Assorted optical solitons of the (1+1)- and (2+1)-dimensional Chiral nonlinear Schrödinger equations using modified extended tanh-function technique

Abstract

In this research article, the (1+1)- and (2+1)-dimensional Chiral nonlinear Schrödinger equations (CNLSEs) are studied, which play an important role in the development of quantum mechanics, particularly in the field of quantum Hall effect. Our primary goal is to obtain the analytical solutions utilizing novel methodology, particularly the modified extended tanh-function technique. We concentrate on the search to solitary wave solutions inside the (1+1)- and (2+1)-dimensional CNLSEs, which are relevant in domains such as optics, electro-magnetic wave propagation, plasma physics, optics and quantum mechanics. Our objective is to increase knowledge of this equation and give insight into the behavior of solitary waves by employing a novel mathematical technique. This will be accomplished by displaying our findings in 2D and 3D graphics. We believe that our results would pave a way for future research generating optical memories based on the nonlinear solitons.

Introduction

Nonlinear partial differential equations (NLPDEs) are utilized to elucidated complex phenomena in nature, scattering and intrinsic attenuation¹, the fractured carbonate reservoirs², an adaptive tuning privacy–preserving approach³, the game-theoretic approach⁴ and the complex Ginzburg–Landau equation⁵. In telecommunications industry, examining the dynamics of optical solitons via optical fibers becomes significant⁶, uncertain high-order nonlinear pure-feedback systems via unified transformation functions⁷, the effect of high-order interactions⁸, the iterated function system⁹ and uncertain nonlinear pure-feedback systems with practical state constraints¹⁰. Several models investigated on solitons such as Van der Waals model through the EShGEEM and the IEEM¹¹. The solitons behavior in optical to the Kudryashov’s quintuple self-phase modulation have been extremely studied by Li et al¹². The complex structures of the Biswas-Milovic equation for power, parabolic and dual parabolic law nonlinearities were investigated in¹³. Author of¹⁴ obtained optical soliton solutions for Schrödinger type nonlinear evolution equations by the $\tan (\phi /2)$-expansion method. In¹⁵, researchers used the same technique to extract dispersive dark optical soliton solutions to Tzitzéica type nonlinear evolution equations arising in nonlinear optics. Qian et al.¹⁶ applied the Hirota’s bilinear methodology to get nonparaxial solitons in a dimensionless coupled nonlinear Schrödinger system with cross-phase modulation. There are a few applications about NLSE, for case, optics, plasma material science, quantum mechanics, biophysics, and profound water etc¹⁷. Quantum field hypothesis is one of the imperative regions of hypothetical material science to consider the waves phenomena¹⁸. NLSE conditions are utilized to decide the values of vitality levels in quantum mechanics¹⁹.

In this work, (1+1)-dimensional CNLS equation with quantum Hall effect is described by the following equation²⁰

$$\begin{aligned} if_t+f_{xx}-i\sigma ({f^*}f_x-f{f_x^*})f=0, \end{aligned}$$

(1)

and the (2+1)-dimensional CNLS equation is given as follows²¹

$$\begin{aligned} if_t+a(f_{xx}+f_{yy})-i(b_1({f^*}f_x-f{f_x^*})+b_2({f^*}f_y-f{f_y^*}))f=0. \end{aligned}$$

(2)

The primary focus of this equation is the field function $f=f(x, y,t)$, where $\sigma ,b_1,b_2$ are the nonlinear coupling constants and a denotes the dispersion terms, and $*$ is linked to the complex conjugate. The Jacobi elliptical function method was employed to obtain cnoidal solutions, the hyperbolic solutions and the trigonometric solutions for the CNLS equations²².

In recent years, the nonlinear models have gained much attention in nonlinear dynamics, caused by considerable implementations of various mathematical problems such as the multiple Exp-function scheme²³, the Hirota’s bilinear method^24,25, a logarithm transformation method²⁶, orthogonal frequency division multiplexing²⁷, non-Newtonian model to investigate the thermal behavior of blood flow²⁸, a Lie analysis²⁹ and the general lump solutions^30,31,32,33. Several exact solutions to nonlinear models, which are gained through many mathematical methods, are excellently effective in numerous physical technology and engineering fields including plasma physics, solid-state physics, space technology, fluids, and optical fiber communication. Methods have been used to derive certain solutions of the NLPDEs, including the maximum external quantum efficiency³⁴, the parametric unified weak vector equilibrium problems³⁵, the back-propagation neural network technique³⁶, the multimodal vision-language learning scheme³⁷ and the complete language-vision interaction network method³⁸. Their implications span a wide array of fields, illustrating the practical importance of nonlinear systems for example, the multimodal hybrid parallel network technique³⁹, the multi-scale channel-spatial scheme⁴⁰, the internet of things technique⁴¹, the novel quaternary surfactant scheme⁴² and the combining the physical mechanism models with the Long Short-Term Memory network⁴³. By analyzing nonlinear systems, scientists continue to explore new connections and methods that improve both theoretical understanding and practical uses such as the Fourier decomposition scheme⁴⁴, the nonlinear descriptor systems⁴⁵, a quasi-Z-source network⁴⁶, single-stage multi-input method^47,48. Nonlinear system has various applications in scientific and engineering discipline for example the symmetrical printed structures method⁴⁹, a lightweight method for intelligent detection⁵⁰, the finite element multi-physics method⁵¹, the higher-order finite dimensional algebra technique⁵², the fractional ornstein-uhlenbeck processes with periodic mean⁵³. In addition, there are many applicable modes including the compositional optimization and structural techniques⁵⁴, mechanical properties of hot dry granite under thermal-mechanical couplings⁵⁵, fluid inverse volumetric modeling⁵⁶, H$\infty$ output feedback design method⁵⁷ and the Hamilton-Jacobi-Issacs inequality⁵⁸. Finally, researchers use systems where have applications in a two-dimensional plasma fluid dynamics model coupled with the current module⁵⁹, constraints for the unique analytic inverse of polynomials of fractional orders⁶⁰, observer design method for nonlinear generalized systems⁶¹, and so forth. The importance of the nonlinear phenomena has grown in a number of newly established categories, including branch of applied mathematics as well as mathematical physics^{62,63,64,65,66}. The investigations on various types of wave propagation generated by some unavoidable geophysical disturbance are parts of extensive mathematical modeling and analysis^{67,68,69,70,71}. The (2+1)-dimensional CNLS equation was investigated the extended direct algebraic and extended trial equation method⁷². The extended Fan sub-equation method was used to solve the CNLS equation⁷³. Based on physics-informed neural networks, gradient-enhanced physics-informed neural networks (gPINNs) add the partial derivative loss term of the independent variable and the physical constraint term was improved the accuracy of network training⁷⁴. The nonlinear state-space model based on the expectation maximization algorithm wherein the decomposition of t-distribution as well as the particle smoother was applied a robust identification strategy⁷⁵. The multiple sets of synchronized systems was termed as a multi-chimera state and has been documented in a network of FitzHugh-Nagumo systems under strong coupling conditions⁷⁶. The robust type-3 fuzzy controller implementation for the path-tracking task of driverless cars during critical driving conditions and subject to exogenous disturbances was introduced in⁷⁷. The closed-form solitary wave solutions for the Chaffee–Infante equation were obtained and has many applications in mass transport and particle diffusion⁷⁸. An improved transient sub-domain analytical model was proposed to analyze the performance of a novel proposed hybrid excitation magnetic lead screw for the WEC system⁷⁹. An optimal control of an affine nonlinear system with unknown system dynamics was considered a new identifier-critic framework was proposed to solve the optimal control problem⁸⁰. The authors of⁸¹ investigated the nonlinear Zakharov system through a comprehensive analysis that integrates logarithmic transformations and the symbolic structures of exponential functions. The Heisenberg ferromagnetic spin chain (HFSC) equation was solved by a logarithmic transformation-based method⁸². The analysis of traveling wave solutions to a special kind of nonlinear Schrödinger equation with logarithmic nonlinearity was investigated and all traveling wave solutions were obtained⁸³. A general form of the multi-wave and soliton molecules in traveling wave form solutions were obtained to the linear structure of Sharma-Tasso-Olver-Burgers equation⁸⁴. The problem of nonfragile finite-time stabilization for linear discrete mean-field stochastic systems was studied⁸⁵.

The different forms of optical pulses in the nonlinear optical fiber to the higher order nonlinear Schrödinger equation were discussed⁸⁶. An inventive Ricatti equation mapping approach was used to solve the dynamic characteristics of the (3+1)-dimensional generalized equation governing shallow water waves⁸⁷. With the same method the (2+1)-dimensional Boussinesq equation has been analyzed to obtain the solitary wave solutions⁸⁸.

The investigation of the W-chirped solitons and modulation instability were addressed in nonlinear optic fibers to the modified cubic-quintic complex Ginzburg-Landau equation with parabolic law nonlinearity⁸⁹. Optical soliton solutions of the generalized non-autonomous nonlinear Schr?dinger equations were obtained using the new Kudryashov’s method⁹⁰. The modulation instability and the dynamics of solitary waves in a higher-order nonlinear Schrödinger equation were studied by the modified sub-equation method⁹¹.

This system makes a physical model that can be exerted to define various categories of nonlinear matters or mechanisms in physical and engineering sciences, optics, and electric communications. In this paper, some solutions including soliton, bright soliton, singular soliton, periodic wave solutions by the modified extended tanh-function method were also obtained.

Inspired by the previous work, the aim of the paper is to investigate the solitons and other form of solutions. The outline of the paper is as follows. In Section “Organized procedure of modified extended tanh-function method”, the modified extended tanh-function method is offered. In Section “Application of METFM”, different forms of solitary wave solutions for the (1+1)- and (2+1)-dimensional CNLS equations are established by the modified extended tanh-function method. Finally, the conclusions are provided in Section “Outcomes and visual presentation”.

Organized procedure of modified extended tanh-function method

A systematic strategy is used to solve a NLPDE in three variables:

Step 1.

$$\begin{aligned} \mathcal {S}_1(f, f_x, f_y, f_t, f_{xx}, f_{yy},...)=0. \end{aligned}$$

(3)

First step involves employing a wave transformation, in which f(x, y, t) can be expressed as $F(\xi )$, where $\xi =l_1x+l_2y-vt$. This transformation serves the purpose of converting the given NLPDE into ODE. Subsequently, we employ the METFM and integrate the concept of the METFM to proceed with the solution process.

$$\begin{aligned} \mathcal {S}_2(F, F', F', F'', F''',...)=0. \end{aligned}$$

(4)

Step 2. We look for Eq. (4) solutions in the form by METFM:

$$\begin{aligned} F(\xi )=\sum _{i=0}^{N}\mu _i\phi (\xi )^i+\sum _{i=1}^{N}\lambda _i\phi (\xi )^{-i}, \end{aligned}$$

(5)

where $\phi (\xi )$ satisfies

$$\begin{aligned} \phi '(\xi )=\varepsilon \sqrt{\sum _{i=0}^{4}w_i\phi (\xi )^{i}},\ \ \ \ \phi ''(\xi )=\frac{\varepsilon ^{2} \left( 4 w_{4} \phi \! \left( \xi \right) ^{3}+3 w_{3} \phi \! \left( \xi \right) ^{2}+2 \phi \! \left( \xi \right) w_{2}+w_{1}\right) }{2}, \end{aligned}$$

(6)

where $\varepsilon =\pm 1$. By applying the balance rule on Eq. (4), the integer N can be estimated. Substituting by Eq. (5) and Eq. (6) into Eq. (4). Then, a polynomial in $\phi (\xi )$ is recovered. Adding all terms with the same powers together and equating them to zero, we obtain a set of nonlinear equations that can be solved by Mathematica software packages to determine the unknown values of $\mu _i$ and $\lambda _i$.

Step 4. From the different possible values of $w_1, w_2, w_3$ and $w_4$, then Eq. (6) has many fundamental solutions as follows:

Case 1: When $w_0 = w_1 = w_3 = 0$, the following solutions are raised:

$$\begin{aligned} \left\{ \begin{array}{ll} \phi (\xi )=\sqrt{-\frac{w_2}{w_4}}\,\, sech(\xi \sqrt{w_2}), & w_2>0,\ \ w_4<0, \\ \phi (\xi )=\sqrt{\frac{w_2}{w_4}}\,\, sec(\xi \sqrt{w_2}), & w_2>0,\ \ w_4>0, \\ \phi (\xi )=\frac{-\varepsilon }{\xi \sqrt{w_4}}, & w_2=0,\ \ w_4>0. \end{array} \right. \end{aligned}$$

(7)

Case 2: When $w_1 = w_3 = 0$, the following solutions are raised:

$$\begin{aligned} \left\{ \begin{array}{llll} \phi (\xi )=\varepsilon \sqrt{-\frac{w_2}{2w_4}}\,\, \tanh (\xi \sqrt{-\frac{w_2}{2}}), & w_2<0,\ \ w_4>0, \ w_0=\frac{w_2^2}{4w_4}, \\ \phi (\xi )=\varepsilon \sqrt{\frac{w_2}{2w_4}}\,\, \tan (\xi \sqrt{\frac{w_2}{2}}), & w_2>0,\ \ w_4>0, \ w_0=\frac{w_2^2}{4w_4}, \\ \phi (\xi )=\sqrt{\frac{-w_2m^2}{(2m^2-1)w_4}}\,\, cn(\xi \sqrt{\frac{w_2}{2m^2-1}}), & w_2>0,\ \ w_4<0, \ w_0=\frac{w_2^2m^2(1-m^2)}{(2m^2-1)^2w_4}, \\ \phi (\xi )=\sqrt{\frac{-m^2w_2}{(2-m^2)w_4}}\,\, dn(\xi \sqrt{\frac{w_2}{2-m^2}}), & w_2>0,\ \ w_4<0, \ w_0=\frac{w_2^2(1-m^2)}{(2-m^2)^2w_4}, \\ \phi (\xi )=\sqrt{\frac{-w_2m^2}{(1+m^2)w_4}}\,\, sn(\xi \sqrt{\frac{-w_2}{1+m^2}}), & w_2<0,\ \ w_4>0, \ w_0=\frac{w_2^2m^2}{(1+m^2)^2w_4}, \\ \end{array} \right. \end{aligned}$$

(8)

where m is the modulus of the Jacobi elliptic functions and $0 < m \le 1$.

Case 3: When $w_0=w_1 = w_4 = 0$, the following solutions are raised:

$$\begin{aligned} \left\{ \begin{array}{ll} \phi (\xi )=\sqrt{-\frac{w_2}{w_3}}\,\, sech^2(\frac{\xi \sqrt{w_2}}{2}), & w_2>0, \\ \phi (\xi )=\sqrt{-\frac{w_2}{w_3}}\,\, sec^2(\frac{\xi \sqrt{-w_2}}{2}), & w_2<0, \\ \phi (\xi )=\frac{4}{\xi ^2w_3}, & w_2=0. \end{array} \right. \end{aligned}$$

(9)

Case 4: When $w_2 = w_4 = 0, w_0\ne 0, w_3>0$, the following solutions are raised:

$$\begin{aligned} \phi (\xi )=\wp \left[ \frac{\xi \sqrt{w_3}}{2},\mathfrak {A_2},\mathfrak {A_3}\right] , \end{aligned}$$

(10)

where $\mathfrak {A_2} = - \frac{4w_1}{w_3}$ and $\mathfrak {A_3} = - \frac{4w_0}{w_3}$ are called Weierstrass elliptic function in-variants.

Case 5: When $w_0=w_1 = w_2 = 0,$ the following solutions are raised:

$$\begin{aligned} \left\{ \begin{array}{ll} \phi (\xi )=\frac{4w_3}{w_3^2\xi ^2-4w_4},\,\,\, w_4\ne 0, \\ \phi (\xi )=\frac{w_3}{2w_4}\,\exp \left( \frac{\varepsilon \xi w_3}{2\sqrt{-w_4}}\right) ,\,\,\, w_4<0. \end{array} \right. \end{aligned}$$

(11)

Case 6: When $w_3 = w_4 = 0,$ the following solutions are raised:

$$\begin{aligned} \left\{ \begin{array}{ll} \phi (\xi )=-\frac{w_1}{2w_2}+\exp (\varepsilon \xi \sqrt{w_2}),\,\, w_2>0, \ w_0=\frac{w_1^2}{4w_2},\\ \phi (\xi )=-\frac{w_1}{2w_2}+\frac{\varepsilon w_1}{2w_2}\,\,\sin (\xi \sqrt{-w_2}),\,\, w_2<0, \ w_0=\frac{w_1^2}{4w_2},\\ \phi (\xi )=-\frac{w_1}{2w_2}+\frac{\varepsilon w_1}{2w_2}\,\,\sinh (2\xi \sqrt{w_2}),\,\, w_2>0, \ w_0=\frac{w_1^2}{4w_2},\\ \phi (\xi )=-\varepsilon \sqrt{-\frac{w_0}{w_2}}\,\,\sin (\xi \sqrt{-w_2}),\,\, w_2<0, \ w_0>0,\ w_1=0,\\ \phi (\xi )=\varepsilon \sqrt{\frac{w_0}{w_2}}\,\,\sinh (\xi \sqrt{w_2}),\,\, w_2>0, \ w_0>0,\ w_1=0. \end{array} \right. \end{aligned}$$

(12)

Case 7: When $w_0= w_1 = 0, w_4>0,$ the following solutions are raised:

$$\begin{aligned} \left\{ \begin{array}{ll} \phi (\xi )=-\frac{w_2sec^2\left( \frac{\xi \sqrt{-w_2}}{2}\right) }{2\varepsilon \sqrt{-w_2w_4} \tan \left( \frac{\xi \sqrt{-w_2}}{2}\right) +w_3},\,\, w_2<0, \\ \phi (\xi )=\frac{w_2sech^2\left( \frac{\xi \sqrt{w_2}}{2}\right) }{2\varepsilon \sqrt{-w_2w_4} \tanh \left( \frac{\xi \sqrt{w_2}}{2}\right) -w_3},\,\, w_2>0, \ \ w_3\ne 2\varepsilon \sqrt{w_2w_4},\\ \phi (\xi )=\frac{\varepsilon }{2}\sqrt{\frac{w_2}{w_4}}\left[ 1+\tanh \left( \frac{\xi \sqrt{w_2}}{2}\right) \right] ,\,\, w_2>0,\ \ w_3=2\varepsilon \sqrt{w_2w_4}. \end{array} \right. \end{aligned}$$

(13)

Application of METFM

In this context, we will look at how the METFM may be used to the (1+1) and (2+1)-dimensional chiral nonlinear Schrödinger equations within the framework of the ideal fluid model. After conducting the integration, we derive the following ODE by replacing the travelling wave variable into Eqs. (1) and (2).

Application of METFM on Eq. (1)

Take the following transformation

$$\begin{aligned} f(x,t)=F(\xi )\,e^{i\eta },\,\, with\,\, \xi =l(x+vt),\,\,\, \eta =s_1x+s_2t+\eta _0. \end{aligned}$$

(14)

Inserting Eq. (14) into Eq. (1), we receive

$$\begin{aligned} v=-2s_1, \end{aligned}$$

(15)

as imaginary part and as real part is obtained below

$$\begin{aligned} lF''+2s_1\sigma \,F^3-(s_1^2+s_2)F=0. \end{aligned}$$

(16)

Equating the terms $F''$ and $F^3$ yields $N=1$. As a result, we have the flexibility to choose $F(\xi )$ from Eq. (5);

$$\begin{aligned} F(\xi )=\mu _0+\mu _1\,\phi (\xi )+\lambda _1\,(\phi (\xi ))^{-1}. \end{aligned}$$

(17)

Where $\mu _0,\mu _1$ and $\lambda _1$ are variables that will be determined. We build a set of algebraic equations involving the variables $\mu _0,\mu _1,\lambda _1,w_1,w_2,w_3$ and $w_4$ by substituting Eq. (17) with Eq. (6) in Eq. (16) and then setting the coefficients of $\phi ^i(\xi )$ to zero.

$$\begin{aligned} \left\{ \begin{array}{ll} -3\,{\epsilon }^{2}l\lambda _{{1}}w_{{1}}-12\,\sigma \,{\lambda _{{1}}}^{2 }\mu _{{0}}s_{{1}}=0,\ \ \ \ \ \ -2\,{\epsilon }^{2}l\lambda _{{1}}w_{{2}}-12\,\sigma \,{\lambda _{{1}}}^{2 }\mu _{{1}}s_{{1}}-12\,\sigma \,\lambda _{{1}}{\mu _{{0}}}^{2}s_{{1}}-2\, \lambda _{{1}}{s_{{1}}}^{2}-2\,\lambda _{{1}}s_{{2}}=0,\\ -{\epsilon }^{2}l\lambda _{{1}}w_{{3}}-{\epsilon }^{2}l\mu _{{1}}w_{{1}}- 24\,\sigma \,\lambda _{{1}}\mu _{{0}}\mu _{{1}}s_{{1}}-4\,\sigma \,{\mu _{{0 }}}^{3}s_{{1}}-2\,\mu _{{0}}{s_{{1}}}^{2}-2\,\mu _{{0}}s_{{2}}=0,\\ -2\,{\epsilon }^{2}l\mu _{{1}}w_{{2}}-12\,\sigma \,\lambda _{{1}}{\mu _{{1} }}^{2}s_{{1}}-12\,\sigma \,{\mu _{{0}}}^{2}\mu _{{1}}s_{{1}}-2\,\mu _{{1}} {s_{{1}}}^{2}-2\,\mu _{{1}}s_{{2}} =0,\\ -4\,{\epsilon }^{2}l\mu _{{1}}w_{{4}}-4\,\sigma \,{\mu _{{1}}}^{3}s_{{1}}=0,\ \ -3\,{\epsilon }^{2}l\mu _{{1}}w_{{3}}-12\,\sigma \,\mu _{{0}}{\mu _{{1}}}^{ 2}s_{{1}}=0,\ \ \ -4\,{\epsilon }^{2}l\lambda _{{1}}w_{{0}}-4\,\sigma \,{\lambda _{{1}}}^{3} s_{{1}}=0. \end{array} \right. \end{aligned}$$

(18)

We use Maple to solve this system of algebraic equations and get following values of those unknown parameters:

$$\begin{aligned} \lambda _1=0,\ \ s_{{1}}=-{\frac{{\epsilon }^{2}lw_{{4}}}{\sigma \,{\mu _{{1}}}^{2}}},\ \ s_{{2}}=-{\frac{l{\epsilon }^{2} \left( -6\,{\sigma }^{2}{\mu _{{0}}}^{2 }{\mu _{{1}}}^{2}w_{{4}}+{\sigma }^{2}{\mu _{{1}}}^{4}w_{{2}}+{\epsilon }^ {2}l{w_{{4}}}^{2} \right) }{{\sigma }^{2}{\mu _{{1}}}^{4}}},\ \ \end{aligned}$$

(19)

$$\begin{aligned} w_{{1}}=-2\,{\frac{\mu _{{0}} \left( 4\,{\mu _{{0}}}^{2}w_{{4}}-{\mu _{{ 1}}}^{2}w_{{2}} \right) }{{\mu _{{1}}}^{3}}},\ \ w_{{3}}=4\,{\frac{\mu _{{0}}w_{{4}}}{\mu _{{1}}}}. \end{aligned}$$

First class of solutions (according to parameters (19)):

$$\begin{aligned} \left\{ \begin{array}{ll} Case\,\, I:\Rightarrow \,\,\mu _0=0,\,\,\, f_{1,1}(\xi ,\eta )=\mu _1\sqrt{-\frac{w_2}{w_4}}\,\, sech(\xi \sqrt{w_2})\,e^{i\eta }, \,\,\, \ w_2>0,\ \ w_4<0, \\ f_{1,2}=\mu _1\sqrt{\frac{w_2}{w_4}}\,\, sec(\xi \sqrt{w_2})\,e^{i\eta }, \ \ \ w_2>0,\ \ w_4>0, \\ f_{1,3}=\mu _1\frac{-\varepsilon }{\xi \sqrt{w_4}}\,e^{i\eta }, \ \ w_2=0,\ \ \ \ \xi =l(x+{\frac{2{\epsilon }^{2}lw_{{4}}}{\sigma \,{\mu _{{1}}}^{2}}}t),\,\,\, {\eta =-{\frac{{\epsilon }^{2}lw_{{4}}}{\sigma \,{\mu _{{1}}}^{2}}}x-{\frac{l{\epsilon }^{2} \left( {\sigma }^{2}{\mu _{{1}}}^{4}w_{ {2}}+{\epsilon }^{2}l{w_{{4}}}^{2} \right) }{{\sigma }^{2}{\mu _{{1}}}^{4 }}} t+\eta _0,} \end{array} \right. \end{aligned}$$

(20)

$$\begin{aligned} \left\{ \begin{array}{ll} Case\,\, II:\Rightarrow \,\,\mu _0=0,\,\,\, f_{1,4}(\xi ,\eta )=\mu _1\varepsilon \sqrt{-\frac{w_2}{2w_4}}\,\, \tanh (\xi \sqrt{-\frac{w_2}{2}})\,e^{i\eta },\,\ \ w_2<0,\ \ w_4>0, \ w_0=\frac{w_2^2}{4w_4}, \\ f_{1,5}(\xi ,\eta )=\mu _1\varepsilon \sqrt{\frac{w_2}{2w_4}}\,\, \tan (\xi \sqrt{\frac{w_2}{2}})\,e^{i\eta },\, \ \ w_2>0,\ \ w_4>0, \ w_0=\frac{w_2^2}{4w_4}, \\ f_{1,6}(\xi ,\eta )=\mu _1\sqrt{\frac{-w_2m^2}{(2m^2-1)w_4}}\,\, cn(\xi \sqrt{\frac{w_2}{2m^2-1}})\,e^{i\eta },\, \ \ w_2>0,\ \ w_4<0, \ w_0=\frac{w_2^2m^2(1-m^2)}{(2m^2-1)^2w_4}, \\ f_{1,7}(\xi ,\eta )=\mu _1\sqrt{\frac{-m^2w_2}{(2-m^2)w_4}}\,\, dn(\xi \sqrt{\frac{w_2}{2-m^2}})\,e^{i\eta },\, \ \ w_2>0,\ \ w_4<0, \ w_0=\frac{w_2^2(1-m^2)}{(2-m^2)^2w_4}, \\ f_{1,8}(\xi ,\eta )=\mu _1\sqrt{\frac{-w_2m^2}{(1+m^2)w_4}}\,\, sn(\xi \sqrt{\frac{-w_2}{1+m^2}})\,e^{i\eta },\, \ \ w_2<0,\ \ w_4>0, \ w_0=\frac{w_2^2m^2}{(1+m^2)^2w_4}, \\ \ \ \ \ \xi =l(x+{\frac{2{\epsilon }^{2}lw_{{4}}}{\sigma \,{\mu _{{1}}}^{2}}}t),\,\,\, \eta =-{\frac{{\epsilon }^{2}lw_{{4}}}{\sigma \,{\mu _{{1}}}^{2}}}x-{\frac{l{\epsilon }^{2} \left( {\sigma }^{2}{\mu _{{1}}}^{4}w_{ {2}}+{\epsilon }^{2}l{w_{{4}}}^{2} \right) }{{\sigma }^{2}{\mu _{{1}}}^{4 }}} t+\eta _0, \end{array} \right. \end{aligned}$$

(21)

where m is the modulus of the Jacobi elliptic functions and $0 < m \le 1$.

Remark 1

When $m\rightarrow 1$, then $cn(\xi , m)\rightarrow { sech}(\xi )$ hence the bright soliton solution (21) will be as

$$\begin{aligned} f_{1,6}(x,t)=\mu _1\sqrt{\frac{-w_2}{w_4}}\,\, { sech}\left( l\sqrt{w_2}(x+{\frac{2{\epsilon }^{2}lw_{{4}}}{\sigma \,{\mu _{{1}}}^{2}}}t)\right) \,e^{i\left[ -{\frac{{\epsilon }^{2}lw_{{4}}}{\sigma \,{\mu _{{1}}}^{2}}}x-{\frac{l{\epsilon }^{2} \left( {\sigma }^{2}{\mu _{{1}}}^{4}w_{ {2}}+{\epsilon }^{2}l{w_{{4}}}^{2} \right) }{{\sigma }^{2}{\mu _{{1}}}^{4 }}} t+\eta _0\right] },\ \ \ \ w_2w_4<0. \end{aligned}$$

(22)

Remark 2

When $m\rightarrow 1$, then $dn(\xi , m)\rightarrow { sech}(\xi )$ hence the bright soliton solution (21) will be as

$$\begin{aligned} f_{1,7}(x,t)=\mu _1\sqrt{\frac{-w_2}{w_4}}\,\, { sech}\left( l\sqrt{w_2}(x+{\frac{2{\epsilon }^{2}lw_{{4}}}{\sigma \,{\mu _{{1}}}^{2}}}t)\right) \,e^{i\left[ -{\frac{{\epsilon }^{2}lw_{{4}}}{\sigma \,{\mu _{{1}}}^{2}}}x-{\frac{l{\epsilon }^{2} \left( {\sigma }^{2}{\mu _{{1}}}^{4}w_{ {2}}+{\epsilon }^{2}l{w_{{4}}}^{2} \right) }{{\sigma }^{2}{\mu _{{1}}}^{4 }}} t+\eta _0\right] },\ \ \ \ w_2w_4<0. \end{aligned}$$

(23)

Remark 3

When $m\rightarrow 1$, then $sn(\xi , m)\rightarrow { tanh}(\xi )$ hence the dark soliton solution (21) will be as

$$\begin{aligned} f_{1,8}(x,t)=\mu _1\sqrt{\frac{-w_2}{2w_4}}\,\, { tanh}\left( l\sqrt{-\frac{w_2}{2}}(x+{\frac{2{\epsilon }^{2}lw_{{4}}}{\sigma \,{\mu _{{1}}}^{2}}}t)\right) \,e^{i\left[ -{\frac{{\epsilon }^{2}lw_{{4}}}{\sigma \,{\mu _{{1}}}^{2}}}x-{\frac{l{\epsilon }^{2} \left( {\sigma }^{2}{\mu _{{1}}}^{4}w_{ {2}}+{\epsilon }^{2}l{w_{{4}}}^{2} \right) }{{\sigma }^{2}{\mu _{{1}}}^{4 }}} t+\eta _0\right] },\ \ \ w_2<0,\ \ \ w_2w_4<0. \end{aligned}$$

(24)

$$\begin{aligned} \left\{ \begin{array}{ll} Case\,\, V:\Rightarrow \,\,\mu _0=0,\,\,\, f_{1,9}(\xi ,\eta )=\mu _1\frac{4w_3}{w_3^2\xi ^2-4w_4}\,e^{i\eta },\,\, \ \ w_4\ne 0, \ \ \ \ \ f_{1,10}(\xi ,\eta )=\mu _1\frac{w_3}{2w_4}\,\exp \left( \frac{\varepsilon \xi w_3}{2\sqrt{-w_4}}\right) \,e^{i\eta },\,\, \ \ w_4<0, \\ \xi =l(x+{\frac{2{\epsilon }^{2}lw_{{4}}}{\sigma \,{\mu _{{1}}}^{2}}}t),\,\,\, {\eta =-{\frac{{\epsilon }^{2}lw_{{4}}}{\sigma \,{\mu _{{1}}}^{2}}}x-{\frac{{l}^{2}{\epsilon }^{4}{w_{{4}}}^{2}}{{\sigma }^{2}{\mu _ {{1}}}^{4}}} t+\eta _0,} \end{array} \right. \end{aligned}$$

(25)

$$\begin{aligned} \left\{ \begin{array}{ll} Case\,\, VII:\Rightarrow \,\,\frac{w_2}{w_4}=\frac{4\mu _0^2}{\mu _1^2},\,\,\, f_{1,11}(\xi ,\eta )=\left( \mu _0-\frac{w_2sec^2\left( \frac{\xi \sqrt{-w_2}}{2}\right) }{2\varepsilon \sqrt{-w_2w_4} \tan \left( \frac{\xi \sqrt{-w_2}}{2}\right) +w_3}\right) \,e^{i\eta },\,\, w_2<0, \\ f_{1,12}(\xi ,\eta )=\left( \mu _0-\mu _1\frac{w_2sech^2\left( \frac{\xi \sqrt{w_2}}{2}\right) }{2\varepsilon \sqrt{-w_2w_4} \tanh \left( \frac{\xi \sqrt{w_2}}{2}\right) -w_3}\right) \,e^{i\eta },\,\, w_2>0, \ \ w_3\ne 2\varepsilon \sqrt{w_2w_4},\\ f_{1,13}(\xi ,\eta )=\left( \mu _0-\mu _1\frac{\varepsilon }{2}\sqrt{\frac{w_2}{w_4}}\left[ 1+\tanh \left( \frac{\xi \sqrt{w_2}}{2}\right) \right] \right) \,e^{i\eta },\,\, w_2>0,\ \ w_3=2\varepsilon \sqrt{w_2w_4},\\ \ \ \ \ \xi =l(x+{\frac{2{\epsilon }^{2}lw_{{4}}}{\sigma \,{\mu _{{1}}}^{2}}}t),\,\,\, {\eta =-{\frac{{\epsilon }^{2}lw_{{4}}}{\sigma \,{\mu _{{1}}}^{2}}}x-{\frac{l{\epsilon }^{2} \left( -6\,{\sigma }^{2}{\mu _{{0}}}^{2}{\mu _{{ 1}}}^{2}w_{{4}}+{\sigma }^{2}{\mu _{{1}}}^{4}w_{{2}}+{\epsilon }^{2}l{w_{ {4}}}^{2} \right) }{{\sigma }^{2}{\mu _{{1}}}^{4}}} t+\eta _0,} \end{array} \right. \end{aligned}$$

(26)

$$\begin{aligned} \mu _1=0,\ \ s_{{1}}=-{\frac{{\epsilon }^{2}lw_{{0}}}{\sigma \,{\lambda _{{1}}}^{2}}},\ \ s_{{2}}=-{\frac{l{\epsilon }^{2} \left( {\sigma }^{2}{\lambda _{{1}}}^{4}w_{{2}} -6\,{\sigma }^{2}{\lambda _{{1}}}^{2}{\mu _{{0}}}^{2}w_{{0}}+{\epsilon }^{ 2}l{w_{{0}}}^{2} \right) }{{\sigma }^{2}{\lambda _{{1}}}^{4}}},\ \ \end{aligned}$$

(27)

$$\begin{aligned} w_{{1}}=4\,{\frac{\mu _{{0}}w_{{0}}}{\lambda _{{1}}}},\ \ w_{{3}}=2\,{\frac{\mu _{{0}} \left( {\lambda _{{1}}}^{2}w_{{2}}-4\,{\mu _{{0}}}^{2}w_{{0}} \right) }{{\lambda _{{1}}}^{3}}}. \end{aligned}$$

Second class of solutions (according to parameters (27)):

$$\begin{aligned} \left\{ \begin{array}{ll} Case\,\, II:\Rightarrow \,\,\mu _0=0,\,\,\, f_{2,1}(\xi ,\eta )=\mu _1\left[ \varepsilon \sqrt{-\frac{w_2}{2w_4}}\,\, \tanh (\xi \sqrt{-\frac{w_2}{2}})\right] ^{-1}\,e^{i\eta },\, \& w_2<0,\ \ w_4>0, \ w_0=\frac{w_2^2}{4w_4}, \\ f_{2,2}(\xi ,\eta )=\mu _1\left[ \varepsilon \sqrt{\frac{w_2}{2w_4}}\,\, \tan (\xi \sqrt{\frac{w_2}{2}})\right] ^{-1}\,e^{i\eta },\, \& w_2>0,\ \ w_4>0, \ w_0=\frac{w_2^2}{4w_4}, \\ f_{2,3}(\xi ,\eta )=\mu _1\left[ \sqrt{\frac{-w_2m^2}{(2m^2-1)w_4}}\,\, cn(\xi \sqrt{\frac{w_2}{2m^2-1}})\right] ^{-1}\,e^{i\eta },\, \& w_2>0,\ \ w_4<0, \ w_0=\frac{w_2^2m^2(1-m^2)}{(2m^2-1)^2w_4}, \\ f_{2,4}(\xi ,\eta )=\mu _1\left[ \sqrt{\frac{-m^2w_2}{(2-m^2)w_4}}\,\, dn(\xi \sqrt{\frac{w_2}{2-m^2}})\right] ^{-1}\,e^{i\eta },\, \& w_2>0,\ \ w_4<0, \ w_0=\frac{w_2^2(1-m^2)}{(2-m^2)^2w_4}, \\ f_{2,5}(\xi ,\eta )=\mu _1\left[ \sqrt{\frac{-w_2m^2}{(1+m^2)w_4}}\,\, sn(\xi \sqrt{\frac{-w_2}{1+m^2}})\right] ^{-1}\,e^{i\eta },\, \& w_2<0,\ \ w_4>0, \ w_0=\frac{w_2^2m^2}{(1+m^2)^2w_4}, \\ \ \ \ \ \xi =l(x+{\frac{2{\epsilon }^{2}lw_{{0}}}{\sigma \,{\lambda _{{1}}}^{2}}}t),\,\,\, \eta =-{\frac{{\epsilon }^{2}lw_{{0}}}{\sigma \,{\lambda _{{1}}}^{2}}}x-{\frac{l{\epsilon }^{2} \left( {\sigma }^{2}{\lambda _{{1}}}^{4}w_{{2}} -6\,{\sigma }^{2}{\lambda _{{1}}}^{2}{\mu _{{0}}}^{2}w_{{0}}+{\epsilon }^{ 2}l{w_{{0}}}^{2} \right) }{{\sigma }^{2}{\lambda _{{1}}}^{4}}} t+\eta _0, \end{array} \right. \end{aligned}$$

(28)

where m is the modulus of the Jacobi elliptic functions and $0 < m \le 1$.

Remark 4

When $m\rightarrow 1$, then $cn(\xi , m)\rightarrow { sech}(\xi )$ hence the bright soliton solution (28) will be as

$$\begin{aligned} f_{2,3}(x,t)=\mu _1\sqrt{\frac{-w_2}{w_4}}\,\, { sech}\left( l\sqrt{w_2}(x+{\frac{2{\epsilon }^{2}lw_{{0}}}{\sigma \,{\lambda _{{1}}}^{2}}}t)\right) \,e^{i\left[ -{\frac{{\epsilon }^{2}lw_{{0}}}{\sigma \,{\lambda _{{1}}}^{2}}}x-{\frac{l{\epsilon }^{2} \left( {\sigma }^{2}{\lambda _{{1}}}^{4}w_{{2}} -6\,{\sigma }^{2}{\lambda _{{1}}}^{2}{\mu _{{0}}}^{2}w_{{0}}+{\epsilon }^{ 2}l{w_{{0}}}^{2} \right) }{{\sigma }^{2}{\lambda _{{1}}}^{4}}} t+\eta _0\right] },\ \ \ \ w_2w_4<0. \end{aligned}$$

(29)

Remark 5

When $m\rightarrow 1$, then $dn(\xi , m)\rightarrow { sech}(\xi )$ hence the bright soliton solution (28) will be as

$$\begin{aligned} f_{2,4}(x,t)=\mu _1\sqrt{\frac{-w_2}{w_4}}\,\, { sech}\left( l\sqrt{w_2}(x+{\frac{2{\epsilon }^{2}lw_{{0}}}{\sigma \,{\lambda _{{1}}}^{2}}}t)\right) \,e^{i\left[ -{\frac{{\epsilon }^{2}lw_{{0}}}{\sigma \,{\lambda _{{1}}}^{2}}}x-{\frac{l{\epsilon }^{2} \left( {\sigma }^{2}{\lambda _{{1}}}^{4}w_{{2}} -6\,{\sigma }^{2}{\lambda _{{1}}}^{2}{\mu _{{0}}}^{2}w_{{0}}+{\epsilon }^{ 2}l{w_{{0}}}^{2} \right) }{{\sigma }^{2}{\lambda _{{1}}}^{4}}} t+\eta _0\right] },\ \ \ \ w_2w_4<0. \end{aligned}$$

(30)

Remark 6

When $m\rightarrow 1$, then $sn(\xi , m)\rightarrow { tanh}(\xi )$ hence the dark soliton solution (28) will be as

$$\begin{aligned} f_{2,5}(x,t)=\mu _1\sqrt{\frac{-w_2}{2w_4}}\,\, { tanh}\left( l\sqrt{-\frac{w_2}{2}}(x+{\frac{2{\epsilon }^{2}lw_{{0}}}{\sigma \,{\lambda _{{1}}}^{2}}}t)\right) \,e^{i\left[ -{\frac{{\epsilon }^{2}lw_{{0}}}{\sigma \,{\lambda _{{1}}}^{2}}}x-{\frac{l{\epsilon }^{2} \left( {\sigma }^{2}{\lambda _{{1}}}^{4}w_{{2}} -6\,{\sigma }^{2}{\lambda _{{1}}}^{2}{\mu _{{0}}}^{2}w_{{0}}+{\epsilon }^{ 2}l{w_{{0}}}^{2} \right) }{{\sigma }^{2}{\lambda _{{1}}}^{4}}} t+\eta _0\right] }, \end{aligned}$$

(31)

$$\begin{aligned} \ \ \ w_2<0,\ \ \ w_2w_4<0. \end{aligned}$$

$$\begin{aligned} \left\{ \begin{array}{ll} Case\,\, IV:\Rightarrow \,\,\,\, f_{2,6}(\xi ,\eta )=\left\{ \mu _0+\mu _1\wp ^{-1}\left[ \frac{\xi \sqrt{w_3}}{2},\mathfrak {A_2},\mathfrak {A_3}\right] \right\} \,e^{i\eta },\ \\ \xi =l(x+{\frac{2{\epsilon }^{2}lw_{{0}}}{\sigma \,{\lambda _{{1}}}^{2}}}t),\,\,\, \eta =-{\frac{{\epsilon }^{2}lw_{{0}}}{\sigma \,{\lambda _{{1}}}^{2}}}x-{\frac{l{\epsilon }^{2} \left( -6\,{\sigma }^{2}{\lambda _{{1}}}^{2}{\mu _{{0}}}^{2}w_{{0}}+{\epsilon }^{ 2}l{w_{{0}}}^{2} \right) }{{\sigma }^{2}{\lambda _{{1}}}^{4}}} t+\eta _0, \end{array} \right. \end{aligned}$$

(32)

where $\mathfrak {A_2} = - \frac{4w_1}{w_3}$ and $\mathfrak {A_3} = - \frac{4w_0}{w_3}$ are called Weierstrass elliptic function in-variants.

$$\begin{aligned} \left\{ \begin{array}{ll} Case\,\, VI:\Rightarrow \,\,\frac{w_2}{w_0}=\frac{4\mu _0^2}{\lambda _1^2},\,\,\, f_{2,7}(\xi ,\eta )=\left[ \mu _0+\mu _1\left( -\frac{w_1}{2w_2}+\exp (\varepsilon \xi \sqrt{w_2})\right) ^{-1}\right] \,e^{i\eta },\,\,\, w_2>0, \ w_0=\frac{w_1^2}{4w_2},\\ f_{2,8}(\xi ,\eta )=\left[ \mu _0+\mu _1\left( -\frac{w_1}{2w_2}+\frac{\varepsilon w_1}{2w_2}\,\,\sin (\xi \sqrt{-w_2})\right) ^{-1}\right] \,e^{i\eta },\,\,\, w_2<0, \ w_0=\frac{w_1^2}{4w_2},\\ f_{2,9}(\xi ,\eta )=\left[ \mu _0+\mu _1\left( -\frac{w_1}{2w_2}+\frac{\varepsilon w_1}{2w_2}\,\,\sinh (2\xi \sqrt{w_2})\right) ^{-1}\right] \,e^{i\eta },\,\,\, w_2>0, \ w_0=\frac{w_1^2}{4w_2},\\ f_{2,10}(\xi ,\eta )=\left[ \mu _0+\mu _1\left( -\varepsilon \sqrt{-\frac{w_0}{w_2}}\,\,\sin (\xi \sqrt{-w_2})\right) ^{-1}\right] \,e^{i\eta },\,\,\, w_2<0, \ w_0>0,\ w_1=0,\\ f_{2,11}(\xi ,\eta )=\left[ \mu _0+\mu _1\left( \varepsilon \sqrt{\frac{w_0}{w_2}}\,\,\sinh (\xi \sqrt{w_2})\right) ^{-1}\right] \,e^{i\eta },\,\,\, w_2>0, \ w_0>0,\,\\ \xi =l(x+{\frac{2{\epsilon }^{2}lw_{{0}}}{\sigma \,{\lambda _{{1}}}^{2}}}t),\,\,\, \eta =-{\frac{{\epsilon }^{2}lw_{{0}}}{\sigma \,{\lambda _{{1}}}^{2}}}x-{\frac{l{\epsilon }^{2} \left( {\sigma }^{2}{\lambda _{{1}}}^{4}w_{{2}} -6\,{\sigma }^{2}{\lambda _{{1}}}^{2}{\mu _{{0}}}^{2}w_{{0}}+{\epsilon }^{ 2}l{w_{{0}}}^{2} \right) }{{\sigma }^{2}{\lambda _{{1}}}^{4}}} t+\eta _0, \end{array} \right. \end{aligned}$$

(33)

$$\begin{aligned} \mu _0=0,\ \ s_{{1}}=-{\frac{{\epsilon }^{2}lw_{{0}}}{\sigma \,{\lambda _{{1}}}^{2}}},\ \ s_{{2}}=-{\frac{l{\epsilon }^{2} \left( {\sigma }^{2}{\lambda _{{1}}}^{4}w_{{2}} -6\,{\sigma }^{2}{\lambda _{{1}}}^{2}{\mu _{{1}}}^{2}w_{{0}}+{\epsilon }^{ 2}l{w_{{0}}}^{2} \right) }{{\sigma }^{2}{\lambda _{{1}}}^{4}}},\ \ w_{{1}}=0,\ \ \ w_{{4}}={\frac{{\mu _{{1}}}^{2}w_{{0}}}{{\lambda _{{1}}}^{2}}}. \end{aligned}$$

(34)

Third class of solutions (according to parameters (34)):

$$\begin{aligned} \left\{ \begin{array}{ll} Case\,\, II:\Rightarrow \,\, f_{3,1}(\xi ,\eta )=[\mu _1\phi (\xi )+\lambda _1\phi ^{-1}(\xi )]\,e^{i\eta },\,\,\,\ \ \phi (\xi )=\varepsilon \sqrt{-\frac{w_2}{2w_4}}\,\, \tanh (\xi \sqrt{-\frac{w_2}{2}}), \& w_2<0,\ \ w_4>0, \ w_0=\frac{w_2^2}{4w_4}, \\ f_{3,2}(\xi ,\eta )=[\mu _1\phi (\xi )+\lambda _1\phi ^{-1}(\xi )]\,e^{i\eta },\,\,\,\ \ \phi (\xi )=\varepsilon \sqrt{\frac{w_2}{2w_4}}\,\, \tan (\xi \sqrt{\frac{w_2}{2}}), \& w_2>0,\ \ w_4>0, \ w_0=\frac{w_2^2}{4w_4}, \\ f_{3,3}(\xi ,\eta )=[\mu _1\phi (\xi )+\lambda _1\phi ^{-1}(\xi )]\,e^{i\eta },\,\,\,\ \ \phi (\xi )=\sqrt{\frac{-w_2m^2}{(2m^2-1)w_4}}\,\, cn(\xi \sqrt{\frac{w_2}{2m^2-1}}), \& w_2>0,\ \ w_4<0, \ w_0=\frac{w_2^2m^2(1-m^2)}{(2m^2-1)^2w_4}, \\ f_{3,4}(\xi ,\eta )=[\mu _1\phi (\xi )+\lambda _1\phi ^{-1}(\xi )]\,e^{i\eta },\,\,\,\ \ \phi (\xi )=\sqrt{\frac{-m^2w_2}{(2-m^2)w_4}}\,\, dn(\xi \sqrt{\frac{w_2}{2-m^2}}), \& w_2>0,\ \ w_4<0, \ w_0=\frac{w_2^2(1-m^2)}{(2-m^2)^2w_4}, \\ f_{3,5}(\xi ,\eta )=[\mu _1\phi (\xi )+\lambda _1\phi ^{-1}(\xi )]\,e^{i\eta },\,\,\,\ \ \phi (\xi )=\sqrt{\frac{-w_2m^2}{(1+m^2)w_4}}\,\, sn(\xi \sqrt{\frac{-w_2}{1+m^2}}), \& w_2<0,\ \ w_4>0, \ w_0=\frac{w_2^2m^2}{(1+m^2)^2w_4}, \\ \xi =l(x+{\frac{2{\epsilon }^{2}lw_{{0}}}{\sigma \,{\lambda _{{1}}}^{2}}}t),\,\,\, \eta =-{\frac{{\epsilon }^{2}lw_{{0}}}{\sigma \,{\lambda _{{1}}}^{2}}}x-{\frac{l{\epsilon }^{2} \left( {\sigma }^{2}{\lambda _{{1}}}^{4}w_{{2}} -6\,{\sigma }^{2}{\lambda _{{1}}}^{2}{\mu _{{1}}}^{2}w_{{0}}+{\epsilon }^{ 2}l{w_{{0}}}^{2} \right) }{{\sigma }^{2}{\lambda _{{1}}}^{4}}} t+\eta _0, \end{array} \right. \end{aligned}$$

(35)

where m is the modulus of the Jacobi elliptic functions and $0 < m \le 1$.

Remark 7

When $m\rightarrow 1$, then $cn(\xi , m)\rightarrow { sech}(\xi )$ hence the combined bright-singular solution (35) will be as

$$\begin{aligned} f_{3,3}(x,t)= & \left\{ \mu _1\sqrt{\frac{-w_2}{w_4}}\,\, { sech}\left( l\sqrt{w_2}(x+{\frac{2{\epsilon }^{2}lw_{{0}}}{\sigma \,{\lambda _{{1}}}^{2}}}t)\right) + \lambda _1\sqrt{\frac{-w_4}{w_2}}\,\, { csch}\left( l\sqrt{w_2}(x+{\frac{2{\epsilon }^{2}lw_{{0}}}{\sigma \,{\lambda _{{1}}}^{2}}}t)\right) \right\} \nonumber \\ & \times \,e^{i\left[ -{\frac{{\epsilon }^{2}lw_{{0}}}{\sigma \,{\lambda _{{1}}}^{2}}}x-{\frac{l{\epsilon }^{2} \left( {\sigma }^{2}{\lambda _{{1}}}^{4}w_{{2}} -6\,{\sigma }^{2}{\lambda _{{1}}}^{2}{\mu _{{1}}}^{2}w_{{0}}+{\epsilon }^{ 2}l{w_{{0}}}^{2} \right) }{{\sigma }^{2}{\lambda _{{1}}}^{4}}} t+\eta _0\right] },\ \ \ \ w_2w_4<0. \end{aligned}$$

(36)

Remark 8

When $m\rightarrow 1$, then $dn(\xi , m)\rightarrow { sech}(\xi )$ hence the combined bright-singular solution (35) will be as

$$\begin{aligned} f_{3,4}(x,t)= & \left\{ \mu _1\sqrt{\frac{-w_2}{w_4}}\,\, { sech}\left( l\sqrt{w_2}(x+{\frac{2{\epsilon }^{2}lw_{{0}}}{\sigma \,{\lambda _{{1}}}^{2}}}t)\right) + \lambda _1\sqrt{\frac{-w_4}{w_2}}\,\, { csch}\left( l\sqrt{w_2}(x+{\frac{2{\epsilon }^{2}lw_{{0}}}{\sigma \,{\lambda _{{1}}}^{2}}}t)\right) \right\} \nonumber \\ & \times \,e^{i\left[ -{\frac{{\epsilon }^{2}lw_{{0}}}{\sigma \,{\lambda _{{1}}}^{2}}}x-{\frac{l{\epsilon }^{2} \left( {\sigma }^{2}{\lambda _{{1}}}^{4}w_{{2}} -6\,{\sigma }^{2}{\lambda _{{1}}}^{2}{\mu _{{1}}}^{2}w_{{0}}+{\epsilon }^{ 2}l{w_{{0}}}^{2} \right) }{{\sigma }^{2}{\lambda _{{1}}}^{4}}} t+\eta _0\right] },\ \ \ \ w_2w_4<0. \end{aligned}$$

(37)

Remark 9

When $m\rightarrow 1$, then $sn(\xi , m)\rightarrow { tanh}(\xi )$ hence the combined dark-singular solution (35) will be as

$$\begin{aligned} f_{3,5}(x,t)= & \left\{ \mu _1\sqrt{\frac{-w_2}{2w_4}}\,\, { tanh}\left( l\sqrt{-\frac{w_2}{2}}(x+{\frac{2{\epsilon }^{2}lw_{{0}}}{\sigma \,{\lambda _{{1}}}^{2}}}t)\right) + \lambda _1\sqrt{\frac{-2w_2}{w_4}}\,\, { coth}\left( l\sqrt{-\frac{w_2}{2}}(x+{\frac{2{\epsilon }^{2}lw_{{0}}}{\sigma \,{\lambda _{{1}}}^{2}}}t)\right) \right\} \nonumber \\ & \times \,e^{i\left[ -{\frac{{\epsilon }^{2}lw_{{0}}}{\sigma \,{\lambda _{{1}}}^{2}}}x-{\frac{l{\epsilon }^{2} \left( {\sigma }^{2}{\lambda _{{1}}}^{4}w_{{2}} -6\,{\sigma }^{2}{\lambda _{{1}}}^{2}{\mu _{{1}}}^{2}w_{{0}}+{\epsilon }^{ 2}l{w_{{0}}}^{2} \right) }{{\sigma }^{2}{\lambda _{{1}}}^{4}}} t+\eta _0\right] },\ \ \ \ w_2<0,\ \ w_2w_4<0. \end{aligned}$$

(38)

$$\begin{aligned} s_{{1}}= & -{\frac{{\epsilon }^{2}lw_{{0}}}{\sigma \,{\lambda _{{1}}}^{2}}},\ \ s_{{2}}=-{\frac{{\epsilon }^{2}lw_{{0}} \left( -8\,{\sigma }^{2}{ \lambda _{{1}}}^{3}\mu _{{1}}-2\,{\sigma }^{2}{\lambda _{{1}}}^{2}{\mu _{{0 }}}^{2}+{\epsilon }^{2}lw_{{0}} \right) }{{\sigma }^{2}{\lambda _{{1}}}^{ 4}}},\ \ w_{{1}}=4\,{\frac{\mu _{{0}}w_{{0}}}{\lambda _{{1}}}},\ \nonumber \\ w_{{2}}= & -2\,{\frac{w_{{0}} \left( \lambda _{{1}}\mu _{{1}}-2\,{\mu _{{0} }}^{2} \right) }{{\lambda _{{1}}}^{2}}},\ \ w_{{3}}=4\,{\frac{\mu _{{0}}\mu _{{1}}w_{{0}}}{{\lambda _{{1}}}^{2}}},\ \ w_{{4}}={\frac{{\mu _{{1}}}^{2}w_{{0}}}{{\lambda _{{1}}}^{2}}}. \end{aligned}$$

(39)

Fourth class of solutions (according to parameters (39)):

$$\begin{aligned} \left\{ \begin{array}{ll} Case\,\, II:\Rightarrow \,\mu _0=0,\, f_{4,1}(\xi ,\eta )=[\mu _1\phi (\xi )+\lambda _1\phi ^{-1}(\xi )]\,e^{i\eta },\,\\ \phi (\xi )=\varepsilon \sqrt{-\frac{w_2}{2w_4}}\,\, \tanh (\xi \sqrt{-\frac{w_2}{2}}), \& w_2<0,\ \ w_4>0, \ w_0=\frac{w_2^2}{4w_4}, \\ f_{4,2}(\xi ,\eta )=[\mu _1\phi (\xi )+\lambda _1\phi ^{-1}(\xi )]\,e^{i\eta },\,\,\ \ \phi (\xi )=\varepsilon \sqrt{\frac{w_2}{2w_4}}\,\, \tan (\xi \sqrt{\frac{w_2}{2}}), \& w_2>0,\ \ w_4>0, \ w_0=\frac{w_2^2}{4w_4}, \\ f_{4,3}(\xi ,\eta )=[\mu _1\phi (\xi )+\lambda _1\phi ^{-1}(\xi )]\,e^{i\eta },\,\,\ \ \phi (\xi )=\sqrt{\frac{-w_2m^2}{(2m^2-1)w_4}}\,\, cn(\xi \sqrt{\frac{w_2}{2m^2-1}}), \& w_2>0,\ \ w_4<0, \ w_0=\frac{w_2^2m^2(1-m^2)}{(2m^2-1)^2w_4}, \\ f_{4,4}(\xi ,\eta )=[\mu _1\phi (\xi )+\lambda _1\phi ^{-1}(\xi )]\,e^{i\eta },\,\,\ \ \phi (\xi )=\sqrt{\frac{-m^2w_2}{(2-m^2)w_4}}\,\, dn(\xi \sqrt{\frac{w_2}{2-m^2}}), \& w_2>0,\ \ w_4<0, \ w_0=\frac{w_2^2(1-m^2)}{(2-m^2)^2w_4}, \\ f_{4,5}(\xi ,\eta )=[\mu _1\phi (\xi )+\lambda _1\phi ^{-1}(\xi )]\,e^{i\eta },\,\,\ \ \phi (\xi )=\sqrt{\frac{-w_2m^2}{(1+m^2)w_4}}\,\, sn(\xi \sqrt{\frac{-w_2}{1+m^2}}), \& w_2<0,\ \ w_4>0, \ w_0=\frac{w_2^2m^2}{(1+m^2)^2w_4}, \\ \xi =l(x+{\frac{2{\epsilon }^{2}lw_{{0}}}{\sigma \,{\lambda _{{1}}}^{2}}}t),\,\,\, \eta =-{\frac{{\epsilon }^{2}lw_{{0}}}{\sigma \,{\lambda _{{1}}}^{2}}}x-{\frac{{\epsilon }^{2}lw_{{0}} \left( -8\,{\sigma }^{2}{ \lambda _{{1}}}^{3}\mu _{{1}}+{\epsilon }^{2}lw_{{0}} \right) }{{\sigma }^{2}{\lambda _{{1}}}^{ 4}}} t+\eta _0, \end{array} \right. \end{aligned}$$

(40)

where m is the modulus of the Jacobi elliptic functions and $0 < m \le 1$.

Remark 10

When $m\rightarrow 1$, then $cn(\xi , m)\rightarrow { sech}(\xi )$ hence the combined bright-singular solution (40) will be as

$$\begin{aligned} f_{4,3}(x,t)= & \left\{ \mu _1\sqrt{\frac{-w_2}{w_4}}\,\, { sech}\left( l\sqrt{w_2}(x+{\frac{2{\epsilon }^{2}lw_{{0}}}{\sigma \,{\lambda _{{1}}}^{2}}}t)\right) + \lambda _1\sqrt{\frac{-w_4}{w_2}}\,\, { csch}\left( l\sqrt{w_2}(x+{\frac{2{\epsilon }^{2}lw_{{0}}}{\sigma \,{\lambda _{{1}}}^{2}}}t)\right) \right\} \nonumber \\ & \times \,e^{i\left[ -{\frac{{\epsilon }^{2}lw_{{0}}}{\sigma \,{\lambda _{{1}}}^{2}}}x-{\frac{{\epsilon }^{2}lw_{{0}} \left( -8\,{\sigma }^{2}{ \lambda _{{1}}}^{3}\mu _{{1}}+{\epsilon }^{2}lw_{{0}} \right) }{{\sigma }^{2}{\lambda _{{1}}}^{ 4}}} t+\eta _0\right] },\ \ \ \ w_2w_4<0. \end{aligned}$$

(41)

Remark 11

When $m\rightarrow 1$, then $dn(\xi , m)\rightarrow { sech}(\xi )$ hence the combined bright-singular solution (40) will be as

$$\begin{aligned} f_{4,4}(x,t)= & \left\{ \mu _1\sqrt{\frac{-w_2}{w_4}}\,\, { sech}\left( l\sqrt{w_2}(x+{\frac{2{\epsilon }^{2}lw_{{0}}}{\sigma \,{\lambda _{{1}}}^{2}}}t)\right) + \lambda _1\sqrt{\frac{-w_4}{w_2}}\,\, { csch}\left( l\sqrt{w_2}(x+{\frac{2{\epsilon }^{2}lw_{{0}}}{\sigma \,{\lambda _{{1}}}^{2}}}t)\right) \right\} \nonumber \\ & \times \,e^{i\left[ -{\frac{{\epsilon }^{2}lw_{{0}}}{\sigma \,{\lambda _{{1}}}^{2}}}x-{\frac{{\epsilon }^{2}lw_{{0}} \left( -8\,{\sigma }^{2}{ \lambda _{{1}}}^{3}\mu _{{1}}+{\epsilon }^{2}lw_{{0}} \right) }{{\sigma }^{2}{\lambda _{{1}}}^{ 4}}} t+\eta _0\right] },\ \ \ \ w_2w_4<0. \end{aligned}$$

(42)

Remark 12

When $m\rightarrow 1$, then $sn(\xi , m)\rightarrow { tanh}(\xi )$ hence the combined dark-singular solution (40) will be as

$$\begin{aligned} f_{4,5}(x,t)= & \left\{ \mu _1\sqrt{\frac{-w_2}{2w_4}}\,\, { tanh}\left( l\sqrt{-\frac{w_2}{2}}(x+{\frac{2{\epsilon }^{2}lw_{{0}}}{\sigma \,{\lambda _{{1}}}^{2}}}t)\right) + \lambda _1\sqrt{\frac{-2w_2}{w_4}}\,\, { coth}\left( l\sqrt{-\frac{w_2}{2}}(x+{\frac{2{\epsilon }^{2}lw_{{0}}}{\sigma \,{\lambda _{{1}}}^{2}}}t)\right) \right\} \nonumber \\ & \times \,e^{i\left[ -{\frac{{\epsilon }^{2}lw_{{0}}}{\sigma \,{\lambda _{{1}}}^{2}}}x-{\frac{{\epsilon }^{2}lw_{{0}} \left( -8\,{\sigma }^{2}{ \lambda _{{1}}}^{3}\mu _{{1}}+{\epsilon }^{2}lw_{{0}} \right) }{{\sigma }^{2}{\lambda _{{1}}}^{ 4}}} t+\eta _0\right] },\ \ \ \ w_2<0,\ \ w_2w_4<0. \end{aligned}$$

(43)

$$\begin{aligned} \left\{ \begin{array}{ll} Case\,\, VI:\Rightarrow \,\mu _1=0,\, f_{4,6}(\xi ,\eta )=[\mu _0+\lambda _1\phi ^{-1}(\xi )]\,e^{i\eta },\,\ \phi (\xi )=-\frac{w_1}{2w_2}+\exp (\varepsilon \xi \sqrt{w_2}),\,\, w_2>0, \ w_0=\frac{w_1^2}{4w_2},\\ f_{4,7}(\xi ,\eta )=[\mu _0+\lambda _1\phi ^{-1}(\xi )]\,e^{i\eta },\,\,\ \ \phi (\xi )=-\frac{w_1}{2w_2}+\frac{\varepsilon w_1}{2w_2}\,\,\sin (\xi \sqrt{-w_2}),\,\, w_2<0, \ w_0=\frac{w_1^2}{4w_2},\\ f_{4,8}(\xi ,\eta )=[\mu _0+\lambda _1\phi ^{-1}(\xi )]\,e^{i\eta },\,\,\ \ \phi (\xi )=-\frac{w_1}{2w_2}+\frac{\varepsilon w_1}{2w_2}\,\,\sinh (2\xi \sqrt{w_2}),\,\, w_2>0, \ w_0=\frac{w_1^2}{4w_2},\\ f_{4,9}(\xi ,\eta )=[\mu _0+\lambda _1\phi ^{-1}(\xi )]\,e^{i\eta },\,\,\ \ \phi (\xi )=\varepsilon \sqrt{\frac{w_0}{w_2}}\,\,\sinh (\xi \sqrt{w_2}),\,\, w_2>0, \ w_0>0,\ \\ \xi =l(x+{\frac{2{\epsilon }^{2}lw_{{0}}}{\sigma \,{\lambda _{{1}}}^{2}}}t),\,\,\, \eta =-{\frac{{\epsilon }^{2}lw_{{0}}}{\sigma \,{\lambda _{{1}}}^{2}}}x-{\frac{{\epsilon }^{2}lw_{{0}} \left( {\epsilon }^{2}lw_{{0}} \right) }{{\sigma }^{2}{\lambda _{{1}}}^{ 4}}} t+\eta _0. \end{array} \right. \end{aligned}$$

(44)

Application of METFM on Eq. (2)

Take the following transformation

$$\begin{aligned} f(x,y,t)=F(\xi )\,e^{i\eta },\,\, with\,\, \xi =l_1x+l_2y-vt,\,\,\, \eta =s_1x+s_2y+s_3t+\eta _0. \end{aligned}$$

(45)

Inserting Eq. (45) into Eq. (2), we receive

$$\begin{aligned} v=2\,a \left( l_{{1}}s_{{1}}+l_{{2}}s_{{2}} \right) , \end{aligned}$$

(46)

as imaginary part and as real part is obtained below

$$\begin{aligned} a(l_1^2+l_2^2)F''+2(s_1b_1+s_2b_2)\,F^3-(a(s_1^2+s_2^2)+s_3)F=0. \end{aligned}$$

(47)

Equating the terms $F''$ and $F^3$ yields $N=1$. As a result, we have the flexibility to choose $F(\xi )$ from Eq. (5);

$$\begin{aligned} F(\xi )=\mu _0+\mu _1\,\phi (\xi )+\lambda _1\,(\phi (\xi ))^{-1}. \end{aligned}$$

(48)

Where $\mu _0,\mu _1$ and $\lambda _1$ are variables that will be determined. We build a set of algebraic equations involving the variables $\mu _0,\mu _1,\lambda _1,w_1,w_2,w_3$ and $w_4$ by substituting Eq. (48) with Eq. (6) in Eq. (47) and then setting the coefficients of $\phi ^i(\xi )$ to zero.

$$\begin{aligned} \left\{ \begin{array}{ll} -3\,a{\epsilon }^{2}{l_{{1}}}^{2}\lambda _{{1}}w_{{1}}-3\,a{\epsilon }^{2 }{l_{{2}}}^{2}\lambda _{{1}}w_{{1}}-12\,b_{{1}}{\lambda _{{1}}}^{2}\mu _{ {0}}s_{{1}}-12\,b_{{2}}{\lambda _{{1}}}^{2}\mu _{{0}}s_{{2}} =0,\\ -a{\epsilon }^{2}{l_{{1}}}^{2}\lambda _{{1}}w_{{3}}-a{\epsilon }^{2}{l_{{ 1}}}^{2}\mu _{{1}}w_{{1}}-a{\epsilon }^{2}{l_{{2}}}^{2}\lambda _{{1}}w_{{ 3}}-a{\epsilon }^{2}{l_{{2}}}^{2}\mu _{{1}}w_{{1}}-24\,b_{{1}}\lambda _{{ 1}}\mu _{{0}}\mu _{{1}}s_{{1}}- 4\,b_{{1}}{\mu _{{0}}}^{3}s_{{1}}-24\,b_{{ 2}}\lambda _{{1}}\mu _{{0}}\mu _{{1}}s_{{2}}-\\ 4\,b_{{2}}{\mu _{{0}}}^{3}s_{ {2}}+2\,a\mu _{{0}}{s_{{1}}}^{2}+2\,a\mu _{{0}}{s_{{2}}}^{2}+2\,\mu _{{0} }s_{{3}} =0,\\ -2\,a{\epsilon }^{2}{l_{{1}}}^{2}\lambda _{{1}}w_{{2}}-2\,a{\epsilon }^{2 }{l_{{2}}}^{2}\lambda _{{1}}w_{{2}}-12\,b_{{1}}{\lambda _{{1}}}^{2}\mu _{ {1}}s_{{1}}-12\,b_{{1}}\lambda _{{1}}{\mu _{{0}}}^{2}s_{{1}}-12\,b_{{2}} {\lambda _{{1}}}^{2}\mu _{{1}}s_{{2}}-12\,b_{{2}}\lambda _{{1}}{\mu _{{0}} }^{2}s_{{2}}+\\ 2\,a\lambda _{{1}}{s_{{1}}}^{2}+2\,a\lambda _{{1}}{s_{{2}}} ^{2}+2\,\lambda _{{1}}s_{{3}} =0,\\ -2\,a{\epsilon }^{2}{l_{{1}}}^{2}\mu _{{1}}w_{{2}}-2\,a{\epsilon }^{2}{l_ {{2}}}^{2}\mu _{{1}}w_{{2}}-12\,b_{{1}}\lambda _{{1}}{\mu _{{1}}}^{2}s_{{ 1}}-12\,b_{{1}}{\mu _{{0}}}^{2}\mu _{{1}}s_{{1}}-12\,b_{{2}}\lambda _{{1} }{\mu _{{1}}}^{2}s_{{2}}-12\,b_{{2}}{\mu _{{0}}}^{2}\mu _{{1}}s_{{2}}+\\ 2\, a\mu _{{1}}{s_{{1}}}^{2}+2\,a\mu _{{1}}{s_{{2}}}^{2}+2\,\mu _{{1}}s_{{3}} =0,\\ -4\,a{\epsilon }^{2}{l_{{1}}}^{2}\mu _{{1}}w_{{4}}-4\,a{\epsilon }^{2}{l_ {{2}}}^{2}\mu _{{1}}w_{{4}}-4\,b_{{1}}{\mu _{{1}}}^{3}s_{{1}}-4\,b_{{2}} {\mu _{{1}}}^{3}s_{{2}} =0,\\ -3\,a{\epsilon }^{2}{l_{{1}}}^{2}\mu _{{1}}w_{{3}}-3\,a{\epsilon }^{2}{l_ {{2}}}^{2}\mu _{{1}}w_{{3}}-12\,b_{{1}}\mu _{{0}}{\mu _{{1}}}^{2}s_{{1}}- 12\,b_{{2}}\mu _{{0}}{\mu _{{1}}}^{2}s_{{2}} =0,\\ -4\,a{\epsilon }^{2}{l_{{1}}}^{2}\lambda _{{1}}w_{{0}}-4\,a{\epsilon }^{2 }{l_{{2}}}^{2}\lambda _{{1}}w_{{0}}-4\,b_{{1}}{\lambda _{{1}}}^{3}s_{{1} }-4\,b_{{2}}{\lambda _{{1}}}^{3}s_{{2}} =0. \end{array} \right. \end{aligned}$$

(49)

We use Maple to solve this system of algebraic equations and get following values of those unknown parameters:

$$\begin{aligned} & \lambda _1=0,\ \ s_{{1}}=-{\frac{a{\epsilon }^{2}{l_{{1}}}^{2}w_{{4}}+a{\epsilon }^{2}{l _{{2}}}^{2}w_{{4}}+b_{{2}}{\mu _{{1}}}^{2}s_{{2}}}{b_{{1}}{\mu _{{1}}}^{ 2}}},\ \ s_2=s_2,\ \ w_{{1}}=-2\,{\frac{\mu _{{0}} \left( 4\,{\mu _{{0}}}^{2}w_{{4}}-{\mu _{{ 1}}}^{2}w_{{2}} \right) }{{\mu _{{1}}}^{3}}},\ \ w_{{3}}=4\,{\frac{\mu _{{0}}w_{{4}}}{\mu _{{1}}}},\nonumber \\ & s_{{3}}=-{\frac{a \left( {\epsilon }^{2} \left( {l_{{1}}}^{2}+{l_{{2}} }^{2} \right) \left( {a}^{2}{\epsilon }^{2} \left( {l_{{1}}}^{2}+{l_{{ 2}}}^{2} \right) {w_{{4}}}^{2}+2\,{\mu _{{1}}}^{2} \left( 3\,{b_{{1}}}^ {2}{\mu _{{0}}}^{2}+ab_{{2}}s_{{2}} \right) w_{{4}}-{b_{{1}}}^{2}{\mu _{ {1}}}^{4}w_{{2}} \right) +{\mu _{{1}}}^{4}{s_{{2}}}^{2} \left( {b_{{1}} }^{2}+{b_{{2}}}^{2} \right) \right) }{{b_{{1}}}^{2}{\mu _{{1}}}^{4}}}. \end{aligned}$$

(50)

First class of solutions (according to parameters (50)):

$$\begin{aligned} \left\{ \begin{array}{ll} Case\,\, I:\Rightarrow \,\,\mu _0=0,\,\,\, f_{1,1}(\xi ,\eta )=\mu _1\sqrt{-\frac{w_2}{w_4}}\,\, sech(\xi \sqrt{w_2})\,e^{i\eta }, \& w_2>0,\ \ w_4<0, \\ f_{1,2}=\mu _1\sqrt{-\frac{w_2}{w_4}}\,\, sec(\xi \sqrt{w_2})\,e^{i\eta }, \& w_2>0,\ \ w_4<0, \ \ \ \ f_{1,3}=\mu _1\frac{-\varepsilon }{\xi \sqrt{w_4}}\,e^{i\eta }, \& w_2=0,\\ \xi =l_1x+l_2y-2\,a \left( -l_{{1}}{\frac{a{\epsilon }^{2}{l_{{1}}}^{2}w_{{4}}+a{\epsilon }^{2}{l _{{2}}}^{2}w_{{4}}+b_{{2}}{\mu _{{1}}}^{2}s_{{2}}}{b_{{1}}{\mu _{{1}}}^{ 2}}}+l_{{2}}s_{{2}} \right) t,\,\,\, \\ {\eta =-{\frac{a{\epsilon }^{2}{l_{{1}}}^{2}w_{{4}}+a{\epsilon }^{2}{l _{{2}}}^{2}w_{{4}}+b_{{2}}{\mu _{{1}}}^{2}s_{{2}}}{b_{{1}}{\mu _{{1}}}^{ 2}}}x+s_2y-{\frac{a \left( {\epsilon }^{2} \left( {l_{{1}}}^{2}+{l_{{2}} }^{2} \right) \left( {a}^{2}{\epsilon }^{2} \left( {l_{{1}}}^{2}+{l_{{ 2}}}^{2} \right) {w_{{4}}}^{2}+2\,ab_{{2}}{\mu _{{1}}}^{2}s_{{2}}w_{{4} }-{b_{{1}}}^{2}{\mu _{{1}}}^{4}w_{{2}} \right) +{\mu _{{1}}}^{4}{s_{{2}} }^{2} \left( {b_{{1}}}^{2}+{b_{{2}}}^{2} \right) \right) }{{b_{{1}}}^ {2}{\mu _{{1}}}^{4}}} t+\eta _0,} \end{array} \right. \end{aligned}$$

(51)

$$\begin{aligned} \left\{ \begin{array}{ll} Case\,\, II:\Rightarrow \,\,\mu _0=0,\,\,\, f_{1,4}(\xi ,\eta )=\mu _1\varepsilon \sqrt{-\frac{w_2}{2w_4}}\,\, \tanh (\xi \sqrt{-\frac{w_2}{2}})\,e^{i\eta },\, \& w_2<0,\ \ w_4>0, \ w_0=\frac{w_2^2}{4w_4}, \\ f_{1,5}(\xi ,\eta )=\mu _1\varepsilon \sqrt{\frac{w_2}{2w_4}}\,\, \tan (\xi \sqrt{\frac{w_2}{2}})\,e^{i\eta },\, \& w_2>0,\ \ w_4>0, \ w_0=\frac{w_2^2}{4w_4}, \\ f_{1,6}(\xi ,\eta )=\mu _1\sqrt{\frac{-w_2m^2}{(2m^2-1)w_4}}\,\, cn(\xi \sqrt{\frac{w_2}{2m^2-1}})\,e^{i\eta },\, \& w_2>0,\ \ w_4<0, \ w_0=\frac{w_2^2m^2(1-m^2)}{(2m^2-1)^2w_4}, \\ f_{1,7}(\xi ,\eta )=\mu _1\sqrt{\frac{-m^2w_2}{(2-m^2)w_4}}\,\, dn(\xi \sqrt{\frac{w_2}{2-m^2}})\,e^{i\eta },\, \& w_2>0,\ \ w_4<0, \ w_0=\frac{w_2^2(1-m^2)}{(2-m^2)^2w_4}, \\ f_{1,8}(\xi ,\eta )=\mu _1\sqrt{\frac{-w_2m^2}{(1+m^2)w_4}}\,\, sn(\xi \sqrt{\frac{-w_2}{1+m^2}})\,e^{i\eta },\, \& w_2<0,\ \ w_4>0, \ w_0=\frac{w_2^2m^2}{(1+m^2)^2w_4}, \\ \xi =l_1x+l_2y-2\,a \left( -l_{{1}}{\frac{a{\epsilon }^{2}{l_{{1}}}^{2}w_{{4}}+a{\epsilon }^{2}{l _{{2}}}^{2}w_{{4}}+b_{{2}}{\mu _{{1}}}^{2}s_{{2}}}{b_{{1}}{\mu _{{1}}}^{ 2}}}+l_{{2}}s_{{2}} \right) t,\,\,\, \\ \eta =-{\frac{a{\epsilon }^{2}{l_{{1}}}^{2}w_{{4}}+a{\epsilon }^{2}{l _{{2}}}^{2}w_{{4}}+b_{{2}}{\mu _{{1}}}^{2}s_{{2}}}{b_{{1}}{\mu _{{1}}}^{ 2}}}x+s_2y-{\frac{a \left( {\epsilon }^{2} \left( {l_{{1}}}^{2}+{l_{{2}} }^{2} \right) \left( {a}^{2}{\epsilon }^{2} \left( {l_{{1}}}^{2}+{l_{{ 2}}}^{2} \right) {w_{{4}}}^{2}+2\,ab_{{2}}{\mu _{{1}}}^{2}s_{{2}}w_{{4} }-{b_{{1}}}^{2}{\mu _{{1}}}^{4}w_{{2}} \right) +{\mu _{{1}}}^{4}{s_{{2}} }^{2} \left( {b_{{1}}}^{2}+{b_{{2}}}^{2} \right) \right) }{{b_{{1}}}^ {2}{\mu _{{1}}}^{4}}} t+\eta _0, \end{array} \right. \end{aligned}$$

(52)

where m is the modulus of the Jacobi elliptic functions and $0 < m \le 1$.

Remark 1

When $m\rightarrow 1$, then $cn(\xi , m)\rightarrow { sech}(\xi )$ hence the bright soliton solution (52) will be as

$$\begin{aligned} f_{1,6}(x,t)= & \mu _1\sqrt{\frac{-w_2}{w_4}}\,\, { sech}\left( \sqrt{w_2}\left[ l_1x+l_2y-2\,a \left( -l_{{1}}{\frac{a{\epsilon }^{2}{l_{{1}}}^{2}w_{{4}}+a{\epsilon }^{2}{l _{{2}}}^{2}w_{{4}}+b_{{2}}{\mu _{{1}}}^{2}s_{{2}}}{b_{{1}}{\mu _{{1}}}^{ 2}}}+l_{{2}}s_{{2}} \right) t\right] \right) \nonumber \\ & \times \,e^{i\left[ -{\frac{a{\epsilon }^{2}{l_{{1}}}^{2}w_{{4}}+a{\epsilon }^{2}{l _{{2}}}^{2}w_{{4}}+b_{{2}}{\mu _{{1}}}^{2}s_{{2}}}{b_{{1}}{\mu _{{1}}}^{ 2}}}x+s_2y-{\frac{a \left( {\epsilon }^{2} \left( {l_{{1}}}^{2}+{l_{{2}} }^{2} \right) \left( {a}^{2}{\epsilon }^{2} \left( {l_{{1}}}^{2}+{l_{{ 2}}}^{2} \right) {w_{{4}}}^{2}+2\,ab_{{2}}{\mu _{{1}}}^{2}s_{{2}}w_{{4} }-{b_{{1}}}^{2}{\mu _{{1}}}^{4}w_{{2}} \right) +{\mu _{{1}}}^{4}{s_{{2}} }^{2} \left( {b_{{1}}}^{2}+{b_{{2}}}^{2} \right) \right) }{{b_{{1}}}^ {2}{\mu _{{1}}}^{4}}} t+\eta _0\right] },\ \ \ w_2w_4<0. \end{aligned}$$

(53)

Remark 2

When $m\rightarrow 1$, then $dn(\xi , m)\rightarrow { sech}(\xi )$ hence the bright soliton solution (52) will be as

$$\begin{aligned} f_{1,7}(x,t)= & \mu _1\sqrt{\frac{-w_2}{w_4}}\,\, { sech}\left( \sqrt{w_2}\left[ l_1x+l_2y-2\,a \left( -l_{{1}}{\frac{a{\epsilon }^{2}{l_{{1}}}^{2}w_{{4}}+a{\epsilon }^{2}{l _{{2}}}^{2}w_{{4}}+b_{{2}}{\mu _{{1}}}^{2}s_{{2}}}{b_{{1}}{\mu _{{1}}}^{ 2}}}+l_{{2}}s_{{2}} \right) t\right] \right) \nonumber \\ & \times \,e^{i\left[ -{\frac{a{\epsilon }^{2}{l_{{1}}}^{2}w_{{4}}+a{\epsilon }^{2}{l _{{2}}}^{2}w_{{4}}+b_{{2}}{\mu _{{1}}}^{2}s_{{2}}}{b_{{1}}{\mu _{{1}}}^{ 2}}}x+s_2y-{\frac{a \left( {\epsilon }^{2} \left( {l_{{1}}}^{2}+{l_{{2}} }^{2} \right) \left( {a}^{2}{\epsilon }^{2} \left( {l_{{1}}}^{2}+{l_{{ 2}}}^{2} \right) {w_{{4}}}^{2}+2\,ab_{{2}}{\mu _{{1}}}^{2}s_{{2}}w_{{4} }-{b_{{1}}}^{2}{\mu _{{1}}}^{4}w_{{2}} \right) +{\mu _{{1}}}^{4}{s_{{2}} }^{2} \left( {b_{{1}}}^{2}+{b_{{2}}}^{2} \right) \right) }{{b_{{1}}}^ {2}{\mu _{{1}}}^{4}}} t+\eta _0\right] },\ \ \ w_2w_4<0. \end{aligned}$$

(54)

Remark 3

When $m\rightarrow 1$, then $sn(\xi , m)\rightarrow { tanh}(\xi )$ hence the dark soliton solution (52) will be as

$$\begin{aligned} f_{1,8}(x,t)= & \mu _1\sqrt{\frac{-w_2}{2w_4}}\,\, { tanh}\left( \sqrt{-\frac{w_2}{2}}\left[ l_1x+l_2y-2\,a \left( -l_{{1}}{\frac{a{\epsilon }^{2}{l_{{1}}}^{2}w_{{4}}+a{\epsilon }^{2}{l _{{2}}}^{2}w_{{4}}+b_{{2}}{\mu _{{1}}}^{2}s_{{2}}}{b_{{1}}{\mu _{{1}}}^{ 2}}}+l_{{2}}s_{{2}} \right) t\right] \right) \, \nonumber \\ & \times e^{i\left[ -{\frac{a{\epsilon }^{2}{l_{{1}}}^{2}w_{{4}}+a{\epsilon }^{2}{l _{{2}}}^{2}w_{{4}}+b_{{2}}{\mu _{{1}}}^{2}s_{{2}}}{b_{{1}}{\mu _{{1}}}^{ 2}}}x+s_2y-{\frac{a \left( {\epsilon }^{2} \left( {l_{{1}}}^{2}+{l_{{2}} }^{2} \right) \left( {a}^{2}{\epsilon }^{2} \left( {l_{{1}}}^{2}+{l_{{ 2}}}^{2} \right) {w_{{4}}}^{2}+2\,ab_{{2}}{\mu _{{1}}}^{2}s_{{2}}w_{{4} }-{b_{{1}}}^{2}{\mu _{{1}}}^{4}w_{{2}} \right) +{\mu _{{1}}}^{4}{s_{{2}} }^{2} \left( {b_{{1}}}^{2}+{b_{{2}}}^{2} \right) \right) }{{b_{{1}}}^ {2}{\mu _{{1}}}^{4}}} t+\eta _0\right] },\nonumber \\ & \ \ \ w_2<0,\ \ \ w_2w_4<0. \end{aligned}$$

(55)

$$\begin{aligned} \left\{ \begin{array}{ll} Case\,\, VI:\Rightarrow \,\,w_{{1}}=2\,{\frac{\mu _{{0}}w_{{2}}}{\mu _{{1}}}},\,\,\, f_{1,9}(\xi ,\eta )=[\mu _0+\lambda _1\phi (\xi )]\,e^{i\eta },\,\ \phi (\xi )=-\frac{w_1}{2w_2}+\exp (\varepsilon \xi \sqrt{w_2}),\,\, w_2>0, \ w_0=\frac{w_1^2}{4w_2},\\ f_{1,10}(\xi ,\eta )=[\mu _0+\lambda _1\phi (\xi )]\,e^{i\eta },\,\,\ \ \phi (\xi )=-\frac{w_1}{2w_2}+\frac{\varepsilon w_1}{2w_2}\,\,\sin (\xi \sqrt{-w_2}),\,\, w_2<0, \ w_0=\frac{w_1^2}{4w_2},\\ f_{1,11}(\xi ,\eta )=[\mu _0+\lambda _1\phi (\xi )]\,e^{i\eta },\,\,\ \ \phi (\xi )=-\frac{w_1}{2w_2}+\frac{\varepsilon w_1}{2w_2}\,\,\sinh (2\xi \sqrt{w_2}),\,\, w_2>0, \ w_0=\frac{w_1^2}{4w_2},\\ f_{1,12}(\xi ,\eta )=[\mu _0+\lambda _1\phi (\xi )]\,e^{i\eta },\,\,\ \ \phi (\xi )=\varepsilon \sqrt{\frac{w_0}{w_2}}\,\,\sinh (\xi \sqrt{w_2}),\,\, w_2>0, \ w_0>0,\ \\ \xi =-{\frac{b_{{2}}s_{{2}}}{b_{{1}}}}x+s_2y-2\,a \left( -l_{{1}}{\frac{b_{{2}}s_{{2}}}{b_{{1}}}}+l_{{2}}s_{{2}} \right) t,\,\,\, \eta =-{\frac{b_{{2}}s_{{2}}}{b_{{1}}}}x+s_2y-{\frac{a \left( -{\epsilon }^{2} \left( {l_{{1}}}^{2}+{l_{{2} }}^{2} \right) {b_{{1}}}^{2}{\mu _{{1}}}^{4}w_{{2}}+{\mu _{{1}}}^{4}{s_{ {2}}}^{2} \left( {b_{{1}}}^{2}+{b_{{2}}}^{2} \right) \right) }{{b_{{1 }}}^{2}{\mu _{{1}}}^{4}}} t+\eta _0. \end{array} \right. \end{aligned}$$

(56)

$$\begin{aligned} \left\{ \begin{array}{ll} Case\,\, VII:\Rightarrow \,\,\frac{w_2}{w_4}=\frac{4\mu _0^2}{\mu _1^2},\,\,\, f_{1,13}(\xi ,\eta )=\left( \mu _0-\frac{w_2sec^2\left( \frac{\xi \sqrt{-w_2}}{2}\right) }{2\varepsilon \sqrt{-w_2w_4} \tan \left( \frac{\xi \sqrt{-w_2}}{2}\right) +w_3}\right) \,e^{i\eta },\,\, w_2<0, \\ f_{1,14}(\xi ,\eta )=\left( \mu _0-\mu _1\frac{w_2sech^2\left( \frac{\xi \sqrt{w_2}}{2}\right) }{2\varepsilon \sqrt{-w_2w_4} \tanh \left( \frac{\xi \sqrt{w_2}}{2}\right) -w_3}\right) \,e^{i\eta },\,\, w_2>0, \ \ w_3\ne 2\varepsilon \sqrt{w_2w_4},\\ f_{1,15}(\xi ,\eta )=\left( \mu _0-\mu _1\frac{\varepsilon }{2}\sqrt{\frac{w_2}{w_4}}\left[ 1+\tanh \left( \frac{\xi \sqrt{w_2}}{2}\right) \right] \right) \,e^{i\eta },\,\, w_2>0,\ \ w_3=2\varepsilon \sqrt{w_2w_4},\\ \frac{w_2}{w_4}=\frac{4\mu _0^2}{\mu _1^2},\,\,\, \ \ \ \ \xi =l_1x+l_2y-2\,a \left( -l_{{1}}{\frac{a{\epsilon }^{2}{l_{{1}}}^{2}w_{{4}}+a{\epsilon }^{2}{l _{{2}}}^{2}w_{{4}}+b_{{2}}{\mu _{{1}}}^{2}s_{{2}}}{b_{{1}}{\mu _{{1}}}^{ 2}}}+l_{{2}}s_{{2}} \right) t,\,\,\, \\ \eta =-{\frac{a{\epsilon }^{2}{l_{{1}}}^{2}w_{{4}}+a{\epsilon }^{2}{l _{{2}}}^{2}w_{{4}}+b_{{2}}{\mu _{{1}}}^{2}s_{{2}}}{b_{{1}}{\mu _{{1}}}^{ 2}}}x+s_2y-{\frac{a \left( {\epsilon }^{2} \left( {l_{{1}}}^{2}+{l_{{2}} }^{2} \right) \left( {a}^{2}{\epsilon }^{2} \left( {l_{{1}}}^{2}+{l_{{ 2}}}^{2} \right) {w_{{4}}}^{2}+2\,ab_{{2}}{\mu _{{1}}}^{2}s_{{2}}w_{{4} }-{b_{{1}}}^{2}{\mu _{{1}}}^{4}w_{{2}} \right) +{\mu _{{1}}}^{4}{s_{{2}} }^{2} \left( {b_{{1}}}^{2}+{b_{{2}}}^{2} \right) \right) }{{b_{{1}}}^ {2}{\mu _{{1}}}^{4}}} t+\eta _0. \end{array} \right. \end{aligned}$$

(57)

$$\begin{aligned} \mu _1=0,\ \ s_{{1}}=-{\frac{a{\epsilon }^{2}{l_{{1}}}^{2}w_{{0}}+a{\epsilon }^{2}{l _{{2}}}^{2}w_{{0}}+b_{{2}}{\lambda _{{1}}}^{2}s_{{2}}}{b_{{1}}{\lambda _ {{1}}}^{2}}},\ \ s_2=s_2,\ \ w_{{1}}=4\,{\frac{\mu _{{0}}w_{{0}}}{\lambda _{{1}}}},\ \ w_{{3}}=2\,{\frac{\mu _{{0}} \left( {\lambda _{{1}}}^{2}w_{{2}}-4\,{\mu _{{0}}}^{2}w_{{0}} \right) }{{\lambda _{{1}}}^{3}}}, \end{aligned}$$

(58)

$$\begin{aligned} s_{{3}}=-{\frac{a \left( {\epsilon }^{2} \left( {l_{{1}}}^{2}+{l_{{2}} }^{2} \right) \left( {a}^{2}{\epsilon }^{2} \left( {l_{{1}}}^{2}+{l_{{ 2}}}^{2} \right) {w_{{0}}}^{2}-{b_{{1}}}^{2}{\lambda _{{1}}}^{2} \left( {\lambda _{{1}}}^{2}w_{{2}}-6\,{\mu _{{0}}}^{2}w_{{0}} \right) + 2\,ab_{{2}}{\lambda _{{1}}}^{2}s_{{2}}w_{{0}} \right) +{\lambda _{{1}}}^ {4}{s_{{2}}}^{2} \left( {b_{{1}}}^{2}+{b_{{2}}}^{2} \right) \right) }{{b_{{1}}}^{2}{\lambda _{{1}}}^{4}}}. \end{aligned}$$

Second class of solutions (according to parameters (58)):

$$\begin{aligned} \left\{ \begin{array}{ll} Case\,\, I:\Rightarrow \,\,\mu _0=0,\,\,or\,\,w_2=0\,\,\, f_{2,1}(\xi ,\eta )=\lambda _1\left( \sqrt{-\frac{w_2}{w_4}}\,\, sech(\xi \sqrt{w_2})\right) ^{-1}\,e^{i\eta }, \& w_2>0,\ \ w_4<0, \\ f_{2,2}=\lambda _1\left( \sqrt{-\frac{w_2}{w_4}}\,\, sec(\xi \sqrt{w_2})\right) ^{-1}\,e^{i\eta }, \& w_2>0,\ \ w_4<0, \ \ \ \ \xi =l_1x+l_2y-2\,a \left( -l_{{1}}{\frac{b_{{2}}s_{{2}}}{b_{{1}}}}+l_{{2}}s_{{2}} \right) t,\,\,\, \\ \eta =-{\frac{b_{{2}}s_{{2}}}{b_{{1}}}}x+s_2y-{\frac{a \left( -{\epsilon }^{2} \left( {l_{{1}}}^{2}+{l_{{2} }}^{2} \right) {b_{{1}}}^{2}{\lambda _{{1}}}^{4}w_{{2}}+{\lambda _{{1}}} ^{4}{s_{{2}}}^{2} \left( {b_{{1}}}^{2}+{b_{{2}}}^{2} \right) \right) }{{b_{{1}}}^{2}{\lambda _{{1}}}^{4}}} t+\eta _0,\ \ \ \ f_{2,3}=\left( \mu _0+\lambda _1\left( \frac{-\varepsilon }{\xi \sqrt{w_4}}\right) ^{-1}\right) \,e^{i\eta }, \& w_2=0,\\ \xi =l_1x+l_2y-2\,a \left( -l_{{1}}{\frac{b_{{2}}s_{{2}}}{b_{{1}}}}+l_{{2}}s_{{2}} \right) t, \ \ \eta =-{\frac{b_{{2}}s_{{2}}}{b_{{1}}}}x+s_2y-{\frac{a{s_{{2}}}^{2} \left( {b_{{1}}}^{2}+{b_{{2}}}^{2} \right) }{{b_{{1}}}^{2}}} t+\eta _0. \end{array} \right. \end{aligned}$$

(59)

$$\begin{aligned} \left\{ \begin{array}{ll} Case\,\, II:\Rightarrow \,\,\mu _0=0,\,\,\, f_{2,4}(\xi ,\eta )=\lambda _1\left[ \varepsilon \sqrt{-\frac{w_2}{2w_4}}\,\, \tanh (\xi \sqrt{-\frac{w_2}{2}})\right] ^{-1}\,e^{i\eta },\, \& w_2<0,\ \ w_4>0, \ w_0=\frac{w_2^2}{4w_4}, \\ f_{2,5}(\xi ,\eta )=\lambda _1\left[ \varepsilon \sqrt{\frac{w_2}{2w_4}}\,\, \tan (\xi \sqrt{\frac{w_2}{2}})\right] ^{-1}\,e^{i\eta },\, \& w_2>0,\ \ w_4>0, \ w_0=\frac{w_2^2}{4w_4}, \\ f_{2,6}(\xi ,\eta )=\lambda _1\left[ \sqrt{\frac{-w_2m^2}{(2m^2-1)w_4}}\,\, cn(\xi \sqrt{\frac{w_2}{2m^2-1}})\right] ^{-1}\,e^{i\eta },\, \& w_2>0,\ \ w_4<0, \ w_0=\frac{w_2^2m^2(1-m^2)}{(2m^2-1)^2w_4}, \\ f_{2,7}(\xi ,\eta )=\lambda _1\left[ \sqrt{\frac{-m^2w_2}{(2-m^2)w_4}}\,\, dn(\xi \sqrt{\frac{w_2}{2-m^2}})\right] ^{-1}\,e^{i\eta },\, \& w_2>0,\ \ w_4<0, \ w_0=\frac{w_2^2(1-m^2)}{(2-m^2)^2w_4}, \\ f_{2,8}(\xi ,\eta )=\lambda _1\left[ \sqrt{\frac{-w_2m^2}{(1+m^2)w_4}}\,\, sn(\xi \sqrt{\frac{-w_2}{1+m^2}})\right] ^{-1}\,e^{i\eta },\, \& w_2<0,\ \ w_4>0, \ w_0=\frac{w_2^2m^2}{(1+m^2)^2w_4}, \\ \xi =l_1x+l_2y-2\,a \left( -l_{{1}}{\frac{a{\epsilon }^{2}{l_{{1}}}^{2}w_{{0}}+a{\epsilon }^{2}{l _{{2}}}^{2}w_{{0}}+b_{{2}}{\lambda _{{1}}}^{2}s_{{2}}}{b_{{1}}{\lambda _ {{1}}}^{2}}}+l_{{2}}s_{{2}} \right) t,\,\,\, \ \ \eta =-{\frac{a{\epsilon }^{2}{l_{{1}}}^{2}w_{{0}}+a{\epsilon }^{2}{l _{{2}}}^{2}w_{{0}}+b_{{2}}{\lambda _{{1}}}^{2}s_{{2}}}{b_{{1}}{\lambda _ {{1}}}^{2}}}x+\\ s_2y-{\frac{a \left( {\epsilon }^{2} \left( {l_{{1}}}^{2}+{l_{{2}} }^{2} \right) \left( {a}^{2}{\epsilon }^{2} \left( {l_{{1}}}^{2}+{l_{{ 2}}}^{2} \right) {w_{{0}}}^{2}-{b_{{1}}}^{2}{\lambda _{{1}}}^{2} \left( {\lambda _{{1}}}^{2}w_{{2}}-6\,{\mu _{{0}}}^{2}w_{{0}} \right) + 2\,ab_{{2}}{\lambda _{{1}}}^{2}s_{{2}}w_{{0}} \right) +{\lambda _{{1}}}^ {4}{s_{{2}}}^{2} \left( {b_{{1}}}^{2}+{b_{{2}}}^{2} \right) \right) }{{b_{{1}}}^{2}{\lambda _{{1}}}^{4}}} t+\eta _0, \end{array} \right. \end{aligned}$$

(60)

where m is the modulus of the Jacobi elliptic functions and $0 < m \le 1$.

Remark 4

When $m\rightarrow 1$, then $cn(\xi , m)\rightarrow { sech}(\xi )$ hence the singular soliton solution (60) will be as

$$\begin{aligned} f_{2,6}(x,t)= & \lambda _1\sqrt{\frac{-w_4}{w_2}}\,\, { csch}\left( \sqrt{w_2}\left[ l_1x+l_2y-2\,a \left( -l_{{1}}{\frac{a{\epsilon }^{2}{l_{{1}}}^{2}w_{{0}}+a{\epsilon }^{2}{l _{{2}}}^{2}w_{{0}}+b_{{2}}{\lambda _{{1}}}^{2}s_{{2}}}{b_{{1}}{\lambda _ {{1}}}^{2}}}+l_{{2}}s_{{2}} \right) t\right] \right) \nonumber \\ & \times \,e^{i\left[ -{\frac{a{\epsilon }^{2}{l_{{1}}}^{2}w_{{0}}+a{\epsilon }^{2}{l _{{2}}}^{2}w_{{0}}+b_{{2}}{\lambda _{{1}}}^{2}s_{{2}}}{b_{{1}}{\lambda _ {{1}}}^{2}}}x+ s_2y-{\frac{a \left( {\epsilon }^{2} \left( {l_{{1}}}^{2}+{l_{{2}} }^{2} \right) \left( {a}^{2}{\epsilon }^{2} \left( {l_{{1}}}^{2}+{l_{{ 2}}}^{2} \right) {w_{{0}}}^{2}-{b_{{1}}}^{2}{\lambda _{{1}}}^{2} \left( {\lambda _{{1}}}^{2}w_{{2}}-6\,{\mu _{{0}}}^{2}w_{{0}} \right) + 2\,ab_{{2}}{\lambda _{{1}}}^{2}s_{{2}}w_{{0}} \right) +{\lambda _{{1}}}^ {4}{s_{{2}}}^{2} \left( {b_{{1}}}^{2}+{b_{{2}}}^{2} \right) \right) }{{b_{{1}}}^{2}{\lambda _{{1}}}^{4}}} t+\eta _0\right] }, \end{aligned}$$

(61)

so that $w_2w_4<0$.

Remark 5

When $m\rightarrow 1$, then $dn(\xi , m)\rightarrow { sech}(\xi )$ hence the singular soliton solution (60) will be as

$$\begin{aligned} f_{2,7}(x,t)= & \lambda _1\sqrt{\frac{-w_4}{w_2}}\,\, { csch}\left( \sqrt{w_2}\left[ l_1x+l_2y-2\,a \left( -l_{{1}}{\frac{a{\epsilon }^{2}{l_{{1}}}^{2}w_{{0}}+a{\epsilon }^{2}{l _{{2}}}^{2}w_{{0}}+b_{{2}}{\lambda _{{1}}}^{2}s_{{2}}}{b_{{1}}{\lambda _ {{1}}}^{2}}}+l_{{2}}s_{{2}} \right) t\right] \right) \nonumber \\ & \times \,e^{i\left[ -{\frac{a{\epsilon }^{2}{l_{{1}}}^{2}w_{{0}}+a{\epsilon }^{2}{l _{{2}}}^{2}w_{{0}}+b_{{2}}{\lambda _{{1}}}^{2}s_{{2}}}{b_{{1}}{\lambda _ {{1}}}^{2}}}x+ s_2y-{\frac{a \left( {\epsilon }^{2} \left( {l_{{1}}}^{2}+{l_{{2}} }^{2} \right) \left( {a}^{2}{\epsilon }^{2} \left( {l_{{1}}}^{2}+{l_{{ 2}}}^{2} \right) {w_{{0}}}^{2}-{b_{{1}}}^{2}{\lambda _{{1}}}^{2} \left( {\lambda _{{1}}}^{2}w_{{2}}-6\,{\mu _{{0}}}^{2}w_{{0}} \right) + 2\,ab_{{2}}{\lambda _{{1}}}^{2}s_{{2}}w_{{0}} \right) +{\lambda _{{1}}}^ {4}{s_{{2}}}^{2} \left( {b_{{1}}}^{2}+{b_{{2}}}^{2} \right) \right) }{{b_{{1}}}^{2}{\lambda _{{1}}}^{4}}} t+\eta _0\right] }, \end{aligned}$$

(62)

so that $w_2w_4<0$.

Remark 6

When $m\rightarrow 1$, then $sn(\xi , m)\rightarrow { tanh}(\xi )$ hence the soliton solution (60) will be as

$$\begin{aligned} f_{2,8}(x,t)= & \lambda _1\sqrt{\frac{-2w_4}{w_2}}\,\, { coth}\left( \sqrt{-\frac{w_2}{2}}\left[ l_1x+l_2y-2\,a \left( -l_{{1}}{\frac{a{\epsilon }^{2}{l_{{1}}}^{2}w_{{0}}+a{\epsilon }^{2}{l _{{2}}}^{2}w_{{0}}+b_{{2}}{\lambda _{{1}}}^{2}s_{{2}}}{b_{{1}}{\lambda _ {{1}}}^{2}}}+l_{{2}}s_{{2}} \right) t\right] \right) \,\nonumber \\\times & e^{i\left[ -{\frac{a{\epsilon }^{2}{l_{{1}}}^{2}w_{{0}}+a{\epsilon }^{2}{l _{{2}}}^{2}w_{{0}}+b_{{2}}{\lambda _{{1}}}^{2}s_{{2}}}{b_{{1}}{\lambda _ {{1}}}^{2}}}x+ s_2y-{\frac{a \left( {\epsilon }^{2} \left( {l_{{1}}}^{2}+{l_{{2}} }^{2} \right) \left( {a}^{2}{\epsilon }^{2} \left( {l_{{1}}}^{2}+{l_{{ 2}}}^{2} \right) {w_{{0}}}^{2}-{b_{{1}}}^{2}{\lambda _{{1}}}^{2} \left( {\lambda _{{1}}}^{2}w_{{2}}-6\,{\mu _{{0}}}^{2}w_{{0}} \right) + 2\,ab_{{2}}{\lambda _{{1}}}^{2}s_{{2}}w_{{0}} \right) +{\lambda _{{1}}}^ {4}{s_{{2}}}^{2} \left( {b_{{1}}}^{2}+{b_{{2}}}^{2} \right) \right) }{{b_{{1}}}^{2}{\lambda _{{1}}}^{4}}} t+\eta _0\right] }, \end{aligned}$$

(63)

so that $w_2<0$ and $w_2w_4<0$.

$$\begin{aligned} \left\{ \begin{array}{ll} Case\,\, III:\Rightarrow \,\, f_{2,9}(\xi ,\eta )=\left( \mu _0+\lambda _1\sqrt{-\frac{w_3}{w_2}}\,\, csch^2(\frac{\xi \sqrt{w_2}}{2})\right) \,e^{i\eta }, \ \ w_2>0, \\ f_{2,10}(\xi ,\eta )=\left( \mu _0+\lambda _1\sqrt{-\frac{w_3}{w_2}}\,\, csc^2(\frac{\xi \sqrt{-w_2}}{2})\right) \,e^{i\eta }, \ \ w_2<0, \ \ \ \xi =l_1x+l_2y-2\,a \left( -l_{{1}}{\frac{b_{{2}}s_{{2}}}{b_{{1}}}}+l_{{2}}s_{{2}} \right) t,\\ \eta =-{\frac{b_{{2}}s_{{2}}}{b_{{1}}}}x+ s_2y-{\frac{a \left( -{\epsilon }^{2} \left( {l_{{1}}}^{2}+{l_{{2} }}^{2} \right) {b_{{1}}}^{2}{\lambda _{{1}}}^{4}w_{{2}}+{\lambda _{{1}}} ^{4}{s_{{2}}}^{2} \left( {b_{{1}}}^{2}+{b_{{2}}}^{2} \right) \right) }{{b_{{1}}}^{2}{\lambda _{{1}}}^{4}}} t+\eta _0. \end{array} \right. \end{aligned}$$

(64)

$$\begin{aligned} \left\{ \begin{array}{ll} Case\,\, IV:\Rightarrow \,\, f_{2,11}(\xi ,\eta )=\left( \mu _0+\lambda _1\wp ^{-1}\left[ \frac{\xi \sqrt{w_3}}{2},\mathfrak {A_2},\mathfrak {A_3}\right] \right) \,e^{i\eta },\\ \xi =l_1x+l_2y-2\,a \left( -l_{{1}}{\frac{a{\epsilon }^{2}{l_{{1}}}^{2}w_{{0}}+a{\epsilon }^{2}{l _{{2}}}^{2}w_{{0}}+b_{{2}}{\lambda _{{1}}}^{2}s_{{2}}}{b_{{1}}{\lambda _ {{1}}}^{2}}}+l_{{2}}s_{{2}} \right) t,\,\,\, \ \ \eta =-{\frac{a{\epsilon }^{2}{l_{{1}}}^{2}w_{{0}}+a{\epsilon }^{2}{l _{{2}}}^{2}w_{{0}}+b_{{2}}{\lambda _{{1}}}^{2}s_{{2}}}{b_{{1}}{\lambda _ {{1}}}^{2}}}x+\\ s_2y-{\frac{a \left( {\epsilon }^{2} \left( {l_{{1}}}^{2}+{l_{{2}} }^{2} \right) \left( {a}^{2}{\epsilon }^{2} \left( {l_{{1}}}^{2}+{l_{{ 2}}}^{2} \right) {w_{{0}}}^{2}+6\,{b_{{1}}}^{2}{\lambda _{{1}}}^{2}{\mu _{{0}}}^{2}w_{{0}}+2\,ab_{{2}}{\lambda _{{1}}}^{2}s_{{2}}w_{{0}} \right) +{\lambda _{{1}}}^{4}{s_{{2}}}^{2} \left( {b_{{1}}}^{2}+{b_{{2 }}}^{2} \right) \right) }{{b_{{1}}}^{2}{\lambda _{{1}}}^{4}}} t+\eta _0, \end{array} \right. \end{aligned}$$

(65)

where $\mathfrak {A_2} = - \frac{4w_1}{w_3}$ and $\mathfrak {A_3} = - \frac{4w_0}{w_3}$ are called Weierstrass elliptic function in-variants.

$$\begin{aligned} \left\{ \begin{array}{ll} Case\,\, VI:\Rightarrow \,\,\frac{w_2}{w_0}=\frac{4\mu _0^2}{\lambda _1^2},\,\,\, f_{2,12}(\xi ,\eta )=[\mu _0+\lambda _1\phi ^{-1}(\xi )]\,e^{i\eta },\,\ \phi (\xi )=-\frac{w_1}{2w_2}+\exp (\varepsilon \xi \sqrt{w_2}),\,\, w_2>0, \ w_0=\frac{w_1^2}{4w_2},\\ f_{2,13}(\xi ,\eta )=[\mu _0+\lambda _1\phi ^{-1}(\xi )]\,e^{i\eta },\,\,\ \ \phi (\xi )=-\frac{w_1}{2w_2}+\frac{\varepsilon w_1}{2w_2}\,\,\sin (\xi \sqrt{-w_2}),\,\, w_2<0, \ w_0=\frac{w_1^2}{4w_2},\\ f_{2,14}(\xi ,\eta )=[\mu _0+\lambda _1\phi ^{-1}(\xi )]\,e^{i\eta },\,\,\ \ \phi (\xi )=-\frac{w_1}{2w_2}+\frac{\varepsilon w_1}{2w_2}\,\,\sinh (2\xi \sqrt{w_2}),\,\, w_2>0, \ w_0=\frac{w_1^2}{4w_2},\\ f_{2,15}(\xi ,\eta )=[\mu _0+\lambda _1\phi ^{-1}(\xi )]\,e^{i\eta },\,\,\ \ \phi (\xi )=\varepsilon \sqrt{\frac{w_0}{w_2}}\,\,\sinh (\xi \sqrt{w_2}),\,\, w_2>0, \ w_0>0,\ \\ \xi =l_1x+l_2y-2\,a \left( -l_{{1}}{\frac{a{\epsilon }^{2}{l_{{1}}}^{2}w_{{0}}+a{\epsilon }^{2}{l _{{2}}}^{2}w_{{0}}+b_{{2}}{\lambda _{{1}}}^{2}s_{{2}}}{b_{{1}}{\lambda _ {{1}}}^{2}}}+l_{{2}}s_{{2}} \right) t,\,\,\, \ \ {\eta =-{\frac{a{\epsilon }^{2}{l_{{1}}}^{2}w_{{0}}+a{\epsilon }^{2}{l _{{2}}}^{2}w_{{0}}+b_{{2}}{\lambda _{{1}}}^{2}s_{{2}}}{b_{{1}}{\lambda _ {{1}}}^{2}}}x+}\\ {s_2y-{\frac{a \left( {\epsilon }^{2} \left( {l_{{1}}}^{2}+{l_{{2}} }^{2} \right) \left( {a}^{2}{\epsilon }^{2} \left( {l_{{1}}}^{2}+{l_{{ 2}}}^{2} \right) {w_{{0}}}^{2}-{b_{{1}}}^{2}{\lambda _{{1}}}^{2} \left( {\lambda _{{1}}}^{2}w_{{2}}-6\,{\mu _{{0}}}^{2}w_{{0}} \right) + 2\,ab_{{2}}{\lambda _{{1}}}^{2}s_{{2}}w_{{0}} \right) +{\lambda _{{1}}}^ {4}{s_{{2}}}^{2} \left( {b_{{1}}}^{2}+{b_{{2}}}^{2} \right) \right) }{{b_{{1}}}^{2}{\lambda _{{1}}}^{4}}} t+\eta _0,} \end{array} \right. \end{aligned}$$

(66)

$$\begin{aligned} \left\{ \begin{array}{ll} Case\,\, VII:\Rightarrow \,\, f_{1,16}(\xi ,\eta )=\left( \mu _0-\lambda _1\left[ \frac{w_2sec^2\left( \frac{\xi \sqrt{-w_2}}{2}\right) }{2\varepsilon \sqrt{-w_2w_4} \tan \left( \frac{\xi \sqrt{-w_2}}{2}\right) +w_3}\right] ^{-1}\right) \,e^{i\eta },\,\, w_2<0, \\ f_{1,17}(\xi ,\eta )=\left( \mu _0+\lambda _1\left[ \frac{w_2sech^2\left( \frac{\xi \sqrt{w_2}}{2}\right) }{2\varepsilon \sqrt{-w_2w_4} \tanh \left( \frac{\xi \sqrt{w_2}}{2}\right) -w_3}\right] ^{-1}\right) \,e^{i\eta },\,\, w_2>0, \ \ w_3\ne 2\varepsilon \sqrt{w_2w_4},\\ f_{1,18}(\xi ,\eta )=\left( \mu _0+\lambda _1\left[ \frac{\varepsilon }{2}\sqrt{\frac{w_2}{w_4}}\left[ 1+\tanh \left( \frac{\xi \sqrt{w_2}}{2}\right) \right] \right] ^{-1}\right) \,e^{i\eta },\,\, w_2>0,\ \ w_3=2\varepsilon \sqrt{w_2w_4},\\ \xi =l_1x+l_2y-2\,a \left( -l_{{1}}{\frac{b_{{2}}s_{{2}}}{b_{{1}}}}+l_{{2}}s_{{2}} \right) t,\,\,\, \\ \eta =-{\frac{b_{{2}}s_{{2}}}{b_{{1}}}}x+ s_2y-{\frac{a \left( -{\epsilon }^{2} \left( {l_{{1}}}^{2}+{l_{{2} }}^{2} \right) {b_{{1}}}^{2}{\lambda _{{1}}}^{4}w_{{2}}+{\lambda _{{1}}} ^{4}{s_{{2}}}^{2} \left( {b_{{1}}}^{2}+{b_{{2}}}^{2} \right) \right) }{{b_{{1}}}^{2}{\lambda _{{1}}}^{4}}} t+\eta _0. \end{array} \right. \end{aligned}$$

(67)

$$\begin{aligned} \mu _0=0,\ \ s_{{1}}=-{\frac{a{\epsilon }^{2}{l_{{1}}}^{2}w_{{0}}+a{\epsilon }^{2}{l _{{2}}}^{2}w_{{0}}+b_{{2}}{\lambda _{{1}}}^{2}s_{{2}}}{b_{{1}}{\lambda _ {{1}}}^{2}}},\ \ s_2=s_2,\ \ w_{{1}}=w_{{3}}=0,\ \ w_{{4}}={\frac{{\mu _{{1}}}^{2}w_{{0}}}{{\lambda _{{1}}}^{2}}}, \end{aligned}$$

(68)

$$\begin{aligned}s_{{3}}=-{\frac{a \left( {\epsilon }^{2} \left( {l_{{1}}}^{2}+{l_{{2}} }^{2} \right) \left( {a}^{2}{\epsilon }^{2} \left( {l_{{1}}}^{2}+{l_{{ 2}}}^{2} \right) {w_{{0}}}^{2}-{b_{{1}}}^{2}{\lambda _{{1}}}^{4}w_{{2}} +6\,{b_{{1}}}^{2}{\lambda _{{1}}}^{3}\mu _{{1}}w_{{0}}+2\,ab_{{2}}{ \lambda _{{1}}}^{2}s_{{2}}w_{{0}} \right) +{\lambda _{{1}}}^{4}{s_{{2}}} ^{2} \left( {b_{{1}}}^{2}+{b_{{2}}}^{2} \right) \right) }{{b_{{1}}}^{ 2}{\lambda _{{1}}}^{4}}}. \end{aligned}$$

Third class of solutions (according to parameters (68)):

$$\begin{aligned} \left\{ \begin{array}{ll} Case\,\, II:\Rightarrow \,\, f_{3,1}(\xi ,\eta )=[\mu _1\phi (\xi )+\lambda _1\phi ^{-1}(\xi )]\,e^{i\eta },\,\, \phi (\xi )=\varepsilon \sqrt{-\frac{w_2}{2w_4}}\,\, \tanh (\xi \sqrt{-\frac{w_2}{2}}) \& w_2<0,\ \ w_4>0, \ w_0=\frac{w_2^2}{4w_4}, \\ f_{3,2}(\xi ,\eta )=[\mu _1\phi (\xi )+\lambda _1\phi ^{-1}(\xi )]\,e^{i\eta },\,\, \phi (\xi )=\varepsilon \sqrt{\frac{w_2}{2w_4}}\,\, \tan (\xi \sqrt{\frac{w_2}{2}}),\, \& w_2>0,\ \ w_4>0, \ w_0=\frac{w_2^2}{4w_4}, \\ f_{3,3}(\xi ,\eta )=[\mu _1\phi (\xi )+\lambda _1\phi ^{-1}(\xi )]\,e^{i\eta },\,\, \phi (\xi )=\sqrt{\frac{-w_2m^2}{(2m^2-1)w_4}}\,\, cn(\xi \sqrt{\frac{w_2}{2m^2-1}}),\, \& w_2>0,\ \ w_4<0, \ w_0=\frac{w_2^2m^2(1-m^2)}{(2m^2-1)^2w_4}, \\ f_{3,4}(\xi ,\eta )=[\mu _1\phi (\xi )+\lambda _1\phi ^{-1}(\xi )]\,e^{i\eta },\,\, \phi (\xi )=\sqrt{\frac{-m^2w_2}{(2-m^2)w_4}}\,\, dn(\xi \sqrt{\frac{w_2}{2-m^2}}),\, \& w_2>0,\ \ w_4<0, \ w_0=\frac{w_2^2(1-m^2)}{(2-m^2)^2w_4}, \\ f_{3,5}(\xi ,\eta )=[\mu _1\phi (\xi )+\lambda _1\phi ^{-1}(\xi )]\,e^{i\eta },\,\, \phi (\xi )=\sqrt{\frac{-w_2m^2}{(1+m^2)w_4}}\,\, sn(\xi \sqrt{\frac{-w_2}{1+m^2}}),\, \& w_2<0,\ \ w_4>0, \ w_0=\frac{w_2^2m^2}{(1+m^2)^2w_4}, \\ \xi =l_1x+l_2y-2\,a \left( -l_{{1}}{\frac{a{\epsilon }^{2}{l_{{1}}}^{2}w_{{0}}+a{\epsilon }^{2}{l _{{2}}}^{2}w_{{0}}+b_{{2}}{\lambda _{{1}}}^{2}s_{{2}}}{b_{{1}}{\lambda _ {{1}}}^{2}}}+l_{{2}}s_{{2}} \right) t,\,\,\, \ \ \eta =-{\frac{a{\epsilon }^{2}{l_{{1}}}^{2}w_{{0}}+a{\epsilon }^{2}{l _{{2}}}^{2}w_{{0}}+b_{{2}}{\lambda _{{1}}}^{2}s_{{2}}}{b_{{1}}{\lambda _ {{1}}}^{2}}}x+\\ s_2y-{\frac{a \left( {\epsilon }^{2} \left( {l_{{1}}}^{2}+{l_{{2}} }^{2} \right) \left( {a}^{2}{\epsilon }^{2} \left( {l_{{1}}}^{2}+{l_{{ 2}}}^{2} \right) {w_{{0}}}^{2}-{b_{{1}}}^{2}{\lambda _{{1}}}^{4}w_{{2}} +6\,{b_{{1}}}^{2}{\lambda _{{1}}}^{3}\mu _{{1}}w_{{0}}+2\,ab_{{2}}{ \lambda _{{1}}}^{2}s_{{2}}w_{{0}} \right) +{\lambda _{{1}}}^{4}{s_{{2}}} ^{2} \left( {b_{{1}}}^{2}+{b_{{2}}}^{2} \right) \right) }{{b_{{1}}}^{ 2}{\lambda _{{1}}}^{4}}} t+\eta _0, \end{array} \right. \end{aligned}$$

(69)

where m is the modulus of the Jacobi elliptic functions and $0 < m \le 1$.

Remark 7

When $m\rightarrow 1$, then $cn(\xi , m)\rightarrow { sech}(\xi )$ hence the bright-singular soliton solution (69) will be as

$$\begin{aligned} f_{3,3}(x,t)= & \left\{ \mu _1\sqrt{\frac{-w_2}{w_4}}\,\, { sech}\left( \sqrt{w_2}\left[ l_1x+l_2y-2\,a \left( -l_{{1}}{\frac{a{\epsilon }^{2}{l_{{1}}}^{2}w_{{0}}+a{\epsilon }^{2}{l _{{2}}}^{2}w_{{0}}+b_{{2}}{\lambda _{{1}}}^{2}s_{{2}}}{b_{{1}}{\lambda _ {{1}}}^{2}}}+l_{{2}}s_{{2}} \right) t\right] \right) \right. \nonumber \\ & +\left. \lambda _1\sqrt{\frac{-w_4}{w_2}}\,\, { csch}\left( \sqrt{w_2}\left[ l_1x+l_2y-2\,a \left( -l_{{1}}{\frac{a{\epsilon }^{2}{l_{{1}}}^{2}w_{{0}}+a{\epsilon }^{2}{l _{{2}}}^{2}w_{{0}}+b_{{2}}{\lambda _{{1}}}^{2}s_{{2}}}{b_{{1}}{\lambda _ {{1}}}^{2}}}+l_{{2}}s_{{2}} \right) t\right] \right) \right\} \nonumber \\ & \times e^{i\left[ -{\frac{a{\epsilon }^{2}{l_{{1}}}^{2}w_{{0}}+a{\epsilon }^{2}{l _{{2}}}^{2}w_{{0}}+b_{{2}}{\lambda _{{1}}}^{2}s_{{2}}}{b_{{1}}{\lambda _ {{1}}}^{2}}}x+ s_2y-{\frac{a \left( {\epsilon }^{2} \left( {l_{{1}}}^{2}+{l_{{2}} }^{2} \right) \left( {a}^{2}{\epsilon }^{2} \left( {l_{{1}}}^{2}+{l_{{ 2}}}^{2} \right) {w_{{0}}}^{2}-{b_{{1}}}^{2}{\lambda _{{1}}}^{4}w_{{2}} +6\,{b_{{1}}}^{2}{\lambda _{{1}}}^{3}\mu _{{1}}w_{{0}}+2\,ab_{{2}}{ \lambda _{{1}}}^{2}s_{{2}}w_{{0}} \right) +{\lambda _{{1}}}^{4}{s_{{2}}} ^{2} \left( {b_{{1}}}^{2}+{b_{{2}}}^{2} \right) \right) }{{b_{{1}}}^{ 2}{\lambda _{{1}}}^{4}}} t+\eta _0\right] }, \end{aligned}$$

(70)

so that $w_2w_4<0$.

Remark 8

When $m\rightarrow 1$, then $dn(\xi , m)\rightarrow { sech}(\xi )$ hence the bright-singular soliton solution (69) will be as

$$\begin{aligned} f_{3,4}(x,t)= & \left\{ \mu _1\sqrt{\frac{-w_2}{w_4}}\,\, { sech}\left( \sqrt{w_2}\left[ l_1x+l_2y-2\,a \left( -l_{{1}}{\frac{a{\epsilon }^{2}{l_{{1}}}^{2}w_{{0}}+a{\epsilon }^{2}{l _{{2}}}^{2}w_{{0}}+b_{{2}}{\lambda _{{1}}}^{2}s_{{2}}}{b_{{1}}{\lambda _ {{1}}}^{2}}}+l_{{2}}s_{{2}} \right) t\right] \right) \right. \nonumber \\ & + \left. \lambda _1\sqrt{\frac{-w_4}{w_2}}\,\, { csch}\left( \sqrt{w_2}\left[ l_1x+l_2y-2\,a \left( -l_{{1}}{\frac{a{\epsilon }^{2}{l_{{1}}}^{2}w_{{0}}+a{\epsilon }^{2}{l _{{2}}}^{2}w_{{0}}+b_{{2}}{\lambda _{{1}}}^{2}s_{{2}}}{b_{{1}}{\lambda _ {{1}}}^{2}}}+l_{{2}}s_{{2}} \right) t\right] \right) \right\} \nonumber \\ & \times e^{i\left[ -{\frac{a{\epsilon }^{2}{l_{{1}}}^{2}w_{{0}}+a{\epsilon }^{2}{l _{{2}}}^{2}w_{{0}}+b_{{2}}{\lambda _{{1}}}^{2}s_{{2}}}{b_{{1}}{\lambda _ {{1}}}^{2}}}x+ s_2y-{\frac{a \left( {\epsilon }^{2} \left( {l_{{1}}}^{2}+{l_{{2}} }^{2} \right) \left( {a}^{2}{\epsilon }^{2} \left( {l_{{1}}}^{2}+{l_{{ 2}}}^{2} \right) {w_{{0}}}^{2}-{b_{{1}}}^{2}{\lambda _{{1}}}^{4}w_{{2}} +6\,{b_{{1}}}^{2}{\lambda _{{1}}}^{3}\mu _{{1}}w_{{0}}+2\,ab_{{2}}{ \lambda _{{1}}}^{2}s_{{2}}w_{{0}} \right) +{\lambda _{{1}}}^{4}{s_{{2}}} ^{2} \left( {b_{{1}}}^{2}+{b_{{2}}}^{2} \right) \right) }{{b_{{1}}}^{ 2}{\lambda _{{1}}}^{4}}} t+\eta _0\right] }, \end{aligned}$$

(71)

so that $w_2w_4<0$.

Remark 9

When $m\rightarrow 1$, then $sn(\xi , m)\rightarrow { tanh}(\xi )$ hence the soliton solution (69) will be as

$$\begin{aligned} f_{3,5}(x,t)= & \left\{ \mu _1\sqrt{\frac{-w_2}{2w_4}}\,\, { tanh}\left( \sqrt{-\frac{w_2}{2}}\left[ l_1x+l_2y-2\,a \left( -l_{{1}}{\frac{a{\epsilon }^{2}{l_{{1}}}^{2}w_{{0}}+a{\epsilon }^{2}{l _{{2}}}^{2}w_{{0}}+b_{{2}}{\lambda _{{1}}}^{2}s_{{2}}}{b_{{1}}{\lambda _ {{1}}}^{2}}}+l_{{2}}s_{{2}} \right) t\right] \right) \right. \nonumber \\ & +\left. \lambda _1\sqrt{\frac{-2w_2}{w_4}}\,\, { coth}\left( \sqrt{-\frac{w_2}{2}}\left[ l_1x+l_2y-2\,a \left( -l_{{1}}{\frac{a{\epsilon }^{2}{l_{{1}}}^{2}w_{{0}}+a{\epsilon }^{2}{l _{{2}}}^{2}w_{{0}}+b_{{2}}{\lambda _{{1}}}^{2}s_{{2}}}{b_{{1}}{\lambda _ {{1}}}^{2}}}+l_{{2}}s_{{2}} \right) t\right] \right) \right\} \nonumber \\ & \times e^{i\left[ -{\frac{a{\epsilon }^{2}{l_{{1}}}^{2}w_{{0}}+a{\epsilon }^{2}{l _{{2}}}^{2}w_{{0}}+b_{{2}}{\lambda _{{1}}}^{2}s_{{2}}}{b_{{1}}{\lambda _ {{1}}}^{2}}}x+ s_2y-{\frac{a \left( {\epsilon }^{2} \left( {l_{{1}}}^{2}+{l_{{2}} }^{2} \right) \left( {a}^{2}{\epsilon }^{2} \left( {l_{{1}}}^{2}+{l_{{ 2}}}^{2} \right) {w_{{0}}}^{2}-{b_{{1}}}^{2}{\lambda _{{1}}}^{4}w_{{2}} +6\,{b_{{1}}}^{2}{\lambda _{{1}}}^{3}\mu _{{1}}w_{{0}}+2\,ab_{{2}}{ \lambda _{{1}}}^{2}s_{{2}}w_{{0}} \right) +{\lambda _{{1}}}^{4}{s_{{2}}} ^{2} \left( {b_{{1}}}^{2}+{b_{{2}}}^{2} \right) \right) }{{b_{{1}}}^{ 2}{\lambda _{{1}}}^{4}}} t+\eta _0\right] }, \end{aligned}$$

(72)

so that $w_2<0$ and $w_2w_4<0$.

$$\begin{aligned} \left\{ \begin{array}{ll} Case\,\, III:\Rightarrow \,\, f_{2,9}(\xi ,\eta )=\left( \mu _0+\lambda _1\sqrt{-\frac{w_3}{w_2}}\,\, csch^2(\frac{\xi \sqrt{w_2}}{2})\right) \,e^{i\eta }, \ \ w_2>0, \\ f_{2,10}(\xi ,\eta )=\left( \mu _0+\lambda _1\sqrt{-\frac{w_3}{w_2}}\,\, csc^2(\frac{\xi \sqrt{-w_2}}{2})\right) \,e^{i\eta }, \ \ w_2<0, \ \ \ \xi =l_1x+l_2y-2\,a \left( -l_{{1}}{\frac{b_{{2}}s_{{2}}}{b_{{1}}}}+l_{{2}}s_{{2}} \right) t,\\ \eta =-{\frac{b_{{2}}s_{{2}}}{b_{{1}}}}x+ s_2y-{\frac{a \left( -{\epsilon }^{2} \left( {l_{{1}}}^{2}+{l_{{2} }}^{2} \right) {b_{{1}}}^{2}{\lambda _{{1}}}^{4}w_{{2}}+{\lambda _{{1}}} ^{4}{s_{{2}}}^{2} \left( {b_{{1}}}^{2}+{b_{{2}}}^{2} \right) \right) }{{b_{{1}}}^{2}{\lambda _{{1}}}^{4}}} t+\eta _0, \end{array} \right. \end{aligned}$$

(73)

$$\begin{aligned} s_{{1}}= & -{\frac{a{\epsilon }^{2}w_{{0}} \left( {l_{{1}}}^{2}+{l_{{2}}} ^{2} \right) +b_{{2}}{\lambda _{{1}}}^{2}s_{{2}}}{b_{{1}}{\lambda _{{1}} }^{2}}},\ \ w_{{1}}=4\,{\frac{\mu _{{0}}w_{{0}}}{\lambda _{{1}}}},\ \ w_{{2}}=-2\,{\frac{w_{{0}} \left( \lambda _{{1}}\mu _{{1}}-2\,{\mu _{{0} }}^{2} \right) }{{\lambda _{{1}}}^{2}}},\nonumber \\ w_{{3}}= & 4\,{\frac{\mu _{{0}}\mu _{{1}}w_{{0}}}{{\lambda _{{1}}}^{2}}},\ \ w_{{4}}={\frac{{\mu _{{1}}}^{2}w_{{0}}}{{\lambda _{{1}}}^{2}}}, \end{aligned}$$

(74)

$$\begin{aligned}s_{{3}}=-{\frac{a \left( {\epsilon }^{2}w_{{0}} \left( {l_{{1}}}^{2}+{ l_{{2}}}^{2} \right) \left( {a}^{2}w_{{0}}{\epsilon }^{2} \left( {l_{{ 1}}}^{2}+{l_{{2}}}^{2} \right) +8\,{b_{{1}}}^{2}{\lambda _{{1}}}^{3}\mu _{{1}}+2\,{b_{{1}}}^{2}{\lambda _{{1}}}^{2}{\mu _{{0}}}^{2}+2\,ab_{{2}}{ \lambda _{{1}}}^{2}s_{{2}} \right) +{b_{{1}}}^{2}{\lambda _{{1}}}^{4}{s_ {{2}}}^{2}+{b_{{2}}}^{2}{\lambda _{{1}}}^{4}{s_{{2}}}^{2} \right) }{{b_ {{1}}}^{2}{\lambda _{{1}}}^{4}}}. \end{aligned}$$

Fourth class of solutions (according to parameters (74)):

$$\begin{aligned} \left\{ \begin{array}{ll} Case\,\, II:\Rightarrow \,\, \mu _0=0,\,\,\, f_{4,1}(\xi ,\eta )=[\mu _1\phi (\xi )+\lambda _1\phi ^{-1}(\xi )]\,e^{i\eta },\,\, \phi (\xi )=\varepsilon \sqrt{-\frac{w_2}{2w_4}}\,\, \tanh (\xi \sqrt{-\frac{w_2}{2}}) \& w_2<0,\ \ w_4>0, \ w_0=\frac{w_2^2}{4w_4}, \\ f_{4,2}(\xi ,\eta )=[\mu _1\phi (\xi )+\lambda _1\phi ^{-1}(\xi )]\,e^{i\eta },\,\, \phi (\xi )=\varepsilon \sqrt{\frac{w_2}{2w_4}}\,\, \tan (\xi \sqrt{\frac{w_2}{2}}),\, \& w_2>0,\ \ w_4>0, \ w_0=\frac{w_2^2}{4w_4}, \\ f_{4,3}(\xi ,\eta )=[\mu _1\phi (\xi )+\lambda _1\phi ^{-1}(\xi )]\,e^{i\eta },\,\, \phi (\xi )=\sqrt{\frac{-w_2m^2}{(2m^2-1)w_4}}\,\, cn(\xi \sqrt{\frac{w_2}{2m^2-1}}),\, \& w_2>0,\ \ w_4<0, \ w_0=\frac{w_2^2m^2(1-m^2)}{(2m^2-1)^2w_4}, \\ f_{4,4}(\xi ,\eta )=[\mu _1\phi (\xi )+\lambda _1\phi ^{-1}(\xi )]\,e^{i\eta },\,\, \phi (\xi )=\sqrt{\frac{-m^2w_2}{(2-m^2)w_4}}\,\, dn(\xi \sqrt{\frac{w_2}{2-m^2}}),\, \& w_2>0,\ \ w_4<0, \ w_0=\frac{w_2^2(1-m^2)}{(2-m^2)^2w_4}, \\ f_{4,5}(\xi ,\eta )=[\mu _1\phi (\xi )+\lambda _1\phi ^{-1}(\xi )]\,e^{i\eta },\,\, \phi (\xi )=\sqrt{\frac{-w_2m^2}{(1+m^2)w_4}}\,\, sn(\xi \sqrt{\frac{-w_2}{1+m^2}}),\, \& w_2<0,\ \ w_4>0, \ w_0=\frac{w_2^2m^2}{(1+m^2)^2w_4}, \\ \xi =l_1x+l_2y-2\,a \left( -l_{{1}}{\frac{a{\epsilon }^{2}w_{{0}} \left( {l_{{1}}}^{2}+{l_{{2}}} ^{2} \right) +b_{{2}}{\lambda _{{1}}}^{2}s_{{2}}}{b_{{1}}{\lambda _{{1}} }^{2}}}+l_{{2}}s_{{2}} \right) t,\,\,\, \ \ \eta =-{\frac{a{\epsilon }^{2}w_{{0}} \left( {l_{{1}}}^{2}+{l_{{2}}} ^{2} \right) +b_{{2}}{\lambda _{{1}}}^{2}s_{{2}}}{b_{{1}}{\lambda _{{1}} }^{2}}}x+\\ s_2y-{\frac{a \left( {\epsilon }^{2}w_{{0}} \left( {l_{{1}}}^{2}+{ l_{{2}}}^{2} \right) \left( {a}^{2}w_{{0}}{\epsilon }^{2} \left( {l_{{ 1}}}^{2}+{l_{{2}}}^{2} \right) +8\,{b_{{1}}}^{2}{\lambda _{{1}}}^{3}\mu _{{1}}+2\,ab_{{2}}{ \lambda _{{1}}}^{2}s_{{2}} \right) +{b_{{1}}}^{2}{\lambda _{{1}}}^{4}{s_ {{2}}}^{2}+{b_{{2}}}^{2}{\lambda _{{1}}}^{4}{s_{{2}}}^{2} \right) }{{b_ {{1}}}^{2}{\lambda _{{1}}}^{4}}} t+\eta _0, \end{array} \right. \end{aligned}$$

(75)

where m is the modulus of the Jacobi elliptic functions and $0 < m \le 1$.

Remark 10

When $m\rightarrow 1$, then $cn(\xi , m)\rightarrow { sech}(\xi )$ hence the bright-singular soliton solution (75) will be as

$$\begin{aligned} f_{4,3}(x,t)= & \left\{ \mu _1\sqrt{\frac{-w_2}{w_4}}\,\, { sech}\left( \sqrt{w_2}\left[ l_1x+l_2y-2\,a \left( -l_{{1}}{\frac{a{\epsilon }^{2}w_{{0}} \left( {l_{{1}}}^{2}+{l_{{2}}} ^{2} \right) +b_{{2}}{\lambda _{{1}}}^{2}s_{{2}}}{b_{{1}}{\lambda _{{1}} }^{2}}}+l_{{2}}s_{{2}} \right) t\right] \right) \right. \nonumber \\ & +\left. \lambda _1\sqrt{\frac{-w_4}{w_2}}\,\, { csch}\left( \sqrt{w_2}\left[ l_1x+l_2y-2\,a \left( -l_{{1}}{\frac{a{\epsilon }^{2}w_{{0}} \left( {l_{{1}}}^{2}+{l_{{2}}} ^{2} \right) +b_{{2}}{\lambda _{{1}}}^{2}s_{{2}}}{b_{{1}}{\lambda _{{1}} }^{2}}}+l_{{2}}s_{{2}} \right) t\right] \right) \right\} \nonumber \\\times & e^{i\left[ -{\frac{a{\epsilon }^{2}w_{{0}} \left( {l_{{1}}}^{2}+{l_{{2}}} ^{2} \right) +b_{{2}}{\lambda _{{1}}}^{2}s_{{2}}}{b_{{1}}{\lambda _{{1}} }^{2}}}x+ s_2y-{\frac{a \left( {\epsilon }^{2}w_{{0}} \left( {l_{{1}}}^{2}+{ l_{{2}}}^{2} \right) \left( {a}^{2}w_{{0}}{\epsilon }^{2} \left( {l_{{ 1}}}^{2}+{l_{{2}}}^{2} \right) +8\,{b_{{1}}}^{2}{\lambda _{{1}}}^{3}\mu _{{1}}+2\,ab_{{2}}{ \lambda _{{1}}}^{2}s_{{2}} \right) +{b_{{1}}}^{2}{\lambda _{{1}}}^{4}{s_ {{2}}}^{2}+{b_{{2}}}^{2}{\lambda _{{1}}}^{4}{s_{{2}}}^{2} \right) }{{b_ {{1}}}^{2}{\lambda _{{1}}}^{4}}} t+\eta _0\right] }, \end{aligned}$$

(76)

so that $w_2w_4<0$.

Remark 11

When $m\rightarrow 1$, then $dn(\xi , m)\rightarrow { sech}(\xi )$ hence the bright-singular soliton solution (75) will be as

$$\begin{aligned} f_{4,4}(x,t)= & \left\{ \mu _1\sqrt{\frac{-w_2}{w_4}}\,\, { sech}\left( \sqrt{w_2}\left[ l_1x+l_2y-2\,a \left( -l_{{1}}{\frac{a{\epsilon }^{2}w_{{0}} \left( {l_{{1}}}^{2}+{l_{{2}}} ^{2} \right) +b_{{2}}{\lambda _{{1}}}^{2}s_{{2}}}{b_{{1}}{\lambda _{{1}} }^{2}}}+l_{{2}}s_{{2}} \right) t\right] \right) \right. \nonumber \\ & + \left. \lambda _1\sqrt{\frac{-w_4}{w_2}}\,\, { csch}\left( \sqrt{w_2}\left[ l_1x+l_2y-2\,a \left( -l_{{1}}{\frac{a{\epsilon }^{2}{l_{{1}}}^{2}w_{{0}}+a{\epsilon }^{2}{l _{{2}}}^{2}w_{{0}}+b_{{2}}{\lambda _{{1}}}^{2}s_{{2}}}{b_{{1}}{\lambda _ {{1}}}^{2}}}+l_{{2}}s_{{2}} \right) t\right] \right) \right\} \nonumber \\\times & e^{i\left[ -{\frac{a{\epsilon }^{2}w_{{0}} \left( {l_{{1}}}^{2}+{l_{{2}}} ^{2} \right) +b_{{2}}{\lambda _{{1}}}^{2}s_{{2}}}{b_{{1}}{\lambda _{{1}} }^{2}}}x+ s_2y-{\frac{a \left( {\epsilon }^{2}w_{{0}} \left( {l_{{1}}}^{2}+{ l_{{2}}}^{2} \right) \left( {a}^{2}w_{{0}}{\epsilon }^{2} \left( {l_{{ 1}}}^{2}+{l_{{2}}}^{2} \right) +8\,{b_{{1}}}^{2}{\lambda _{{1}}}^{3}\mu _{{1}}+2\,ab_{{2}}{ \lambda _{{1}}}^{2}s_{{2}} \right) +{b_{{1}}}^{2}{\lambda _{{1}}}^{4}{s_ {{2}}}^{2}+{b_{{2}}}^{2}{\lambda _{{1}}}^{4}{s_{{2}}}^{2} \right) }{{b_ {{1}}}^{2}{\lambda _{{1}}}^{4}}} t+\eta _0\right] }, \end{aligned}$$

(77)

so that $w_2w_4<0$.

Remark 12

When $m\rightarrow 1$, then $sn(\xi , m)\rightarrow { tanh}(\xi )$ hence the soliton solution (75) will be as

$$\begin{aligned} f_{4,5}(x,t)= & \left\{ \mu _1\sqrt{\frac{-w_2}{2w_4}}\,\, { tanh}\left( \sqrt{-\frac{w_2}{2}}\left[ l_1x+l_2y-2\,a \left( -l_{{1}}{\frac{a{\epsilon }^{2}w_{{0}} \left( {l_{{1}}}^{2}+{l_{{2}}} ^{2} \right) +b_{{2}}{\lambda _{{1}}}^{2}s_{{2}}}{b_{{1}}{\lambda _{{1}} }^{2}}}+l_{{2}}s_{{2}} \right) t\right] \right) \right. \nonumber \\ & + \left. \lambda _1\sqrt{\frac{-2w_2}{w_4}}\,\, { coth}\left( \sqrt{-\frac{w_2}{2}}\left[ l_1x+l_2y-2\,a \left( -l_{{1}}{\frac{a{\epsilon }^{2}w_{{0}} \left( {l_{{1}}}^{2}+{l_{{2}}} ^{2} \right) +b_{{2}}{\lambda _{{1}}}^{2}s_{{2}}}{b_{{1}}{\lambda _{{1}} }^{2}}}+l_{{2}}s_{{2}} \right) t\right] \right) \right\} \nonumber \\ & \times e^{i\left[ -{\frac{a{\epsilon }^{2}w_{{0}} \left( {l_{{1}}}^{2}+{l_{{2}}} ^{2} \right) +b_{{2}}{\lambda _{{1}}}^{2}s_{{2}}}{b_{{1}}{\lambda _{{1}} }^{2}}}x+ s_2y-{\frac{a \left( {\epsilon }^{2}w_{{0}} \left( {l_{{1}}}^{2}+{ l_{{2}}}^{2} \right) \left( {a}^{2}w_{{0}}{\epsilon }^{2} \left( {l_{{ 1}}}^{2}+{l_{{2}}}^{2} \right) +8\,{b_{{1}}}^{2}{\lambda _{{1}}}^{3}\mu _{{1}}+2\,ab_{{2}}{ \lambda _{{1}}}^{2}s_{{2}} \right) +{b_{{1}}}^{2}{\lambda _{{1}}}^{4}{s_ {{2}}}^{2}+{b_{{2}}}^{2}{\lambda _{{1}}}^{4}{s_{{2}}}^{2} \right) }{{b_ {{1}}}^{2}{\lambda _{{1}}}^{4}}} t+\eta _0\right] }, \end{aligned}$$

(78)

so that $w_2<0$ and $w_2w_4<0$.

$$\begin{aligned} \left\{ \begin{array}{ll} Case\,\, V:\Rightarrow \,\,\,\,\mu _1=0,\ \ f_{4,6}(\xi ,\eta )=[\mu _0+\lambda _1\phi ^{-1}(\xi )]\,e^{i\eta },\,\, \phi (\xi )=\frac{4w_3}{w_3^2\xi ^2-4w_4},\,\, w_4\ne 0, \\ f_{4,7}(\xi ,\eta )=[\mu _0+\lambda _1\phi ^{-1}(\xi )]\,e^{i\eta },\,\,\, \phi (\xi )=\frac{w_3}{2w_4}\,\exp \left( \frac{\varepsilon \xi w_3}{2\sqrt{-w_4}}\right) ,\,\, w_4<0, \\ \xi =l_1x+l_2y-2\,a \left( -l_{{1}}{\frac{a{\epsilon }^{2}w_{{0}} \left( {l_{{1}}}^{2}+{l_{{2}}} ^{2} \right) +b_{{2}}{\lambda _{{1}}}^{2}s_{{2}}}{b_{{1}}{\lambda _{{1}} }^{2}}}+l_{{2}}s_{{2}} \right) t,\\ \eta =-{\frac{b_{{2}}s_{{2}}}{b_{{1}}}}x+ s_2y-{\frac{a \left( {\epsilon }^{2}w_{{0}} \left( {l_{{1}}}^{2}+{ l_{{2}}}^{2} \right) \left( {a}^{2}w_{{0}}{\epsilon }^{2} \left( {l_{{ 1}}}^{2}+{l_{{2}}}^{2} \right) +2\,{b_{{1}}}^{2}{\lambda _{{1}}}^{2}{ \mu _{{0}}}^{2}+2\,ab_{{2}}{\lambda _{{1}}}^{2}s_{{2}} \right) +{\lambda _{{1}}}^{4}{s_{{2}}}^{2} \left( {b_{{1}}}^{2}+{b_{{2}}}^{2} \right) \right) }{{b_{{1}}}^{2}{\lambda _{{1}}}^{4}}} t+\eta _0, \end{array} \right. \end{aligned}$$

(79)

Fig. 1

Plot of bright soliton solution (20) for the solutions (a, c) $\Re (f)$, (b, d) $\Im (f)$, where $f=f_{1,1}(x,t)$.

Fig. 2

Plot of singular periodic solution (20) for the solutions (a, c) $\Re (f)$, (b, d) $\Im (f)$, where $f=f_{1,2}(x,t)$.

Fig. 3

Plot of dark soliton solution (21) for the solutions (a, c) $\Re (f)$, (b, d) $\Im (f)$, where $f=f_{1,4}(x,t)$.

Fig.4

Plot of periodic solution (21) for the solutions (a, c) $\Re (f)$, (b, d) $\Im (f)$, where $f=f_{1,5}(x,t)$.

Outcomes and visual presentation

Our research encompassed a thorough investigation into the CNLS equations derived from a quantum mechanics model in a two-dimensional space with time evolution. We obtained a variety of solutions by employing METFM. Using the METFM, we discovered bell-shaped and shock solitons. In this section, we provide 3D and density diagrams of some selected solutions to show the physical behavior of some extracted solutions.

Fig. 5

Plot of dark soliton solution (40) for the solutions (a, c) $\Re (f)$, (b, d) $\Im (f)$, where $f=f_{4,1}(x,t)$.

Fig. 6

Plot of periodic solution (40) for the solutions (a, c) $\Re (f)$, (b, d) $\Im (f)$, where $f=f_{4,2}(x,t)$.

(1+1)-dimensional CNLSE:

Figure 1 shows the solution with bright soliton type of Eq. (20) $f_{1,1}(\xi ,\eta )$ when assuming $w_2 = 2, w_4 = -1, \sigma = 2, \mu _0 = 1, \mu _1 = 2, l = 2, \epsilon = 1, \eta _0 = 1.$ Also, Fig. 2 shows the solution with singular periodic type of Eq. (20) $f_{1,2}(\xi ,\eta )$ when assuming $w_2 = 2, w_4 = 1, \sigma = 2, \mu _0 = 1, \mu _1 = 2, l = 2, \epsilon = 1, \eta _0 = 1$. Figure 3 shows the solution with dark soliton type of Eq. (21) $f_{1,4}(\xi ,\eta )$ when assuming $w_2 = -2, w_4 = 1, \sigma = 2, \mu _0 = 1, \mu _1 = 2, l = 2, \epsilon = 1, \eta _0 = 1$. In addition, Fig. 4 presents the solution with periodic type of Eq. (21) $f_{1,5}(\xi ,\eta )$ when assuming $w_2 = 2, w_4 = 1, \sigma = 2, \mu _0 = 1, \mu _1 = 2, l = 2, \epsilon = 1, \eta _0 = 1$. Figure 5 shows the solution with dark-singular soliton type of Eq. (40) $f_{4,1}(\xi ,\eta )$ when assuming $w_2 = -2, w_4 = 1, \sigma = 2, \mu _0 = 0, \mu _1 = 2, \lambda _1=2,l = 2, \epsilon = 1, \eta _0 = 1$. In addition, Fig. 6 presents the solution with periodic-singular type of Eq. (40) $f_{4,2}(\xi ,\eta )$ when assuming $w_2 = 2, w_4 = 1, \sigma = 2, \mu _0 = 0, \mu _1 = 2, \lambda _1=-2,l = 2, \epsilon = 1, \eta _0 = 1$.

Fig. 7

Plot of bright soliton solution (51) for the solutions (a, c) $\Re (f)$, (b, d) $\Im (f)$, where $f=f_{1,1}(x,t)$.

Fig. 8

Plot of periodic solution (51) for the solutions (a, c) $\Re (f)$, (b, d) $\Im (f)$, where $f=f_{1,2}(x,t)$.

Fig. 9

Plot of dark soliton solution (52) for the solutions (a, c) $\Re (f)$, (b, d) $\Im (f)$, where $f=f_{1,4}(x,t)$.

Fig. 10

Plot of periodic solution (52) for the solutions (a, c) $\Re (f)$, (b, d) $\Im (f)$, where $f=f_{1,5}(x,t)$.

Fig. 11

Plot of periodic-singular solution (66) for the solutions (a, c) $\Re (f)$, (b, d) $\Im (f)$, where $f=f_{2,13}(x,t)$.

Fig. 12

Plot of soliton solution (66) for the solutions (a, c) $\Re (f)$, (b, d) $\Im (f)$, where $f=f_{2,14}(x,t)$.

(2+1)-dimensional CNLSE:

Figure 7 shows the solution with bright soliton type of Eq. (51) $f_{1,1}(\xi ,\eta )$ when assuming $w_2 = 2, w_4 = -1, s_2=2, a=2,b_1=1,b_2=2,l_1=1,l_2=2,\sigma = 2, \mu _0 = 1, \mu _1 = 2, \epsilon = 1, \eta _0 = 1$. Also, Fig. 8 shows the solution with singular periodic type of Eq. (51) $f_{1,2}(\xi ,\eta )$ when assuming $w_2 = 2, w_4 = -1, s_2=2, a=2,b_1=1,b_2=2,l_1=1,l_2=2,\sigma = 2, \mu _0 = 1, \mu _1 = 2, \epsilon = 1, \eta _0 = 1$. Figure 9 shows the solution with dark soliton type of Eq. (52) $f_{1,4}(\xi ,\eta )$ when assuming $w_2 = -0.25, w_4 = 1, s_2=1, a=2,b_1=1,b_2=1,l_1=1,l_2=2,\sigma = 2, \mu _0 = 1, \mu _1 = 2, \epsilon = 1, \eta _0 = 1$. In addition, Fig. 10 presents the solution with periodic type of Eq. (52) $f_{1,5}(\xi ,\eta )$ when assuming $w_2 = 0.25, w_4 = 1, s_2=1, a=2,b_1=1,b_2=1,l_1=1,l_2=2,\sigma = 2, \mu _0 = 1, \mu _1 = 2, \epsilon = 1, \eta _0 = 1$. Figure 11 shows the solution with periodic type of Eq. (66) $f_{2,13}(\xi ,\eta )$ when assuming $w_2 = -2, w_4 = 1, s_2=1, a=2,b_1=1,b_2=1,l_1=1,l_2=2,\sigma = 2, \mu _0 = 1, \lambda _1 = 2, \epsilon = 1, \eta _0 = 1$. In addition, Fig. 12 presents the solution with soliton type of Eq. (66) $f_{2,14}(\xi ,\eta )$ when assuming $w_2 = 0.1, w_4 = 1, s_2=1, a=1,b_1=1,b_2=1,l_1=1,l_2=1,\sigma = 1, \mu _0 = 1, \lambda _1 = 1, \epsilon = 1, \eta _0 = 1$.

Conclusion

The modified extended tanh-function technique was effectively applied to study optical solitons and other wave solutions in the (1+1)- and (2+1)-dimensional Chiral nonlinear Schrödinger equations. Different kinds of solutions were extracted, like (bright, dark, combo bright-dark, singular) soliton, hyperbolic, combo hyperbolic, combo periodic, singular periodic, rational and exponential. Additionally, the graphical simulations were provided to illustrate the nature of these derived results. These findings were the great significance for the further development of optical communication systems. Furthermore, this reliable and efficient technique can be used to solve a variety of physical and applied science models. We deduced that the proposed methods were powerful with observation to find analytical results to nonlinear evolution models. Our implemented methods can be used in the future to achieve more unprecedented outcomes to various kinds of nonlinear dynamical equations. This research sets the foundation for future investigations and experiments on even more complex systems that don’t follow straightforward rules, and how these can be used in real-life situations.