1 Introduction

Partial differential equations, especially NLSEs, can widely be used to model optical phenomena as used for phenomena in various areas. Generating the model for a physical event and obtaining exact, analytical or numerical solutions for the model are the main research subjects for many researchers. One of these models is optical modeling, which has come to the fore in recent years. There are some examples of optical modeling like optical image denoising (Qiao and Zou 2013), shock waves (Guner 2017), optical fiber pulses (Kudryashov 2020; Samir et al. 2021), light rays (Ren et al. 2019), electromagnetic analysis of dispersive media (Li et al. 2019), magneto-optic wave guides with different non-linearity (Ekici et al. 2017), thirring optical solitons (Bakodah et al. 2017), phase-shift controlling of solitons (Liu et al. 2019), 1-soliton solutions Biswas (2009), soliton transmission of optical fibers (Biswas and Arshed 2018).

Besides these models, with the help of computer-aided mathematical calculation programs, a wide variety of analytical and very efficient numerical methods have been developed in the recent years. For example, modified extended tanh method (Cinar et al. 2021), Sardar sub-equation method (Esen et al. 2021), extended rational sin-cos and sinh-cosh methods (Mahak and Akram 2019; Cinar et al. (2021; Mahak and Akram 2019; Cinar et al. 2021), Riccati Bernoulli sub-ODE method (Ozdemir et al. 2021), generated exponential function method (Srivastava et al. 2019), wavelet metods (Secer and Cinar 2020; Cinar et al. 2021), modified Kudryashov method (Srivastava et al. 2020) and the extended sinh-Gordon method (Dutta et al. 2020).Laplace-Adomian Decomposition (Gonzalez-Gaxiola 2022), the improved Adomian decomposition scheme (Al Qarni et al. 2022), Lie symmetry analysis (Bansal et al. 2018), analytic soliton solution (Liu et al. 2019), stationary soliton (Adem et al. 2021), sine-Gordon equation approach (Yildirim et al. 2021a, b), Modified Kudryashov’s method (Biswas et al. 2018), improved modified extended tanh-function method (Mirzazadeh et al. 2017), extended trial equation and extended \(\frac{G^{'}}{G}\)-expansion scheme (Ekici et al. 2016), unified Riccati equation, new mappig scheme (Zayed et al. 2021), semi-inverse variational principle (Zayed et al. 2021), the extended auxiliary equation approach and Jacobi’s elliptic function method (Bansal et al. 2018); Yıldırım et al. 2022), the Kudryashov’s integration scheme (Arnous et al. 2022; Ozisik et al. 2022).

In this work, we deal with the system of coupled NLSEs defined as Agrawal and Liao (1995); Baskonus et al. (2018).

$$\begin{aligned}{} & {} {{\eta }_{1}}{{u}_{tt}}+i\left( {{u}_{x}}+{{\eta }_{2}}{{v}_{t}} \right) +{{\eta }_{3}}{{\left| u \right| }^{2}}u+{{\eta }_{4}}v=0, \end{aligned}$$
(1)
$$\begin{aligned}{} & {} \quad {{\eta }_{1}}{{v}_{tt}}+i\left( {{v}_{x}}+{{\eta }_{2}}{{u}_{t}} \right) +{{\eta }_{3}}{{\left| v \right| }^{2}}v+{{\eta }_{4}}u=0, \end{aligned}$$
(2)

where \(i=\sqrt{-1}\), \(u=u(x,t)\) and \(v=v(x,t)\) are envelopes of the field, x is the co-ordinate of propagation, \(\frac{1}{{\eta }_{2}}\) is the group-velocity mismatch, \({\eta }_{1}\) is the group-velocity dispersion, \({\eta }_{4}\) is the linear coupling coefficient and \({\eta }_{3}\) is defined as \({\eta }_{3}=\frac{2 \pi m_{2}}{k B_{e f f}}\), where \(m_{2}\) is the nonlinear refractive index, k is the wavelength and \(B_{e f f}\) denotes effective mode area of each wavelength. To obtain the solution of the equation systems in this study, we used the computer algebra system as in previous studies (Guzel and Bayram 2006; Cinar et al. 2021; Akinlar et al. 2014).

Here, eMETEM has been applied to the dual-core optical equations in order to gain novel solutions. Some related researches on the method are the space-time fractional equal width and modified equal width equation (Raslan et al. 2017), modified Benjamin-Bona-Mahony and Zakharov-Kuznetsov equations (Taghizadeh and Mirzazadeh 2012).

The organization of the paper is as follows: Sect. 2 explains the algorithm of eMETEM (Ozisik et al. 2022). The application of the method to the dual-core optical fiber equations is added to the Sect. 3. The obtained results and illustrative figures are given in Sect. 4 and the conclusion of the study is included in the final section.

2 Method

Step 1: Consider the following NLPDE (Nonlinear Partial Differential Equation) and the traveling wave transformations, respectively:

$$\begin{aligned}{} & {} F\left( \mathcal {M},\frac{\partial \mathcal {M}}{\partial t},\frac{\partial \mathcal {M}}{\partial x},\frac{{{\partial }^{2}}\mathcal {M}}{\partial {{t}^{2}}},\frac{{{\partial }^{2}}\mathcal {M}}{\partial {{x}^{2}}},\frac{{{\partial }^{2}}\mathcal {M}}{\partial x\partial t},\ldots \right) =0, \end{aligned}$$
(3)
$$\begin{aligned}{} & {} \quad \mathcal {M}\left( x,t \right) ={e}^{i\theta } M\left( \xi \right) , \quad \theta ={{\beta }_{1}}x-{{\beta }_{2}}t,\quad \xi ={{p}_{1}}x+{{p}_{2}}t, \end{aligned}$$
(4)

where \(\beta _{1}\), \(\beta _{2}\), \(p_1\) and \(p_2\) are real constants. Substituting Eq. (4) into Eq. (3), we obtain the following NLODE:

$$\begin{aligned} P\left( M\left( \xi \right) ,{M}'\left( \xi \right) ,{M}''\left( \xi \right) ,\ldots \right) =0, \end{aligned}$$
(5)

where \(^\prime\) denotes the derivative of \(M\left( \xi \right)\) with respect to \(\xi\).

Step 2: Suppose that Eq. (5) admits the following equation as solution:

$$\begin{aligned} M\left( \xi \right) ={{A}_{0}}+\sum \limits _{i=1}^{m}{{{A}_{i}}{{\kappa }^{i}}(\xi )}+\sum \limits _{i=1}^{m}{{{B}_{i}}{{\kappa }^{-i}}(\xi )}, \end{aligned}$$
(6)

where \({{A}_{0}}, A_1, \ldots ,{{A}_{m}},{{B}_{1}},\ldots ,{{B}_{m}}\) are real constants to be computed later (\(A_{m}\) and \(B_{m}\) should not be zero, simultaneously). We can find \(m \in \mathbb {Z}^{+}\) by applying the balancing rule in Eq. (5). The \(\kappa (\xi )\) satisfies the following Riccati differential equation:

$$\begin{aligned} \frac{d\kappa (\xi )}{d\xi }={{\omega }}\mp {{\left[ \kappa (\xi ) \right] }^{2}}, \end{aligned}$$
(7)

where \(\omega\) is a real constant.

Step 3: Utilize the solutions of Eq. (7) are given in Table 1.

Table 1 The analytical solutions of Riccati Eq. (7)
Full size table

Step 4: Substituting Eq. (6) and its related derivatives into Eq. (5) and considering Eq. (7), one can get a polynomial in \(\kappa (\xi )\). Gathering the coefficients of \(\kappa (\xi )\) with the same power and equating each coefficient to zero, we get a system of algebraic equations.

Step 5: The unknowns \({{A}_{0}},A_1, \ldots ,{{A}_{m}},{{B}_{1}},{{B}_{2}},\ldots ,{{B}_{m}},{{\beta }_{1}}\), \({{\beta }_{2}}\), \(p_1, p_2\) and \(\omega\) can be found by solving the set of algebraic equations in Step 4. Substituting the \(\kappa _{i} (\xi )\) funtions given in the Table 1. into Eq. (6) and considering Eq. (4), the solutions of the NLPDE in Eq. (3) are found.

3 Application of the method

Let us consider Eq. (1) and the following wave transformations:

$$\begin{aligned} u(x,t)={{e}^{i\theta }}U(\xi ),\;v(x,t)={{e}^{i\theta }}V(\xi ),\;\xi =\delta x-\lambda t,\;\theta ={{\beta }_{1}}x-{{\beta }_{2}}t, \end{aligned}$$
(8)

where \(\delta , \lambda , \beta _1\) and \(\beta _2\) are non-zero real values. \(\theta ,\beta _1, \beta _2, \delta\) and \(\lambda\) stand for the phase component, the frequency, the wave number and the velocity of the wave, respectively.

Substituting the wave transformations in Eq. (8) into Eq. (1), we obtain the following equations from the real and the imaginary parts, respectively:

$$\begin{aligned}{} & {} \left( {{\beta }_{2}}{{\eta }_{2}}+{{\eta }_{4}} \right) V\left( \xi \right) -\left( {{\beta }_{2}}^{2}{{\eta }_{1}}+{{\beta }_{1}} \right) U\left( \xi \right) +{{\eta }_{3}}{{\left( U\left( \xi \right) \right) }^{3}}+{{\eta }_{1}}{{\lambda }^{2}}\frac{{{\text {d}}^{2}}}{\text {d}{{\xi }^{2}}}U\left( \xi \right) =0 , \end{aligned}$$
(9)
$$\begin{aligned}{} & {} \quad \left( 2\lambda {{\beta }_{2}}{{\eta }_{1}}+\delta \right) \frac{\text {d}}{\text {d}\xi }U\left( \xi \right) -{{\eta }_{2}}\lambda \frac{\text {d}}{\text {d}\xi }V\left( \xi \right) =0. \end{aligned}$$
(10)

Similarly, substituting Eq. (8) into Eq. (2), we obtain the following equations from the real and the imaginary parts, respectively:

$$\begin{aligned}{} & {} \left( {{\beta }_{2}}{{\eta }_{2}}+{{\eta }_{4}} \right) U\left( \xi \right) -\left( {{\beta }_{2}}^{2}{{\eta }_{1}}+{{\beta }_{1}} \right) V\left( \xi \right) +{{\eta }_{3}}{{\left( V\left( \xi \right) \right) }^{3}}+{{\eta }_{1}}{{\lambda }^{2}}\frac{{{\text {d}}^{2}}}{\text {d}{{\xi }^{2}}}V\left( \xi \right) =0, \end{aligned}$$
(11)
$$\begin{aligned}{} & {} \quad \left( 2\lambda {{\beta }_{2}}{{\eta }_{1}}+\delta \right) \frac{\text {d}}{\text {d}\xi }V\left( \xi \right) -{{\eta }_{2}}\lambda \frac{\text {d}}{\text {d}\xi }U\left( \xi \right) =0. \end{aligned}$$
(12)

By considering the imaginary parts equations in eqs. (10) and (12), it can be seen that the system has a non-trivial solution \(U\left( \xi \right) =V\left( \xi \right) .\) So, from Eqs. (10) or (12), we get:

$$\begin{aligned} \lambda =-\frac{\delta }{2{{\beta }_{2}}{{\eta }_{1}}-{{\eta }_{2}}}. \end{aligned}$$
(13)

Taking Eqs. (9) or (11) into consideration, one can get:

$$\begin{aligned} \left( -{{\beta }_{2}}^{2}{{\eta }_{1}}+{{\beta }_{2}}{{\eta }_{2}}-{{\beta }_{1}}+{{\eta }_{4}} \right) U\left( \xi \right) +{{\eta }_{3}}{{\left( U\left( \xi \right) \right) }^{3}}+{{\eta }_{1}}{{\lambda }^{2}}\frac{{{\text {d}}^{2}}}{\text {d}{{\xi }^{2}}}U\left( \xi \right) =0. \end{aligned}$$
(14)

Balancing the terms \({U}^{\prime \prime }\) and \({{U}^{3}}\) in Mahak and Akram (2019), we get \(m=1\). So, Eq. (6) turns into following form:

$$\begin{aligned} U(\xi )={{A}_{0}}+{{A}_{1}}\kappa (\xi )+{{B}_{1}}\frac{1}{\kappa (\xi )}, \end{aligned}$$
(15)

where \(A_{1}\) and \(B_{1}\) should not be zero, simultaneously. Substituting the Eq. (15) and its necessary derivatives into Eq. (14), we get a polynomial form in \(\kappa (\xi )\). Gathering the each term according to same power of \(\kappa ^{i}(\xi )\) and equating each coefficients to zero, we get the following system as:

$$\begin{aligned} \begin{aligned}&{{\kappa }^{-3}}(\xi ): \quad {{B}_{1}}\left( 2{{\eta }_{3}}{{\left( {{\eta }_{1}}{{\beta }_{2}}-\left( 1/2 \right) {{\eta }_{2}} \right) }^{2}}{{B}_{1}}^{2}+{{\delta }^{2}}{{\eta }_{1}}{{\omega }^{2}} \right) =0, \\&{{\kappa }^{-2}}(\xi ): \quad {{\eta }_{3}}{{A}_{0}}{{B}_{1}}^{2}=0, \\&{{\kappa }^{-1}}(\xi ): \quad \left( \begin{aligned}&-2{{\beta }_{2}}^{4}{{\eta }_{1}}^{3}+4{{\beta }_{2}}^{3}{{\eta }_{1}}^{2}{{\eta }_{2}} \\&-2{{\eta }_{1}}\left( \left( \left( -3{{A}_{0}}^{2}-3{{A}_{1}}{{B}_{1}} \right) {{\eta }_{3}}+{{\beta }_{1}}-{{\eta }_{4}} \right) {{\eta }_{1}}+\left( 5/4 \right) {{\eta }_{2}}^{2} \right) {{\beta }_{2}}^{2} \\&+2\left( \left( \left( -3{{A}_{0}}^{2}-3{{A}_{1}}{{B}_{1}} \right) {{\eta }_{3}}+{{\beta }_{1}}-{{\eta }_{4}} \right) {{\eta }_{1}}+\left( 1/4 \right) {{\eta }_{2}}^{2} \right) {{\eta }_{2}}{{\beta }_{2}} \\&+{{\delta }^{2}}{{\eta }_{1}}w-\left( 1/2 \right) \left( \left( -3{{A}_{0}}^{2}-3{{A}_{1}}{{B}_{1}} \right) {{\eta }_{3}}+{{\beta }_{1}}-{{\eta }_{4}} \right) {{\eta }_{2}}^{2} \\ \end{aligned} \right) {{B}_{1}} =0,\\&{{\kappa }^{0}}(\xi ): \quad {{A}_{0}}\left( -{{\eta }_{1}}{{\beta }_{2}}^{2}+{{\eta }_{2}}{{\beta }_{2}}+\left( {{A}_{0}}^{2}+6{{A}_{1}}{{B}_{1}} \right) {{\eta }_{3}}+{{\eta }_{4}}-{{\beta }_{1}} \right) =0, \\&{{\kappa }^{1}}(\xi ): \quad \left( \begin{aligned}&-2{{\beta }_{2}}^{4}{{\eta }_{1}}^{3}+4{{\beta }_{2}}^{3}{{\eta }_{1}}^{2}{{\eta }_{2}} \\&-2{{\eta }_{1}}\left( \left( \left( -3{{A}_{0}}^{2}-3{{A}_{1}}{{B}_{1}} \right) {{\eta }_{3}}+{{\beta }_{1}}-{{\eta }_{4}} \right) {{\eta }_{1}}+\left( 5/4 \right) {{\eta }_{2}}^{2} \right) {{\beta }_{2}}^{2} \\&+2\left( \left( \left( -3{{A}_{0}}^{2}-3{{A}_{1}}{{B}_{1}} \right) {{\eta }_{3}}+{{\beta }_{1}}-{{\eta }_{4}} \right) {{\eta }_{1}}+\left( 1/4 \right) {{\eta }_{2}}^{2} \right) {{\eta }_{2}}{{\beta }_{2}} \\&+{{\delta }^{2}}{{\eta }_{1}}w-\left( 1/2 \right) \left( \left( -3{{A}_{0}}^{2}-3{{A}_{1}}{{B}_{1}} \right) {{\eta }_{3}}+{{\beta }_{1}}-{{\eta }_{4}} \right) {{\eta }_{2}}^{2} \\ \end{aligned} \right) {{A}_{1}} =0, \\&{{\kappa }^{2}}(\xi ): \quad {{\eta }_{3}}{{A}_{0}}{{A}_{1}}^{2}=0, \\&{{\kappa }^{3}}(\xi ): \quad \left( 2{{A}_{1}}^{2}{{\beta }_{2}}^{2}{{\eta }_{1}}^{2}{{\eta }_{3}}+\left( -2{{A}_{1}}^{2}{{\beta }_{2}}{{\eta }_{2}}{{\eta }_{3}}+{{\delta }^{2}} \right) {{\eta }_{1}}+\left( 1/2 \right) {{A}_{1}}^{2}{{\eta }_{2}}^{2}{{\eta }_{3}} \right) {{A}_{1}} =0. \\ \end{aligned} \end{aligned}$$
(16)

By solving this system, we get many solution sets. But we present some of the sets as follows:

$$\begin{aligned} DSet_{1}&=\left\{ \begin{array}{l} {\omega =\frac{\left( \eta _{1} \beta _{2} {}^{2} -\eta _{2} \beta _{2} +\beta _{1} -\eta _{4} \right) \left( 2 \eta _{1} \beta _{2} -\eta _{2} \right) ^{2} }{8 \delta ^{2} \eta _{1} } ,A_{0} =0,} \\ {A_{1} =\frac{\delta \sqrt{-2\eta _{1} \eta _{3} } }{\eta _{3} \left( 2 \eta _{1} \beta _{2} -\eta _{2} \right) } ,B_{1} =\frac{\sqrt{2} \left( \eta _{1} \beta _{2} {}^{2} -\eta _{2} \beta _{2} +\beta _{1} -\eta _{4} \right) \left( 2\eta _{1} \beta _{2} -\eta _{2} \right) }{8 \delta \sqrt{-\eta _{1} \eta _{3} } }} \end{array} \right\} , \\ DSet_{2}&=\left\{ \begin{array}{l} {\beta _{1} =\frac{-4 \beta _{2} {}^{4} \eta _{1} {}^{3} +\left( 8 \beta _{2} {}^{3} \eta _{2} +4 \beta _{2} {}^{2} \eta _{4} \right) \eta _{1} {}^{2} +\Omega _{0} \eta _{1} +\eta _{2} {}^{2} \left( \eta _{2} \beta _{2} +\eta _{4} \right) }{\left( 2 \eta _{1} \beta _{2} -\eta _{2} \right) ^{2} } , } \\ {A_{0} =0, A_{1} =\frac{\delta \sqrt{-2\eta _{1} \eta _{3} } }{\eta _{3} \left( 2 \eta _{1} \beta _{2} -\eta _{2} \right) } ,B_{1} =\frac{w\delta \sqrt{-2\eta _{1} \eta _{3} } }{\eta _{3} \left( 2 \eta _{1} \beta _{2} -\eta _{2} \right) } } \end{array}\right\} ,\\ DSet_{3}&= \left\{ \begin{array}{ll} {A_{0} =0,A_{1} = \frac{16w\eta _{1} {}^{2} \delta ^{2} }{\sqrt{\eta _{1} \eta _{3} \left( \Omega _{2} -\Omega _{3} \right) w} \left( \Omega _{4} -\Omega _{3} \right) } ,\, \beta _{2} =\frac{2 \eta _{2} +\sqrt{2\Omega _{4} -2 \Omega _{3} } }{4 \eta _{1} },} \\ {B_{1} =-\frac{\sqrt{\eta _{1} \eta _{3} \left( \Omega _{2} -\Omega _{3} \right) w} }{4 \eta _{1} \eta _{3} } } \end{array} \right\} ,\\ DSet_{4}&=\left\{ \begin{array}{ll} {A_{0} =0,A_{1} =\frac{16\delta ^{2} \eta _{1} {}^{2} w}{\sqrt{\eta _{1} \eta _{3} w\left( \Omega _{2} +\Omega _{3} \right) } \left( \Omega _{3} +\Omega _{4} \right) } ,\, \beta _{2} =\frac{\sqrt{2\Omega _{3} +2\Omega _{4} } +2 \eta _{2} }{4 \eta _{1} },} \\ {B_{1} =-\frac{\sqrt{\eta _{1} \eta _{3} w\left( \Omega _{2} +\Omega _{3} \right) } }{4 \eta _{1} \eta _{3} }} \end{array} \right\} , \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} \Omega _{0}&=\left( -5 \beta _{2} {}^{2} \eta _{2} {}^{2} -4 \delta ^{2} w-4 \beta _{2} \eta _{2} \eta _{4} \right) , \\ \Omega _{1}&=\left( -64 \delta ^{2} w+16 \beta _{1}^{2}-32 \beta _{1} \eta _{4}+16 \eta _{4}^{2}\right) \eta _{1}^{2}, \Omega _{2}=4 \beta _{1} \eta _{1}-4 \eta _{4} \eta _{1}-\eta _{2}^{2},\\ \Omega _{3}&=\sqrt{\Omega _{1}+8 \eta _{2}^{2}\left( \eta _{4}-\beta _{1}\right) \eta _{1}+\eta _{2}^{4}}, \Omega _{4}=\left( 4 \eta _{4}-4 \beta _{1}\right) \eta _{1}+\eta _{2}^{2}, \end{aligned} \end{aligned}$$
(17)

Substituting the \(\kappa _{j}(\xi )\) in Table 1 into Eq. (15) and using the Eq. (8), the following solution functions \(u_j(x,t)=v_j(x,t)\) for the NLSE system in eqs. (1) and (2) are obtained in the general form for \(j=1,2,\ldots ,15\):

$$\begin{aligned}{} & {} u_{1} \mathrm{(x,t)}=\chi \left( A_{0} -A_{1} \sqrt{-\omega } \tanh \left( P_{\omega xt}\right) -\frac{B_{1} }{\sqrt{-\omega } \tanh \left( P_{\omega xt}\right) } \right) , \end{aligned}$$
(18)
$$\begin{aligned}{} & {} \quad u_{2} \mathrm{(x,t)}=\chi \left( A_{0} -A_{1} \sqrt{-\omega } \textrm{coth}\left( P_{\omega xt}\right) -\frac{B_{1} }{\sqrt{-\omega } \textrm{coth}\left( P_{\omega xt}\right) } \right) , \end{aligned}$$
(19)
$$\begin{aligned}{} & {} \quad u_{3} \mathrm{(x,t)}=\chi \left( \begin{array}{l} {A_{0} -A_{1} \sqrt{-\omega } \left( \tanh \left( 2 P_{\omega xt}\right) +i\mu \textrm{sech}\left( 2 P_{\omega xt}\right) \right) } \\ {-\frac{B_{1} }{\sqrt{-\omega } \left( \tanh \left( 2 P_{\omega xt}\right) +i\mu \textrm{sech}\left( 2 P_{\omega xt}\right) \right) } } \end{array}\right) , \end{aligned}$$
(20)
$$\begin{aligned}{} & {} \quad u_{4} \mathrm{(x,t)}=\chi \left( A_{0} +A_{1} \frac{\omega -\sqrt{-\omega } \tanh \left( P_{\omega xt}\right) }{1+\sqrt{-\omega } \tanh \left( P_{\omega xt}\right) } +B_{1} \frac{1+\sqrt{-\omega } \tanh \left( P_{\omega xt}\right) }{\omega -\sqrt{-\omega } \tanh \left( P_{\omega xt}\right) } \right) , \end{aligned}$$
(21)
$$\begin{aligned}{} & {} \quad u_{5} \mathrm{(x,t)}=\chi \left( A_{0} +A_{1} \frac{\sqrt{-\omega } \left( 5-4 \cosh \left( 2 P_{\omega xt}\right) \right) }{3+4 \sinh \left( 2 P_{\omega xt}\right) } +B_{1} \frac{3+4 \sinh \left( 2 P_{\omega xt}\right) }{\sqrt{-\omega } \left( 5-4 \cosh \left( 2 P_{\omega xt}\right) \right) } \right) , \end{aligned}$$
(22)
$$\begin{aligned}{} & {} \quad u_{6} \mathrm{(x,t)}=\chi \left( \begin{array}{l} {A_{0} +A_{1} \frac{\mu \sqrt{-\left( a^{2} +b^{2} \right) \omega } -a\sqrt{-\omega } \cosh \left( 2 P_{\omega xt}\right) }{a\sinh \left( 2 \sqrt{-\omega } \eta \left( x-vt\right) \right) +b} } \\ {+B_{1} \frac{a\sinh \left( 2 P_{\omega xt}\right) +b}{\mu \sqrt{-\left( a^{2} +b^{2} \right) \omega } -a\sqrt{-\omega } \cosh \left( 2 P_{\omega xt}\right) } } \end{array}\right) , \end{aligned}$$
(23)
$$\begin{aligned}{} & {} \quad u_{7} \mathrm{(x,t)}=\chi \left( \begin{array}{l} {A_{0} +A_{1} \left( \mu \sqrt{-\omega } -2 \frac{\mu a\sqrt{-\omega } }{a+\cosh \left( 2 P_{\omega xt}\right) -\mu \sinh \left( 2 P_{\omega xt}\right) } \right) } \\ {+\frac{B_{1} }{\mu \sqrt{-\omega } -2 \frac{\mu a\sqrt{-\omega } }{a+\cosh \left( 2 P_{\omega xt}\right) -\mu \sinh \left( 2 P_{\omega xt}\right) } } } \end{array}\right) , \end{aligned}$$
(24)
$$\begin{aligned}{} & {} \quad u_{8} \mathrm{(x,t)}=\chi \left( A_{0} +A_{1} \sqrt{\omega } \tan \left( Q_{\omega xt}\right) +\frac{B_{1} }{\sqrt{\omega } \tan \left( Q_{\omega xt}\right) } \right) , \end{aligned}$$
(25)
$$\begin{aligned}{} & {} \quad u_{9} \mathrm{(x,t)}=\chi \left( A_{0} -A_{1} \sqrt{\omega } \cot \left( Q_{\omega xt}\right) -\frac{B_{1} }{\sqrt{\omega } \cot \left( Q_{\omega xt}\right) } \right) , \end{aligned}$$
(26)
$$\begin{aligned}{} & {} \quad u_{10} \mathrm{(x,t)}=\chi \left( A_{0} +A_{1} \sqrt{\omega } \left( \tan \left( 2 Q_{\omega xt}\right) +\mu \sec \left( 2 Q_{\omega xt}\right) \right) +\frac{B_{1} }{G_0} \right) , \end{aligned}$$
(27)

where \(G_0=\sqrt{\omega } \tan \left( 2 Q_{\omega xt}\right) +\mu \sec \left( 2 Q_{\omega xt}\right) .\)

$$\begin{aligned}{} & {} u_{11} \mathrm{(x,t)}=\chi \left( A_{0} -A_{1} \frac{\sqrt{\omega } \left( 1-\tan \left( Q_{\omega xt}\right) \right) }{1+\tan \left( Q_{\omega xt}\right) } -B_{1} \frac{1+\tan \left( Q_{\omega xt}\right) }{\sqrt{\omega } \left( 1-\tan \left( Q_{\omega xt}\right) \right) } \right) , \end{aligned}$$
(28)
$$\begin{aligned}{} & {} \quad u_{12} \mathrm{(x,t)}=\chi \left( A_{0} +A_{1} \frac{\sqrt{\omega } \left( 4-5 \cos \left( 2 Q_{\omega xt}\right) \right) }{3+5 \sin \left( 2 Q_{\omega xt}\right) } +B_{1} \frac{3+5 \sin \left( 2 Q_{\omega xt}\right) }{\sqrt{\omega } \left( 4-5 \cos \left( 2 Q_{\omega xt}\right) \right) } \right) , \end{aligned}$$
(29)
$$\begin{aligned}{} & {} \quad u_{13} \mathrm{(x,t)}=\chi \left( {A_{0} +A_{1} \frac{\mu G_1}{a\sin \left( 2 Q_{\omega xt}\right) +b} +B_{1} \frac{a\sin \left( 2 Q_{\omega xt}\right) +b}{\mu \sqrt{\left( a^{2} -b^{2} \right) \omega } -a\sqrt{\omega } \cos \left( 2 Q_{\omega xt}\right) } } \right) , \end{aligned}$$
(30)

where \(G_1= \sqrt{\left( a^{2} -b^{2} \right) \omega } -a\sqrt{\omega } \cos \left( 2 Q_{\omega xt}\right) .\)

$$\begin{aligned}{} & {} u_{14} \mathrm{(x,t)}=\chi \left( \begin{array}{l} {A_{0} +A_{1} \left( i\mu \sqrt{\omega } -\frac{2 i\mu a\sqrt{\omega } }{a+\cos \left( 2 Q_{\omega xt}\right) -i\mu \sin \left( 2 Q_{\omega xt}\right) } \right) } \\ {+\frac{B_{1} }{\left( i\mu \sqrt{\omega } -\frac{2 i\mu a\sqrt{\omega } }{a+\cos \left( 2 Q_{\omega xt}\right) -i\mu \sin \left( 2 Q_{\omega xt}\right) } \right) } } \end{array}\right) , \end{aligned}$$
(31)
$$\begin{aligned}{} & {} \quad u_{15} \mathrm{(x,t)}=-\frac{1}{\delta x-\lambda t}, \end{aligned}$$
(32)

where \(\chi \mathrm{=e}^{i\left( -t\beta _{2} +x\beta _{1} \right) } ,\, \lambda =\frac{\delta }{\eta _{2} -2 \eta _{1} \beta _{2} }, P_{\omega xt}=\sqrt{-\omega }(\delta x - \lambda t), Q_{\omega xt}=\sqrt{\omega }(\delta x - \lambda t)\) and \(v_{j} \mathrm{(x,t)}=u_{j} \mathrm{(x,t),} \, \mathrm{(j=1,2,\ldots ,15)}\).

4 Results and discussion

In this section, we have presented various graphical illustrations of selected some of the solution functions obtained in the study. In Fig. 1, we give the some graphical illustrations of \(u_{3} (x,t)\) in Eq. (20) for the parameters \(\eta _{1} =-0.25, \eta _{2} =0.3, \eta _{3} =0.75, \eta _{4} =0.15, \beta _{1} =0.85, \beta _{2} =1.25,\delta =2\) and \(\mu =1\) with \(DSet_{1}\). The Fig. 1a–c give the representation of anti-peaked soliton. The Fig. 1c also reflects traveling wave property. The Fig. 1d–f belong to the imaginary part of \(u_{3} (x,t)\), and a periodic bright-dark soliton with different amplitudes is formed for the investigated case.

In Fig. 2, the some plots of \(u_{3} (x,t)\) in Eq. (20) are demonstrated for the parameters \(\omega =-0.4, \eta _{1} =0.65, \eta _{2} =0.15, \eta _{3} =0.25, \eta _{4} =0.5, \beta _{2} =0.6, \delta =0.85\) and \(\mu =1\) with \(DSet_{2}\). The Fig. 2a–c represents a bright soliton. The Fig. 2c also reflects the traveling wave feature. The Fig. 2d–f belong to the imaginary part of \(u_{3} (x,t)\), producing a lump-shaped soliton for the case under consideration.

In Fig. 3, the some views of \(u_{6} (x,t)\) in Eq. (23) are illustrated for the parameters \(\omega =-0.4, \eta =1, \eta _{1} =1.65, \eta _{2} =1.55, \eta _{3} =2.25, \eta _{4} =2.5, \beta _{1} =0.25, \delta =0.2\) and \(\mu =1\) with \(DSet_{3}\). When the Fig. 3a–c are viewed from the x-axis direction, they demonstrate a bright soliton without traveling wave feature. The Fig. 3d–f belong to the imaginary part of \(u_{6} (x,t)\), and a degenerate dark-bright-like image is formed for the examined situation.

In Fig. 4, we present some portraits of \(u_{14} (x,t)\) in Eq. (31) by selecting the parameters \(\omega =0.04, \eta _{1} =0.65, \eta _{2} =0.15, \eta _{3} =0.25, \eta _{4} =0.5, \beta _{2} =0.6, \delta =0.85,\, a=1.2,\, b=0.6\) and \(\mu =1\) with \(DSet_{2}\). The Fig. 4a–c have a periodic bright soliton behavior. The Fig. 4c shows the traveling wave feature. The Fig. 4d–f belong to the imaginary part of \(u_{14} (x,t)\) and produce a rogue wave style soliton for the case under consideration.

In Fig. 5, the some silhouettes of \(u_{14} (x,t)\) in Eq. (31) are given for the parameters \(\omega =0.00004, \eta _{1} =0.65, \eta _{2} =0.15, \eta _{3} =0.25, \eta _{4} =0.5, \beta _{2} =0.6, \delta =0.85,\, a=1.2,\, b=0.6\) and \(\mu =1\) with \(DSet_{2}\). The Fig. 5a–c illustrate a compacton-style soliton without a traveling wave image. In a sense, it can also be called a parabolic soliton. The Fig. 5d–f belong to the imaginary part of \(u_{14} (x,t)\), producing periodic bright-dark solitons of variable amplitude.

In Fig. 6, the some depictions of \(u_{14} (x,t)\) in Eq. (31) are presented for the parameters \(\omega =0.4, \eta _{1} =1.65, \eta _{2} =1.55, \eta _{3} =2.25, \eta _{4} =2.5, \beta _{1} =0.25, \delta =0.2,\, a=1.2,\, b=0.6\) and \(\mu =1\) with \(DSet_{4}\). The Fig. 6a–c represent a periodic bright soliton. The Fig. 6d–f belong to the imaginary part of \(u_{14} (x,t)\) and show variable types of degenerate lump-like soliton.

In Fig. 7, we investigate the effect of the group velocity dispersion on the \(u_{3} (x,t)\) in Eq. (20) for the parameters \(\eta _{2}=0.3, \eta _{3}=0.75, \beta _{1}=0.85, \beta _{2}=1.25, \delta =2\) and \(\mu =1\) with \(DSet_{1}\). In Fig. 7a, taking \(\eta _4=-0.15\) and choosing \(\eta _1<0\), the chromatic dispersion term coefficient \(\eta _1\) are selected as \(-0.0025, -0.050, -0.075, -0.100, -0.125, -0.150, -0.175, -0.200, -0.225 and -0.250\), respectively. When the obtained figures are examined, if \(\eta _1<0\) and \(\eta _1\) increase, the lower peak of the wave remains on the horizontal x-axis, but the soliton moves to the right. At the same time, there is an increase in the vertical amplitude of the soliton.

In Fig. 7b the same examination is made for \(\eta _1>0\) and \(\eta _4=0.95\), the lower peak of the soliton stays on the horizontal axis due to the increase in \(\eta _1>0\) and \(\eta _1\), and the peak moves horizontally depending on the changing \(\eta _1\) values. However, this movement is not just a movement to the right or the left, as we obtained in our previous examination, the movement varies. The reason for this variability can be explained by the difficulty in controlling chromatic dispersion in optical wave propagation. Moreover, the increase in \(\eta _1\) values causes a decrease in the vertical amplitude of the soliton and also affects the shape of the soliton. For example, when \(\eta _1=0.135\), the soliton often refers to a V-type soliton.

In Fig. 8, the effect of the chromatic dispersion on the \(u_{3} (x,t)\) in Eq. (20) is investigated for the parameters \(\omega =-0.4, \eta _{2}=0.15, \eta _{3}=0.25, \eta _{4}=0.95, \beta _{2}=0.6, \delta =0.85\) and \(\mu =1\) with \(DSet_{2}\). In Fig. 8a, an examination is made for \(\eta _1<0\). As a result of the examination, when \(\eta _1<0\) and \(\eta _1\) increase, the soliton maintains its general shape in terms of species, that is, the bright soliton character continues, the skirts of the soliton remain on the horizontal axis, but the soliton moves to the right. At the same time, there is a change in the vertical amplitude of the soliton. It is not possible to characterize this change as a regular increase or decrease. As can be seen from the Fig. 8a, there is an increase for some values and a decrease for some values.

In Fig. 8b, while \(\eta _1>0\) and \(\eta _1\) increase, an examination is made. As a result of the examination, the soliton still maintains its bright soliton feature, and the skirts of the soliton remain on the x-axis. However, there are dramatic changes in the position and amplitude of the soliton. When the Fig. 8b is examined in detail, it is observed that the change in the position and amplitude (pulse beats) of the soliton is random. The randomness can also be explained as the difficulty in controlling the GVD. For example, although \(\eta _1=-0.125\) in Fig. 8a, it is chosen as 0.135 instead of 0.125 in Fig. 8b. Moreover, in this case, the \(\eta _4\) value is taken as 0.95 instead of 0.15. The reason for this is that if \(\eta _1=0.125, \eta _4=0.15\) are taken, the soliton turns into a completely different shape. Thus, in a sense, the GVD is controlled by the nonlinear term within the perturbation term. This situation alone shows how difficult it is to control the terms group velocity dispersion, chromatic dispersion, and inter-modal dispersion in optical fibers. Therefore, studies on these situations are important and a great need.

Fig. 1
figure 1

The some graphical illustrations of \(u_{3} (x,t)\) given in Eq. (20) a 3D view of \(|u_{3}(x,t)|\), b the contour view of \(|u_{3}(x,t)|\), c 2D view of \(|u_{3}(x,t)|\), d 3D view of \(Im\left( u_{3}(x,t)\right)\), e the contour of \(Im\left( u_{3}(x,t)\right)\), f the 2D view of \(Im\left( u_{3}(x,t)\right)\)

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Fig. 2
figure 2

The some graphical illustrations of \(u_{3} (x,t)\) given in Eq. (20) a 3D view of \(|u_{3}(x,t)|\), b the contour view of \(|u_{3}(x,t)|\), c 2D view of \(|u_{3}(x,t)|\), d3D view of \(Im\left( u_{3}(x,t)\right)\), d the contour of \(Im\left( u_{3}(x,t)\right)\), d the 2D view of \(Im\left( u_{3}(x,t)\right)\)

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Fig. 3
figure 3

The some views of \(u_{6} (x,t)\) given in Eq. (23) a 3D view of \(|u_{6}(x,t)|\), b the contour view of \(|u_{6}(x,t)|\), c 2D view of \(|u_{6}(x,t)|\), d 3D view of \(Im\left( u_{6}(x,t)\right)\), e the contour of \(Im\left( u_{6}(x,t)\right)\), f the 2D view of \(Im\left( u_{6}(x,t)\right)\)

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Fig. 4
figure 4

The some portraits of \(u_{14} (x,t)\) given in Eq. (31) a 3D view of \(|u_{14}(x,t)|\), b the contour view of \(|u_{14}(x,t)|\), c 2D view of \(|u_{14}(x,t)|\), d 3D view of \(Im\left( u_{14}(x,t)\right)\), e the contour of \(Im\left( u_{14}(x,t)\right)\), f the 2D view of \(Im\left( u_{14}(x,t)\right)\)

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Fig. 5
figure 5

The some silhouettes of \(u_{14} (x,t)\) given in Eq. (31) a 3D view of \(|u_{14}(x,t)|\), b the contour view of \(|u_{14}(x,t)|\), c 2D view of \(|u_{14}(x,t)|\), d 3D view of \(Im\left( u_{14}(x,t)\right)\), e the contour of \(Im\left( u_{14}(x,t)\right)\), f the 2D view of \(Im\left( u_{14}(x,t)\right)\)

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Fig. 6
figure 6

The some depictions of \(u_{14} (x,t)\) given in Eq. (31) a 3D view of \(|u_{14}(x,t)|\), b the contour view of \(|u_{14}(x,t)|\), c 2D view of \(|u_{14}(x,t)|\), d 3D view of \(Im\left( u_{14}(x,t)\right)\), e the contour of \(Im\left( u_{14}(x,t)\right)\), f the 2D view of \(Im\left( u_{14}(x,t)\right)\)

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Fig. 7
figure 7

The effect of the coefficient of the GVD on the \(u_{3} (x,t)\) in Eq. (20) by selected parameters \(\eta _{2}=0.3, \eta _{3}=0.75, \beta _{1}=0.85, \beta _{2}=1.25, \delta =2\) and \(\mu =1\) with \(DSet_{1}\) a \(\eta _1<0\) and \(\eta _4=-0.15\), b \(\eta _1>0\) and \(\eta _4=0.95\)

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Fig. 8
figure 8

The effect of the coefficient of the GVD on the \(u_{3} (x,t)\) in Eq. (20) by selected parameters \(\omega =-0.4, \eta _{2}=0.15, \eta _{3}=0.25, \eta _{4}=0.95, \beta _{2}=0.6, \delta =0.85\) and \(\mu =1\) with \(DSet_2\) a \(\eta _1<0\), b \(\eta _1>0\)

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5 Conclusion

In this study, the nonlinear Schrödinger equation, which has an important place in modeling soliton transmission in optical fibers, has been discussed and successfully investigated. As a result of the examination, many soliton solutions and graphics have been obtained and interpreted in detail, and the comments made with 3D, contour and 2D graphics have been demonstrated. The study is not only about obtaining the soliton solutions of the NLSE equation, but also the eMETEM method has been successfully applied. As a more important contribution of this paper, the effect of the coefficient of the GVD dispersion term on the soliton propagation has been investigated. Considering the effect of the term coefficient of the GVD dispersion on soliton transmission in optical fibers and the difficulty in controlling this effect, the obtained results in the study might contribute to future studies in this field.