \sec \left( \sqrt{-\Omega }\,\xi \right) \right) \right) \right) \right] ,\nonumber \\\end{aligned}$$

(2.163)

$$\begin{aligned} Q(x,t)= & {} \left[ a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{({2\,\epsilon }/{5}})-k\,\mu }\, \left( d+\frac{1}{2(v-1)} \left( -\lambda +\sqrt{-\Omega } \,\left( \tan \left( \,k\sqrt{\Omega }\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \right. \right. \right. \right. \right. \nonumber \\&\left. \left. \left. \left. \left. \pm \, \sec \left( k\,\sqrt{\Omega }\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \right) \right) \right) \right] \right] ^{{1}/{2}}\, \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) },\end{aligned}$$

(2.164)

$$\begin{aligned}&u(\xi )=a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{{2\,\epsilon }/{5}})-k\,\mu }\, \left( d-\frac{1}{2(v-1)} \left( \lambda +\sqrt{-\Omega } \,\left( \cot \left( \sqrt{-\Omega }\,\xi \right) \pm \, \csc \left( \sqrt{-\Omega }\,\xi \right) \right) \right) \right) \right] ,\nonumber \\\end{aligned}$$

(2.165)

$$\begin{aligned} Q(x,t)= & {} \left[ a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{({2\,\epsilon }/{5})}-k\,\mu }\, \left( d-\frac{1}{2(v-1)}\right. \right. \right. \left( \lambda +\sqrt{-\Omega } \,\left( \cot \left( \,k\,\sqrt{\Omega }\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \right. \right. \nonumber \\&\left. \left. \left. \left. \left. \pm \,\csc \left( \,k\,\sqrt{\Omega }\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha } \right) \right) \right) \right) \right) \right] \right] ^{{1}/{2}}\, \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) },\end{aligned}$$

(2.166)

$$\begin{aligned}&u(\xi )=a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{({2\,\epsilon }/{5}})-k\,\mu }\,\left( d+\frac{1}{4(v-1)}\right. \right. \left( -2\,\lambda +\sqrt{-\Omega } \,\left( \tan \left( \frac{\sqrt{-\Omega }}{4}\,\xi \right) \right. \right. \left. \left. \left. \left. -\cot \left( \frac{\sqrt{-\Omega }}{4}\,\xi \right) \right) \right) \right) \right] ,\nonumber \\ \end{aligned}$$

(2.167)

$$\begin{aligned} Q(x,t)= & {} \left[ a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{({2\,\epsilon }/{5}})-k\,\mu }\, \left( d+\frac{1}{4(v-1)} \left( -2\,\lambda +\sqrt{-\Omega } \,\left( \tan \left( \frac{k\,\sqrt{\Omega }}{4}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \right. \right. \right. \right. \right. \nonumber \\&\left. \left. \left. \left. \left. -\cot \left( \frac{k\,\sqrt{\Omega }}{4}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \right) \right) \right) \right] \right] ^{{1}/{2}}\, \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) },\end{aligned}$$

(2.168)

$$\begin{aligned}&u(\xi )=a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{({2\,\epsilon }/{5}})-k\,\mu }\,\left( d+\frac{1}{2(v-1)} \left( -\lambda +\frac{\pm \sqrt{-\Omega \,\left( A^{2}-B^{2}\right) }-A\,\sqrt{-\Omega }\, \cos \left( \sqrt{-\Omega }\,\xi \right) }{A\, \sin \left( \sqrt{-\Omega }\, \xi \right) +B}\right) \right) \right] ,\nonumber \\\end{aligned}$$

(2.169)

$$\begin{aligned} Q(x,t)= & {} \left[ a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{({2\,\epsilon }/{5}})-k\,\mu }\,\left( d+\frac{1}{2(v-1)}\right. \right. \right. \nonumber \\&\quad \left. \left. \left. \times \left( -\lambda +\frac{\pm \sqrt{-\Omega \,\left( A^{2}-B^{2}\right) }-A\,\sqrt{-\Omega }\, \cos \left( k\,\sqrt{\Omega }\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) }{A\, \sin \left( k\,\sqrt{\Omega }\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) +B}\right) \right) \right] \right] ^{{1}/{2}}\,\mathrm { e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) },\nonumber \\\end{aligned}$$

(2.170)

$$\begin{aligned}&u(\xi )=a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{({2\,\epsilon }/{5}})-k\,\mu }\,\left( d+\frac{1}{2(v-1)} \left( -\lambda +\frac{\pm \sqrt{-\Omega \,\left( A^{2}-B^{2}\right) }+A\,\sqrt{-\Omega }\, \cos \left( \sqrt{-\Omega }\,\xi \right) }{A\, \sin \left( \sqrt{-\Omega }\, \xi \right) +B}\right) \right) \right] ,\nonumber \\\end{aligned}$$

(2.171)

$$\begin{aligned} Q(x,t)= & {} \left[ a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{({2\,\epsilon }/{5}})-k\,\mu }\, \left( d+\frac{1}{2(v-1)}\right. \right. \right. \nonumber \\&\left. \left. \left. \times \left( -\lambda +\frac{\pm \sqrt{-\Omega \,\left( A^{2}-B^{2}\right) }+A\,\sqrt{-\Omega }\, \cos \left( k\,\sqrt{\Omega }\,i\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) }{A\, \sin \left( k\,\sqrt{\Omega }\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) +B}\right) \right) \right] \right] ^{{1}/{2}}\, \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) },\nonumber \\\end{aligned}$$

(2.172)

$$\begin{aligned}&u(\xi )=a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{({2\,\epsilon }/{5}})-k\,\mu }\,\left( d-\frac{2\,\mu \, \cos \left( \frac{\sqrt{-\Omega }}{2}\,\xi \right) }{\sqrt{-\Omega }\, \sin \left( \frac{\sqrt{-\Omega }}{2}\,\xi \right) +\lambda \,\cos \left( \frac{\sqrt{-\Omega }}{2}\,\xi \right) }\right) \right] ,\end{aligned}$$

(2.173)

$$\begin{aligned} Q(x,t)= & {} \left[ a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{{(2\,\epsilon }/{5}})-k\,\mu }\left( d-\frac{2\,\mu \, \cos \left( \frac{\sqrt{-\Omega }}{2}\,i\,k\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) }{\sqrt{-\Omega }\, \sin \left( \frac{k\,\sqrt{-\Omega }}{2}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) +\lambda \,\cos \left( \frac{k\,\sqrt{\Omega }}{2}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) }\right) \right] \right] ^{{1}/{2}}\nonumber \\&\times \,\mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) },\end{aligned}$$

(2.174)

$$\begin{aligned}&u(\xi )=a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{{(2\,\epsilon }/{5}})-k\,\mu }\,\right. \left. \left( d+\frac{2\,\mu \, \sin \left( \frac{\sqrt{-\Omega }}{2}\,\xi \right) }{\sqrt{-\Omega }\, \cos \left( \frac{\sqrt{-\Omega }}{2}\,\xi \right) -\lambda \,\sin \left( \frac{\sqrt{-\Omega }}{2}\,\xi \right) }\right) \right] ,\end{aligned}$$

(2.175)

$$\begin{aligned} Q(x,t)= & {} \left[ a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{({2\,\epsilon }/{5}})-k\,\mu } \left( d+\frac{2\,\mu \, \sin \left( \frac{\sqrt{-\Omega }}{2}\,i\,k\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) }{\sqrt{-\Omega }\, \cos \left( \frac{k\,\sqrt{\Omega }}{2}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) -\lambda \,\sin \left( \frac{k\,\sqrt{\Omega }}{2}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) }\right) \right] \right] ^{{1}/{2}}\nonumber \\&\times \,\mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) }, \end{aligned}$$

(2.176)

$$\begin{aligned}&u(\xi )=a_{0} \left[ 1-\frac{\sqrt{\frac{2\,\epsilon }{5}} }{d\, \sqrt{\frac{2\,\epsilon }{5}}-k\,\mu }\left( d-\frac{2\,\mu \, \cos \left( {\sqrt{-\Omega }}\,\xi \right) }{\sqrt{-\Omega }\, \sin \left( {\sqrt{-\Omega }}\,\xi \right) +\lambda \,\cos \left( {\sqrt{-\Omega }}\,\xi \right) \pm \,\sqrt{-\Omega }}\right) \right] ,\end{aligned}$$

(2.177)

$$\begin{aligned} Q(x,t)= & {} \left[ a_{0} \left[ 1-\frac{\sqrt{\frac{2\,\epsilon }{5}} }{d\, \sqrt{({2\,\epsilon }/{5}})-k\,\mu }\,\right. \right. \nonumber \\&\left. \left. \times \left( d-\frac{2\,\mu \, \cos \left( {k\,\sqrt{\Omega }}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) }{\sqrt{-\Omega }\, \sin \left( {k\,\sqrt{\Omega }}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) +\lambda \,\cos \left( {k\,\sqrt{\Omega }}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \pm \,\sqrt{-\Omega }}\right) \right] \right] ^{{1}/{2}}\nonumber \\&\times \,\mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) },\end{aligned}$$

(2.178)

$$\begin{aligned}&u(\xi )=a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{({2\,\epsilon }/{5}})-k\,\mu }\left( d+\frac{2\,\mu \, \sin \left( {\sqrt{-\Omega }}\,\xi \right) }{\sqrt{-\Omega }\, \cos \left( {\sqrt{-\Omega }}\,\xi \right) -\lambda \,\sin \left( {\sqrt{-\Omega }}\,\xi \right) \pm \,\sqrt{-\Omega }}\right) \right] ,\end{aligned}$$

(2.179)

$$\begin{aligned} Q(x,t)= & {} \left[ a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{({2\,\epsilon }/{5}})-k\,\mu }\,\right. \right. \nonumber \\&\quad \left. \left. \times \left( d+\frac{2\,\mu \, \sin \left( {k\,\sqrt{\Omega }}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) }{\sqrt{-\Omega }\, \cos \left( k\,{\sqrt{\Omega }}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) -\lambda \,\sin \left( {k\,\sqrt{\Omega }}\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \pm \,\sqrt{-\Omega }}\right) \right] \right] ^{{1}/{2}}\, \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) },\nonumber \\ \end{aligned}$$

(2.180)

where A, B are arbitrary real constants and $A^{2}-B^{2}>0$.

When $\mu =0$ and $\lambda (v-1)\ne 0$, we have

$$\begin{aligned}&u(\xi )=a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{({2\,\epsilon }/{5}})-k\,\mu }\left( d-\frac{\lambda \,k}{(v-1)\left( k+\cosh (\lambda \,\xi )-\sinh (\lambda \,\xi )\right) }\right) \right] ,\end{aligned}$$

(2.181)

$$\begin{aligned} Q(x,t)= & {} \left[ a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{({2\,\epsilon }/{5}})-k\,\mu }\,\right. \right. \nonumber \\&\quad \times \left. \left. \left( d-\frac{\lambda \,k}{(v-1)\,\left( k+\cosh \left( i\,\lambda \,k\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) -\sinh \left( i\,\lambda \,k\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \right) }\right) \right] \right] ^{{1}/{2}}\, \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) },\nonumber \\\end{aligned}$$

(2.182)

$$\begin{aligned}&u(\xi )=a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{({2\,\epsilon }/{5}})-k\,\mu }\,\right. \left. \left( d-\frac{\lambda \,\left( \cosh (\lambda \,\xi )+\sinh (\lambda \,\xi )\right) }{(v-1)\,\left( k+\cosh (\lambda \,\xi )+\sinh (\lambda \,\xi )\right) }\right) \right] ,\end{aligned}$$

(2.183)

$$\begin{aligned} Q(x,t)= & {} \left[ a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{({2\,\epsilon }/{5}})-k\,\mu }\,\right. \right. \nonumber \\&\quad \left. \left. \times \left( d-\frac{\lambda \,\left( \cosh \left( i\,\lambda \,k\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) +\sinh \left( i\,\lambda \,k\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \right) }{(v-1)\,\left( k+\cosh \left( i\,\lambda \,k\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) +\sinh \left( i\,\lambda \,k\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) \right) \right) }\right) \right] \right] ^{{1}/{2}}\, \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) },\nonumber \\\end{aligned}$$

(2.184)

$$\begin{aligned}&u(\xi )=a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{({2\,\epsilon }/{5}})-k\,\mu }\left( d-\frac{1}{(v-1)\,\xi + C}\right) \right] , \end{aligned}$$

(2.185)

$$\begin{aligned} Q(x,t)= & {} \left[ a_{0} \left[ 1-\frac{\sqrt{{2\,\epsilon }/{5}} }{d\, \sqrt{({2\,\epsilon }/{5}})-k\,\mu }\left( d+\frac{i}{k\,(v-1)\,\left( x+\frac{2\,\omega \,t^{\alpha }}{\alpha }\right) + C}\right) \right] \right] ^{{1}/{2}}\, \mathrm {e}^{i\,\left( \omega \,x+\frac{\epsilon \, t^{\alpha }}{\alpha }\right) }, \end{aligned}$$

(2.186)

where C, k are arbitrary constants.

Note that all the obtained results have been checked with Maple 2017 by putting them back into the original equation and the results are found to be correct.

3 Conclusion

In this research, we succeeded in applying new auxiliary equation method and novel $\left( {G'}/{G}\right)$-expansion method on nonlinear complex fractional Schrödinger equation. We obtained new form of solitary travelling wave solutions on this model that can be observed when a comparison is made between our solutions and that obtained in [37,38,39],  that were reported with specific coefficients of the equation as a special case to the results of this paper. This improves the superiority of our methods. Therefore, this paper gives a generalised flavour to nonlinear complex fractional Schrödinger equation in the study of optical solitons. Thus, the paper encourages us to carry out further research, especially with the inclusion of perturbation terms of KE equations. Those results will be soon published.