Optical wave propagation in magneto-optic waveguides with generalized anti-cubic model

Abstract

This study investigates optical solitons in magneto-optic waveguides while preserving the generalized anti-cubic structure of nonlinear self-phase modulation. A novel ${\varphi }^6$ -model expansion technique is proposed, and its application in magneto-optic waveguides is explored. The ${\varphi }^6$ approach provides an accurate and efficient method to solve wave propagation problems in magneto-optic media, enabling improved modeling of light behavior by incorporating the combined effects of electric and magnetic fields within a unified framework. To address the challenges arising from the nonlinear magneto-optic effects on waveguide characteristics, the method yields precise dispersion relations, field distributions, and transmission properties. This technique facilitates the design of magneto-optic waveguides with tailored characteristics for next-generation photonic systems. By offering a reliable and accurate modeling framework, the ${\varphi }^6$ approach contributes to the development of more precise and efficient magneto-optic devices, advancing photonic integration, communication, and sensing technologies.

Introduction

It is commonly known that complex processes in a variety of scientific domains, including physics, biology, chemistry, mechanics, and others, are described by nonlinear partial differential equations (NLPDEs). One can gain a better understanding of these events by studying the exact solution of NLPDEs, which are mathematical models of the phenomena.^1,2 Many efficient techniques to obtain precise solutions of NLPDEs have been proposed in the last few decades, including the inverse scattering method, ¹ the tanh function method,^3,4 the Hirota bilinear transform method, ⁵ the truncated Painleve expansion method, ⁶ the Backlund transform method, ⁷ the exp-function method,^8–10 the methods for expanding Jacobi elliptic functions,^11–13 the generalized Riccati equation,^14–16 the sub-ODE (ordinary differential equation) method,^17,18 the extended auxiliary equation method,^17,19 the soliton ansatz method^20,21 and others are listed. Magneto-optic waveguides, which use magnetic fields to control light propagation, are crucial parts of contemporary photonic and optoelectronic technologies.

The nonlinear Schrödinger equation (NLSE) for magneto-optic waveguide have been examined analytically by numerous scholars. Numerous authors addressed this model utilizing a wide range of integration strategies. One of the fastest-growing subfields of nonlinear fiber optics study is optical solitons. ²² Metamaterials and couplers are two examples of optical devices that exhibit the dynamics of these soliton molecules. An other type of fabric that sends data bits over continents is optical couplers. There are various kinds of optical couplers, including multiple-core and twin-core couplers. ²³ To solve this kind of NLSE, they employed a variety of approaches. In 2012, Chen ²⁴ explained the evolution of a complex wave in a medium where dispersion and nonlinearity interact the NLSE. It estimate phenomena like optical soliton, pulse propagation and modulaton in nonlinear systems like magneto-optic waveguide. In 2012, Agrawal ²⁵ described the propagation of slowly changing optical pulse envelops in nonlinear media that was the initially developed standard form of NLSE. The equation includes consideration of Kerr-type nonlinearity and second-order dispersion, these two aspects are essential in optical waveguide and fiber.

The idea of Russell’s in 1844 “great wave of translation” gave rise to the soliton, a powerful, enduring wave with a de Broglie wavelength that find uses in physics, telecommunications, mathematical physics, and among other domains. ²⁶ In 2017, Haider ²⁷ analyzed another method that was very helpful for waveguides with weak guidance was perturbation theory. It makes possible to estimate magneto-optic effects and modal properties without having to solve the entire wave equation. Faraday rotation in thin-film waveguides was examined by using the perturbation theory. ²⁸ Non-reciprocal devices like optical circulators and isolators, which are crucial for avoiding back reflections in laser systems and optical networks, depend on magneto-optic waveguides. In 2017, Bi et al., ²⁹ worked on chip magneto-optic isolator that was the use of garnet materials. In 2019, Wang et al. ³⁰ by using magnetic field that detect the changes in light propagation is the sensing application that magneto-optic waveguides were also used. In practical applications of magneto-optic waveguides needs to reduce optical losses, integrate with photonic platform and enhance magneto-optic waveguides. The NLSE supports both dark and bright solutions for optical solution of magneto-optic waveguide with anticubic law nonlinearity. Innovation in optical soliton is extensive. In 2020, Rezazadeh et al. ³¹ and in 2014, Zhou et al. ³² explained the effects of nonlinearity and dispersion on solutions for NLSE using the ansatz approach. Since 1972, due to their high GHz frequency range tunability and versatility the grain film optical waveguides have get the popularity. For the creation of dynamic devices like intensifiers and channel waveguides, the research on nonlinear magneto-optical interaction is important. In conjunction with an artificially produced magnetic field, magneto-optic waveguides improve data transport and lessen soliton mess. The experimental realization of solitons in anti-cubic nonlinear magneto-optic waveguides still faces many obstacles despite the tremendous advancements made, future work should concentrate on creating materials with low optical losses and strong anti-cubic nonlinearity, as well as investigating novel material systems like two-dimensional materials and topological insulators. Rabie et al. ³³ constructed the new soliton solutions for a concatenation model using modified extended tanh-function method. Mathanaranjan et al. ³⁴ explored the chirped optical solitons for the nonlinear Schrödinger equation using the new extended auxiliary equation method.

Several recent works have applied advanced analytical techniques to nonlinear differential equations. For example, the study by Hashemi ³⁵ employed the variable-coefficient third-degree generalized Abel equation method to obtain exact solutions of complex nonlinear wave models. Such auxiliary-equation approaches provide useful tools for handling higher-order nonlinearities. In continuation of these developments, the present study adopts the ${\varphi }^6$ -model expansion method within the generalized anti-cubic (GAC) model framework, offering an alternative and efficient way to derive exact optical wave solutions in magneto-optic waveguides.

In integrated optical systems, these waveguides are essential components that are commonly utilized in sensors, modulators, and isolators.^36–38 NLPDEs resulting from the interaction between electromagnetic fields and the waveguide medium control the propagation of light in magneto-optic waveguides. Analytical solutions to these equations are difficult, particularly when higher-order nonlinearities are included. The ${\varphi }^6$ offers a sophisticated mathematical framework for investigating solutions to these kinds of nonlinear systems.^39–41 In order to describe more complicated physical phenomena including multi-stable states, domain walls, and soliton solutions, the ${\varphi }^6$ technique uses a sixth-order polynomial potential in the field equations. It offers a precise analytical method for comprehending the soliton production, in the context of magneto-optic waveguides.^42–46 The use of the ${\varphi }^6$ approach^47–51 soliton solutions for magneto-optic waveguide equations is presented in this article. The resulting solutions offer a better understanding of the waveguides’ nonlinear dynamics and practical photonics implications by shedding light on their physical behavior under various operating situations. This article’s goal is to use the so-called novel ${\varphi }^6$ -model expansion approach of the resonant NLSE for the first time.

The following linked NLSE controls the propagation of optical pulses in nonlinear optical magneto-optic materials with GAC law of nonlinearity.^31,51,52

$$\begin{array}{rl}& \iota {\psi }_t+a_1{\psi }_{xx}+[{\sigma }_1|\psi {|}^{-(2n+2)}+{\sigma }_2|\psi {|}^{2n}+{\sigma }_3|\psi {|}^{(2n+2)}+{\sigma }_4|\varphi {|}^{-(2n+2)}+{\sigma }_5|\varphi {|}^{2n}+{\sigma }_6|\varphi {|}^{(2n+2)}]\psi \\ & ={\sigma }_7\varphi +\iota [{\sigma }_8{\psi }_x+{\sigma }_9(|\psi {|}^{2n}\psi {)}_x+{\sigma }_{10}(|\psi {|}^{2n}{)}_x\psi +{\sigma }_{11}|\psi {|}^{2n}{\psi }_x]\end{array}$$

(1)

and

$$\begin{array}{rl}& \iota {\varphi }_t+a_2{\varphi }_{xx}+[{\delta }_1|\varphi {|}^{-(2n+2)}+{\delta }_2|\varphi {|}^{2n}+{\delta }_3|\varphi {|}^{(2n+2)}+{\delta }_4|\psi {|}^{-(2n+2)}+{\delta }_5|\psi {|}^{2n}+{\delta }_6|\psi {|}^{(2n+2)]\varphi }\\ & ={\delta }_7\psi +\iota [{\delta }_8{\varphi }_x+{\delta }_9(|\varphi {|}^{2n}\varphi {)}_x+{\delta }_{10}(|\varphi {|}^{2n}{)}_x\varphi +{\delta }_{11}|\varphi {|}^{2n}{\varphi }_x]\end{array}$$

(2)

Here, the nonlinearity parameter $n$ represents the GAC, where $-1<n<3$ . The system reduces to the coupled NLSE with anti-cubic law when $n=1$ . The self-phase modulation is indicated by the coefficients ${\sigma }_i$ and ${\delta }_i$ for $i=1,2,3$ while the cross-phase modulation parameters are indicated by the coefficients ${\sigma }_i$ and ${\delta }_i$ for $i=4,5,6$ . On the other side, the inter-modal dispersions and the self-steepening coefficients are denoted by the coefficients ${\sigma }_i$ and ${\delta }_i$ for $i=10,11$ , respectively.

When analyzing magneto-optic waveguide the ${\varphi }^6$ -model expansion technique suggest in this study shows much higher accuracy and convergence efficiency compared to the previously published method and results. Prior research which relied on the expansion of the ${\varphi }^4$ -model and traditional coupled-mode approach,^53,54 only produced accurate results under mild nonlinear settings. When strong nonlinear and magneto-optic waveguide interactions were taken into account, the accuracy of the results drastically decreased. By incorporating higher-order nonlinear elements, the ${\varphi }^6$ -model expansion created in this study can more precisely represent multistable behaviors, field distributions, and propagation constant even in strongly nonlinear regimes. The resulting results show faster convergence with fewer computational steps and better agreement with full wave numerical simulations and experimental observations published in the literature.^55,56

Higher effectiveness is shown for the suggested ${\varphi }^6$ -model expansion approach^47,48 when compared to other methods, including the ${\varphi }^4$ -model and classical coupled-mode approach. It provides more accurate descriptions of magneto-optic waveguide propagation properties, especially when there are strong nonlinear and magneto-optic interactions. Higher-order terms allow the model to effectively represent complicated behavior that lower-order models are unable to such multistability and nonlinear mode coupling. Additionally, the ${\varphi }^6$ approach maintains the numerical stability across a significantly wider range of parameters while exhibiting faster convergence with fewer computational steps. Consequently, the suggested approach offers a more accurate and effective analytical tool for researching magneto-optic waveguide phenomena.

The motivation behind this research arises from the increasing importance of soliton-based communication systems and the demand for precise mathematical soliton structures in nonlinear media.

The motivation of this study is to investigate optical wave propagation in magneto-optic waveguides under GAC nonlinearity, which is not well explored in existing literature. The ${\varphi }^6$ -model expansion method is used to obtain new exact wave solutions, helping to better understand how magneto-optic effects influence the behavior and propagation of nonlinear optical waves. Recent improvement in exact solution methods and analytical technique has signigicantly expanded the toolbox for extracting soliton solutions and treating nonlinear partial differential equations that are pertinent to optical and magneto-optic waveguides. Furthermore, a number of researchers demonstrate how well ${\varphi }^6$ -model expansion approach work to produce new families of hyperbolic, elliptic, and singular solitons.^57–59 These analytical methods have also been expanded for models of linked nonlinear Schrödinger types and magneto-optic systems, producing explicit forms of solitons that are directly applicable to modeling the magneto-optic waveguide phenomena. Applications of higher-order expansions such as ${\varphi }^6$ -model to substantially nonlinear regimes is justified by more general reviews and analytical investigations that highlight the improved convergence insights that contemporary expansion methods offer.^60,61

This work introduces a novel GAC for magneto-optic optical waveguides by using ${\varphi }^6$ -model expansion approach, deriving new solitary wave solutions that have not been reported before. The combined anti-cubic and ${\varphi }^6$ -model expansion approach nonlinearities reveal new propagation dynamics and tunability via external magnetic fields. The ${\varphi }^6$ -model expansion is used to derive a higher-order effective potential, offering a new mathematical framework for nonlinear magneto-optic wave propagation. Graphical analysis reveals new stability regions controlled by magnetization and anti-cubic strength, enabling tunable soliton engineering. The GAC term produces novel solitary wave profiles not observed in cubic–quintic systems, revealing new propagation dynamics in magneto-optic media.

The remaining part of this article is organized as follows. In the “Introduction” section, the introduction and mathematical model are presented. The “An explanation of the new expansion strategy for the ${\varphi }^6$ model” section describes the applied method and the “Solving equation and its various Jacobi elliptic function-based solutions” section describes the derivation of solutions. In the “Graphical discussion” section, the graphical analysis and discussion are provided. Finally, the “Conclusion” section concludes the article with key findings.

An explanation of the new expansion strategy for the ${\varphi }^6$ model

Assume that the form of a NLPDE is taken as follows:

$$E(\psi ,{\psi }_t,{\psi }_x,{\psi }_{xx},{\psi }_{tt},\dots .)=0$$

(3)

where $\psi =\psi (x,t)$ is an unknown function, the highest-order derivatives and nonlinear terms are involved in the polynomial $E$ , $\psi (x,t)$ and its partial derivatives. The following is a description of the primary steps of the employed method. We take the wave transformation $\psi (\eta )=\psi (x,t)$ , $\eta =x-\alpha t$ . To reduce equation (3) to the subsequent nonlinear ODE, where $\alpha$ is a nonzero constant

$$G(\psi ,{\psi }^{{}^{\prime }},{\psi }^{{}^{″}},{\psi }^{{}^{‴}},\dots .)=0$$

(4)

where $G$ , $\psi (\eta )$ is a polynomial, and its derivatives are ${\psi }^{{}^{\prime }}(\eta )$ , ${\psi }^{{}^{″}}(\eta )$ , and so forth.

We presume that the new formal solution to equation (4) exists as follows:

$$\psi (\eta )=\sum _{i=0}^{2N}{\chi }_i{\rho }^i(\eta )$$

(5)

where ${\chi }_i$ , $(i=0,\dots ,2N)$ are constants to be found later, while $\rho (\eta )$ satisfies the well-known auxiliary nonlinear ODE as follows:

$${\rho }^{\prime 2}(\eta )=c_0+c_2{\rho }^2(\eta )+c_4{\rho }^4(\eta )+c_6{\rho }^6(\eta )$$

(6)

$${\rho }^{″}(\eta )=c_2\rho (\eta )+2c_4{\rho }^3(\eta )+3c_6{\rho }^5(\eta )$$

(7)

where $c_i$ is a real constant (i = 0, 2, 4, 6). By balancing the highest-order derivatives with the highest nonlinear terms in equation (4), we may find the positive integer N in equation (5). The answer to equation (7) is well known.

$$\rho (\eta )=\frac{{\mathrm{\Delta }}^2(\eta )}{\sqrt{e_1{\mathrm{\Delta }}^2(\eta )+e_2}}$$

(8)

where $\mathrm{\Delta }(\eta$ ) is the Jacobian elliptic equation solution and $(e_1{\mathrm{\Delta }}^2(\eta )+e_2)>0$

$${\mathrm{\Delta }}^{\prime 2}(\eta )=k_0+k_2{\mathrm{\Delta }}^2(\eta )+k_4{\mathrm{\Delta }}^4(\eta )$$

(9)

and $k_j$ $(j=0,2,4)$ are variables that will be ascertained later, whereas $e_1$ and $e_2$ are provided by the following equations:

$$e_1=\frac{c_4(k_2-c_2)}{{(k_2-c_2)}^2-2k_2(k_2-c_2)+3k_0{k}_4}$$

(10)

$$e_2=\frac{3c_4{k}_0}{{(k_2-c_2)}^2-2k_2(k_2-c_2)+3k_0{k}_4}$$

(11)

under the situation of constraint

$$3c_6{(3k_0{k}_4-(k_2^2-c_2^2))}^2+c_4^2(k_2-c_2)(9k_0{k}_4-(k_2-c_2)(c_2+2k_2))=0$$

(12)

It is commonly known that the following Jacobi elliptic solutions exist for equation (9).

In this Table 1, $\mathrm{sn}(\eta )=\mathrm{sn}(\eta ,p)$ , $\mathrm{cd}(\eta )=\mathrm{cd}(\eta ,p)$ , $\mathrm{cn}(\eta )=\mathrm{cn}(\eta ,p)$ , $\mathrm{dn}(\eta )=\mathrm{dn}(\eta ,p)$ , $\mathrm{ns}(\eta )=\mathrm{ns}(\eta ,p)$ , $\mathrm{cs}(\eta )=\mathrm{cs}(\eta ,p)$ , $\mathrm{ds}(\eta )=\mathrm{ds}(\eta ,p)$ , $\mathrm{sc}(\eta )=\mathrm{sc}(\eta ,p)$ , and $\mathrm{sd}(\eta )=\mathrm{sd}(\eta ,p)$ are the Jacobi elliptic functions with the modulus $0<p<1$ . These functions degenerate into hyperbolic functions, when $p⟶1$ as follows:

Table 1.

shows the Jacobi elliptic functions.

Sr. No.	$k_0$	$k_2$	$k_4$	$\mathrm{\Delta }(\eta )$
1	$1$	$-(1+p^2)$	$p^2$	$\mathrm{sn}(\eta ),\mathrm{cd}(\eta )$
2	$1-p^2$	$2p^2-1$	$-p^2$	$\mathrm{cn}(\eta )$
3	$p^2-1$	$2-p^2$	$-1$	$\mathrm{dn}(\eta )$
4	$p^2$	$-(1+p^2)$	$1$	$\mathrm{ns}(\eta ),\mathrm{dc}(\eta )$
5	$-p^2$	$2p^2-1$	$1-p^2$	$\mathrm{nc}(\eta )$
6	$-1$	$2-p^2$	$-(1-p^2)$	$\mathrm{nd}(\eta )$
7	$1$	$2-p^2$	$1-p^2$	$\mathrm{sc}(\eta )$
8	$1$	$2p^2-1$	$-p^2(1-p^2)$	$\mathrm{sd}(\eta )$
9	$1-p^2$	$2-p^2$	$1$	$\mathrm{cs}(\eta )$
10	$p^{(}1-p^2)$	$2p^2-1$	$1$	$\mathrm{ds}(\eta )$
11	$\frac{1-p^2}{4}$	$\frac{p^2+1}{2}$	$\frac{1-p^2}{4}$	$\mathrm{nc}(\eta )\pm \mathrm{sc}(\eta )$ or $\frac{\mathrm{cn}(\eta )}{1\pm \mathrm{sn}(\eta )}$
12	$\frac{-(1-p^2{)}^2}{4}$	$\frac{p^2+1}{2}$	$\frac{-1}{4}$	$p\mathrm{cn}(\eta )\pm \mathrm{dn}(\eta )$
13	$\frac{1}{4}$	$\frac{1-2p^2}{2}$	$\frac{1}{4}$	$\frac{\mathrm{sn}(\eta )}{1\pm \mathrm{cn}(\eta )}$
14	$\frac{1}{4}$	$\frac{1+p^2}{2}$	$\frac{(1-p^2{)}^2}{4}$	$\frac{\mathrm{sn}(\eta )}{\mathrm{cn}(\eta )\pm \mathrm{dn}(\eta )}$

$\mathrm{sn}(\eta ,1)=\tanh (\eta )$ , $\mathrm{cn}(\eta ,1)=\mathrm{sech}(\eta )$ , $\mathrm{dn}(\eta ,1)=\mathrm{sech}(\eta )$ , $\mathrm{ns}(\eta ,1)=\coth (\eta )$ , $\mathrm{cs}(\eta ,1)=\mathrm{csch}(\eta )$ , $\mathrm{ds}(\eta ,1)=\mathrm{csch}(\eta )$ , $\mathrm{sc}(\eta ,1)=\sinh (\eta )$ , $\mathrm{sd}(\eta ,1)=\sinh (\eta )$ , $\mathrm{nc}(\eta ,1)=\cosh (\eta )$ , $\mathrm{cd}(\eta ,1)=1$ , and into trigonometric functions, when $p⟶0$ , as follows:

$\mathrm{sn}(\eta ,0)=\sin (\eta )$ , $\mathrm{cd}(\eta ,0)=\cos (\eta )$ , $\mathrm{cn}(\eta ,0)=\cos (\eta )$ , $\mathrm{ns}(\eta ,0)=\csc (\eta )$ , $\mathrm{cs}(\eta ,0)=\cot (\eta )$ , $\mathrm{ds}(\eta ,0)=\csc (\eta )$ , $\mathrm{sc}(\eta ,0)=\tan (\eta )$ , $\mathrm{sd}(\eta ,0)=\sin (\eta )$ , $\mathrm{nc}(\eta ,0)=\sec (\eta )$ , $\mathrm{dn}(\eta ,0)=1$ .

Equations (8) and (9) can be substituted into equation (5) to obtain the Jacobi elliptic function solutions of equation (3).

Solving equation and its various Jacobi elliptic function-based solutions

In this part, we apply the novel ${\varphi }^6$ -model expansion method to solve equations (1) and (2). To achieve this, we presume that the formal solution to equations (1) and (2) exists.

$$\psi (x,t)=a(\eta )e^{\iota \beta (x,t)}$$

(13)

$$\varphi (x,t)=b(\eta )e^{\iota \beta (x,t)}$$

(14)

where $\beta (x,t)=-Rx+\gamma t+{\mathrm{\Theta }}_0$ , $\eta =x-\alpha t$ and $\iota =\sqrt{-1}$ . Where $R$ , $\gamma$ , and ${\mathrm{\Theta }}_0$ are constants and $a(\eta )$ and $b(\eta )$ is a real function. After dividing the imaginary and real components of equations (1) and (2) and substituting equations (13) and (14) into it, then the imaginary componentes are as follows:

$$\alpha +a_{1}2R+{\sigma }_8+((2n+1){\sigma }_9+2n{\sigma }_{10}+{\sigma }_{11})a(\eta {)}^{2n})=0$$

(15)

and

$$\alpha +a_{2}2R+{\delta }_8+((2n+1){\delta }_9+2n{\delta }_{10}+{\delta }_{11})b(\eta {)}^{2n})=0$$

(16)

$$((2n+1){\sigma }_9+2n{\sigma }_{10}+{\sigma }_{11})=0$$

(17)

and

$$((2n+1){\delta }_9+2n{\delta }_{10}+{\delta }_{11})=0$$

(18)

$${\sigma }_8={\delta }_8=c$$

(19)

and

$$a_1=a_2=u$$

(20)

Then, equations (15) and (16) become as follows:

$$\alpha =-(2Ru+c)$$

(21)

where $u$ is a real constant parameter.

$$\begin{array}{rl}& u{a}^{{}^{″}}-(\gamma +u{R}^2)a+{\sigma }_1{a}^{-(2n+1)}+{\sigma }_2{a}^{(2n+1)}+{\sigma }_3{a}^{(2n+3)}+{\sigma }_{4}a{b}^{-(2n+2)}+{\sigma }_{5}a{b}^{2n}\\ & +{\sigma }_{6}a{b}^{(2n+2)}-{\sigma }_{7}b-R{\sigma }_8=0\end{array}$$

(22)

and

$$\begin{array}{rl}& u{b}^{{}^{″}}-(\gamma +u{R}^2)b+{\delta }_1{b}^{-(2n+1)}+{\delta }_2{b}^{(2n+1)}+{\delta }_3{b}^{(2n+3)}+{\delta }_{4}b{a}^{-(2n+2)}+{\delta }_{5}b{a}^{2n}\\ & +{\delta }_{6}b{a}^{(2n+2)}-{\delta }_{7}a-R{\sigma }_8=0\end{array}$$

(23)

where $\eta =x+(2Ru+c)t$ and $u\ne 0$ . To balance equations (22) and (23), we take the transformation. Assume the transformation as follows:

$$a(\eta )=A^{\frac{1}{n}}(\eta )$$

(24)

$$b(\eta )=B^{\frac{1}{n}}(\eta )$$

(25)

when $A(\eta )$ and $B(\eta )$ are the new functions of $\eta$ . When we enter equations (24) and (25) into equations (22) and (23), we obtain as follows:

$$u(M{A}^{{}^{\prime 2}}+A{A}^{{}^{″}})-n(\gamma +u{R}^2)A^2+n{\sigma }_2{A}^4+n{\sigma }_5{A}^2{B}^2=0$$

(26)

and

$$u(M{B}^{{}^{\prime 2}}+B{B}^{{}^{″}})-n(\gamma +u{R}^2)B^2+n{\delta }_2{B}^4+n{\delta }_5{B}^2{A}^2=0$$

(27)

Consider the formal solution to equations (26) and (27) exists.

$$\psi (\eta )=\sum _{i=0}^{2N}{\chi }_i{\rho }^i(\eta )$$

(28)

$$\varphi (\eta )=\sum _{i=0}^{2N}{\kappa }_i{\rho }^i(\eta )$$

(29)

$N=1$ is obtained by balancing $A{A}^{{}^{″}}$ and $B{B}^{{}^{″}}$ with $A^2{B}^2$ and $B^2{A}^2$ in equations (26) and (27). Thus

$$\psi (\eta )={\chi }_0+{\chi }_1\rho (\eta )+{\chi }_2{\rho }^2(\eta )$$

(30)

$$\varphi (\eta )={\kappa }_0+{\kappa }_1\rho (\eta )+{\kappa }_2{\rho }^2(\eta )$$

(31)

where ${\chi }_2\ne 0$ and ${\kappa }_2\ne 0$ and ${\chi }_0$ , ${\chi }_1$ , ${\chi }_2$ , ${\kappa }_0$ , ${\kappa }_1$ , and ${\kappa }_2$ are constants that will be found later. When equations (6), (7), (30), and (31) are substituted into equations (26) and (27), and the coefficients of the same powers are obtained through Mathematica, then we have the result ${\chi }_0=-\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}}{2\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}$ , ${\chi }_1=0$ , ${\chi }_2=-\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}}{\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}$ , ${\kappa }_0=-\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}}{2\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}$ , ${\kappa }_1=0$ , ${\kappa }_2=-\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}}{\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}$ , $c_0=\frac{c_4^3}{32c_6^2}$ , $c_2=\frac{5c_4^2}{16c_6}$ , and $\gamma =-\frac{c_4^2(M+1)u}{4c_{6}n}-R^{2}u$ .

The following several new exact solutions to equations (1) and (2) are now available:

Case $1$ : If $k_0=1$ , $k_2=-(p^2+1)$ , $k_4=p^2$ , then $\mathrm{\Delta }(\eta )=\text{sn}(\eta )$ and we have the Jacobi elliptic function solution

$$\begin{array}{rl}{\psi }_{1,1}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}\text{sn}(\eta {)}^4}{\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}(\frac{768c_4{c}_6^2}{25c_4^4-256c_6^2(p^4-p^2+1)}}\\ & -\frac{16c_4{c}_6(16c_6(p^2+1)+5c_4^2)\text{sn}(\eta {)}^2}{25c_4^4-256c_6^2(p^4-p^2+1)})-\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}}{2\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\end{array}$$

(32)

and

$$\begin{array}{rl}{\varphi }_{1,1}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}\text{sn}(\eta {)}^4}{\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}(\frac{768c_4{c}_6^2}{25c_4^4-256c_6^2(p^4-p^2+1)}}\\ & -\frac{16c_4{c}_6(16c_6(p^2+1)+5c_4^2)\text{sn}(\eta {)}^2}{25c_4^4-256c_6^2(p^4-p^2+1)})-\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}}{2\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\end{array}$$

(33)

Here $e_1$ and $e_2$ are auxiliary constants introduced in the expansion method. These constants arise during the balancing procedure and are used to simplify the analytic form of the Jacobi elliptic function solutions, where $e_1$ and $e_2$ are as follows:

$$e_1=-\frac{16c_4{c}_6(16c_6(p^2+1)+5c_4^2)}{25c_4^4-256c_6^2(p^4-p^2+1)}$$

(34)

and

$$e_2=\frac{768c_4{c}_6^2}{25c_4^4-256c_6^2(p^4-p^2+1)}$$

(35)

under the situation of constraint

$$\begin{array}{rl}& c_4^2(-\frac{5c_4^2}{16c_6}-p^2-1)(9p^2-(-\frac{5c_4^2}{16c_6}-p^2-1)(\frac{5c_4^2}{16c_6}+2(-p^2-1)))\\ & +3c_6{(\frac{25c_4^4}{256c_6^2}+3p^2-{(-p^2-1)}^2)}^2=0\end{array}$$

(36)

if $p⟶0$ , then we have periodic wave solution

$$\begin{array}{rl}{\psi }_{1,2}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}{\sin }^4(x-\alpha t)}{\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}(\frac{768c_4{c}_6^2}{25c_4^4-256c_6^2}-\frac{16c_4{c}_6(5c_4^2+16c_6){\sin }^2(x-\alpha t)}{25c_4^4-256c_6^2})}\\ & -\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}}{2\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\end{array}$$

(37)

and

$$\begin{array}{rl}{\varphi }_{1,2}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}{\sin }^4(x-\alpha t)}{\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}(\frac{768c_4{c}_6^2}{25c_4^4-256c_6^2}-\frac{16c_4{c}_6(5c_4^2+16c_6){\sin }^2(x-\alpha t)}{25c_4^4-256c_6^2})}\\ & -\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}}{2\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\end{array}$$

(38)

under the situation of constraint

$$3{(\frac{25c_4^4}{256c_6^2}-1)}^2{c}_6-c_4^2{(-\frac{5c_4^2}{16c_6}-1)}^2(\frac{5c_4^2}{16c_6}-2)=0$$

(39)

if $p⟶1$ , then we have periodic wave solution

$$\begin{array}{rl}{\psi }_{1,3}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}{\tanh }^4(x-\alpha t)}{\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}(\frac{768c_4{c}_6^2}{25c_4^4-256c_6^2}-\frac{16c_4{c}_6(5c_4^2+32c_6){\tanh }^2(x-\alpha t)}{25c_4^4-256c_6^2})}\\ & -\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}}{2\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\end{array}$$

(40)

$$\begin{array}{rl}{\varphi }_{1,3}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}{\tanh }^4(x-\alpha t)}{\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}(\frac{768c_4{c}_6^2}{25c_4^4-256c_6^2}-\frac{16c_4{c}_6(5c_4^2+32c_6){\tanh }^2(x-\alpha t)}{25c_4^4-256c_6^2})}\\ & -\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}}{2\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\end{array}$$

(41)

under the situation of constraint

$$(-\frac{5c_4^2}{16c_6}-2)(9-(-\frac{5c_4^2}{16c_6}-2)(\frac{5c_4^2}{16c_6}-4))c_4^2+3c_6{(\frac{25c_4^4}{256c_6^2}-1)}^2=0$$

(42)

Case $2$ : If $k_0=1-p^2$ , $k_2=2p^2-1$ , $k_4=-p^2$ then $\mathrm{\Delta }(\eta )=\text{cn}(\eta )$ and we have the Jacobi elliptic function solution

$$\begin{array}{rl}{\psi }_{1,4}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\text{cn}(\eta {)}^4\sqrt{{\sigma }_5-{\delta }_2}}{\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}(\frac{768c_4{c}_6^2(p^2-1)}{256c_6^2(p^4-p^2+1)-25c_4^4}}\\ & -\frac{16c_4{c}_6(16c_6(1-2p^2)+5c_4^2)\text{cn}(\eta {)}^2}{25c_4^4-256c_6^2(p^4-p^2+1)})-\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}}{2\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\end{array}$$

(43)

$$\begin{array}{rl}{\varphi }_{1,4}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\text{cn}(\eta {)}^4\sqrt{{\sigma }_2-{\delta }_5}}{\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}(\frac{768c_4{c}_6^2(p^2-1)}{256c_6^2(p^4-p^2+1)-25c_4^4}}\\ & -\frac{16c_4{c}_6(16c_6(1-2p^2)+5c_4^2)\text{cn}(\eta {)}^2}{25c_4^4-256c_6^2(p^4-p^2+1)})-\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}}{2\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\end{array}$$

(44)

where $e_1$ and $e_2$ are as follows:

$$e_1=-\frac{16c_4{c}_6(16c_6(1-2p^2)+5c_4^2)}{25c_4^4-256c_6^2(p^4-p^2+1)}$$

(45)

and

$$e_2=\frac{768c_4{c}_6^2(p^2-1)}{256c_6^2(p^4-p^2+1)-25c_4^4}$$

(46)

under the situation of constraint

$$\begin{array}{rl}& c_4^2(-\frac{5c_4^2}{16c_6}+2p^2-1)(-(-\frac{5c_4^2}{16c_6}+2p^2-1)(\frac{5c_4^2}{16c_6}+2(2p^2-1))-9(1-p^2)p^2)\\ & +3c_6{(\frac{25c_4^4}{256c_6^2}-{(2p^2-1)}^2-3p^2(1-p^2))}^2=0\end{array}$$

(47)

if $p⟶0$ , then we have periodic wave solution

$$\begin{array}{rl}{\psi }_{1,5}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}{\cos }^4(x-\alpha t)}{\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}(-\frac{16c_4{c}_6(5c_4^2+16c_6){\cos }^2(x-\alpha t)}{25c_4^4-256c_6^2}-\frac{768c_4{c}_6^2}{256c_6^2-25c_4^4})}\\ & -\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}}{2\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\end{array}$$

(48)

and

$$\begin{array}{rl}{\varphi }_{1,5}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}{\cos }^4(x-\alpha t)}{\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}(-\frac{16c_4{c}_6(5c_4^2+16c_6){\cos }^2(x-\alpha t)}{25c_4^4-256c_6^2}-\frac{768c_4{c}_6^2}{256c_6^2-25c_4^4})}\\ & -\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}}{2\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\end{array}$$

(49)

under the situation of constraint

$$3{(\frac{25c_4^4}{256c_6^2}-1)}^2{c}_6-c_4^2{(-\frac{5c_4^2}{16c_6}-1)}^2(\frac{5c_4^2}{16c_6}-2)=0$$

(50)

if $p⟶1$ , then we have periodic wave solution

$$\begin{array}{rl}{\psi }_{1,6}(x,t)=& \frac{(25c_4^4-256c_6^2)\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}{\text{sech}}^2(x-\alpha t)}{8c_4(5c_4^2-16c_6)\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\\ & -\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}}{2\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\end{array}$$

(51)

$$\begin{array}{rl}{\varphi }_{1,6}(x,t)=& \frac{(25c_4^4-256c_6^2)\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}{\text{sech}}^2(x-\alpha t)}{8c_4(5c_4^2-16c_6)\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\\ & -\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}}{2\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\end{array}$$

(52)

under the situation of constraint

$$3{(\frac{25c_4^4}{256c_6^2}-1)}^2{c}_6-c_4^2{(1-\frac{5c_4^2}{16c_6})}^2(\frac{5c_4^2}{16c_6}+2)=0$$

(53)

Case $3$ : If $k_0=p^2-1$ , $k_2=2-p^2$ , $k_4=-1$ then $\mathrm{\Delta }(\eta )=\text{dn}(\eta )$ and we have the Jacobi elliptic function solution

$$\begin{array}{rl}{\psi }_{1,7}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}\text{dn}(\eta {)}^4}{\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}(-\frac{16c_4{c}_6(16c_6(p^2-2)+5c_4^2)\text{dn}(\eta {)}^2}{25c_4^4-256c_6^2(p^4-p^2+1)}}\\ & -\frac{768c_4{c}_6^2(p^2-1)}{256c_6^2(p^4-p^2+1)-25c_4^4})-\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}}{2\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\end{array}$$

(54)

$$\begin{array}{rl}{\varphi }_{1,7}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}\text{dn}(\eta {)}^4}{\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}(-\frac{16c_4{c}_6(16c_6(p^2-2)+5c_4^2)\text{dn}(\eta {)}^2}{25c_4^4-256c_6^2(p^4-p^2+1)}}\\ & -\frac{768c_4{c}_6^2(p^2-1)}{256c_6^2(p^4-p^2+1)-25c_4^4})-\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}}{2\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\end{array}$$

(55)

where $e_1$ and $e_2$ are as follows:

$$e_1=-\frac{16c_4{c}_6(16c_6(p^2-2)+5c_4^2)}{25c_4^4-256c_6^2(p^4-p^2+1)}$$

(56)

and

$$e_2=-\frac{768c_4{c}_6^2(p^2-1)}{256c_6^2(p^4-p^2+1)-25c_4^4}$$

(57)

under the situation of constraint

$$\begin{array}{rl}& c_4^2(-\frac{5c_4^2}{16c_6}-p^2+2)(-(-\frac{5c_4^2}{16c_6}-p^2+2)(\frac{5c_4^2}{16c_6}+2(2-p^2))-9(p^2-1))\\ & +3c_6{(\frac{25c_4^4}{256c_6^2}-{(2-p^2)}^2-3(p^2-1))}^2=0\end{array}$$

(58)

if $p⟶1$ , then we have periodic wave solution

$$\begin{array}{rl}{\psi }_{1,8}(x,t)=& \frac{(25c_4^4-256c_6^2)\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}{\text{sech}}^2(x-\alpha t)}{8c_4(5c_4^2-16c_6)\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\\ & -\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}}{2\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\end{array}$$

(59)

$$\begin{array}{rl}{\varphi }_{1,8}(x,t)=& \frac{(25c_4^4-256c_6^2)\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}{\text{sech}}^2(x-\alpha t)}{8c_4(5c_4^2-16c_6)\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\\ & -\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}}{2\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\end{array}$$

(60)

under the situation of constraint

$$3{(\frac{25c_4^4}{256c_6^2}-1)}^2{c}_6-c_4^2{(1-\frac{5c_4^2}{16c_6})}^2(\frac{5c_4^2}{16c_6}+2)=0$$

(61)

Case $4$ : If $k_0=p^2$ , $k_2=-(1-p^2)$ , $k_4=1$ then $\mathrm{\Delta }(\eta )=\text{ns}(\eta )$ and we have the Jacobi elliptic function solution

$$\begin{array}{rl}{\psi }_{1,9}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}\text{ns}(\eta {)}^4}{\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}(-\frac{16c_4{c}_6(5c_4^2-16c_6(p^2-1))\text{ns}(\eta {)}^2}{25c_4^4-256c_6^2(p^4-5p^2+1)}}\\ & -\frac{768c_4{c}_6^2{p}^2}{256c_6^2(p^4-5p^2+1)-25c_4^4})-\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}}{2\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\end{array}$$

(62)

$$\begin{array}{rl}{\varphi }_{1,9}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}\text{ns}(\eta {)}^4}{\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}(-\frac{16c_4{c}_6(5c_4^2-16c_6(p^2-1))\text{ns}(\eta {)}^2}{25c_4^4-256c_6^2(p^4-5p^2+1)}}\\ & -\frac{768c_4{c}_6^2{p}^2}{256c_6^2(p^4-5p^2+1)-25c_4^4})-\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}}{2\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\end{array}$$

(63)

where $e_1$ and $e_2$ are as follows:

$$e_1=-\frac{16c_4{c}_6(5c_4^2-16c_6(p^2-1))}{25c_4^4-256c_6^2(p^4-5p^2+1)}$$

(64)

and

$$e_2=-\frac{768c_4{c}_6^2{p}^2}{256c_6^2(p^4-5p^2+1)-25c_4^4}$$

(65)

under the situation of constraint

$$\begin{array}{rl}& c_4^2(-\frac{5c_4^2}{16c_6}+p^2-1)(9p^2-(-\frac{5c_4^2}{16c_6}+p^2-1)(\frac{5c_4^2}{16c_6}+2(p^2-1)))\\ & +3c_6{(\frac{25c_4^4}{256c_6^2}+3p^2-{(p^2-1)}^2)}^2=0\end{array}$$

(66)

if $p⟶0$ , then we have periodic wave solution

$$\begin{array}{rl}{\psi }_{1,10}(x,t)=& \frac{(25c_4^4-256c_6^2)\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}{\csc }^2(x-\alpha t)}{8c_4\sqrt{c_6}(5c_4^2+16c_6)\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\\ & -\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}}{2\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\end{array}$$

(67)

and

$$\begin{array}{rl}{\varphi }_{1,10}(x,t)=& \frac{(25c_4^4-256c_6^2)\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}{\csc }^2(x-\alpha t)}{8c_4\sqrt{c_6}(5c_4^2+16c_6)\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\\ & -\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}}{2\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\end{array}$$

(68)

under the situation of constraint

$$3{(\frac{25c_4^4}{256c_6^2}-1)}^2{c}_6-c_4^2{(-\frac{5c_4^2}{16c_6}-1)}^2(\frac{5c_4^2}{16c_6}-2)=0$$

(69)

if $p⟶1$ , then we have periodic wave solution

$$\begin{array}{rl}{\psi }_{1,11}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}{\coth }^4(x-\alpha t)}{\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}(-\frac{80c_6{c}_4^3{\coth }^2(x-\alpha t)}{25c_4^4+768c_6^2}-\frac{768c_6^2{c}_4}{-25c_4^4-768c_6^2})}\\ & -\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}}{2\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\end{array}$$

(70)

$$\begin{array}{rl}{\varphi }_{1,11}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}{\coth }^4(x-\alpha t)}{\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}(-\frac{80c_6{c}_4^3{\coth }^2(x-\alpha t)}{25c_4^4+768c_6^2}-\frac{768c_6^2{c}_4}{-25c_4^4-768c_6^2})}\\ & -\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}}{2\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\end{array}$$

(71)

under the situation of constraint

$$3{(\frac{25c_4^4}{256c_6^2}+3)}^2{c}_6-\frac{5c_4^4(\frac{25c_4^4}{256c_6^2}+9)}{16c_6}=0$$

(72)

Case $5$ : If $k_0=-p^2$ , $k_2=2p^2-1$ , $k_4=1-p^2$ then $\mathrm{\Delta }(\eta )=\text{nc}(\eta )$ and we have the Jacobi elliptic function solution

$$\begin{array}{rl}{\psi }_{1,12}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}\text{nc}(\eta {)}^4}{\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}(\frac{768c_4{c}_6^2{p}^2}{256c_6^2(p^4-p^2+1)-25c_4^4}}\\ & -\frac{16c_4{c}_6(16c_6(1-2p^2)+5c_4^2)\text{nc}(\eta {)}^2}{25c_4^4-256c_6^2(p^4-p^2+1)})-\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}}{2\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\end{array}$$

(73)

$$\begin{array}{rl}{\varphi }_{1,12}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}\text{nc}(\eta {)}^4}{\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}(\frac{768c_4{c}_6^2{p}^2}{256c_6^2(p^4-p^2+1)-25c_4^4}}\\ & -\frac{16c_4{c}_6(16c_6(1-2p^2)+5c_4^2)\text{nc}(\eta {)}^2}{25c_4^4-256c_6^2(p^4-p^2+1)})-\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}}{2\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\end{array}$$

(74)

where $e_1$ and $e_2$ are as follows:

$$e_1=-\frac{16c_4{c}_6(16c_6(1-2p^2)+5c_4^2)}{25c_4^4-256c_6^2(p^4-p^2+1)}$$

(75)

and

$$e_2=\frac{768c_4{c}_6^2{p}^2}{256c_6^2(p^4-p^2+1)-25c_4^4}$$

(76)

under the situation of constraint

$$\begin{array}{rl}& c_4^2(-\frac{5c_4^2}{16c_6}+2p^2-1)(-(-\frac{5c_4^2}{16c_6}+2p^2-1)(\frac{5c_4^2}{16c_6}+2(2p^2-1))-9(1-p^2)p^2)\\ & +3c_6{(\frac{25c_4^4}{256c_6^2}-{(2p^2-1)}^2-3p^2(1-p^2))}^2=0\end{array}$$

(77)

if $p⟶0$ , then we have periodic wave solution

$$\begin{array}{rl}{\psi }_{1,13}(x,t)=& \frac{(25c_4^4-256c_6^2)\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}{\sec }^2(x-\alpha t)}{8c_4\sqrt{c_6}(5c_4^2+16c_6)\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\\ & -\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}}{2\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\end{array}$$

(78)

and

$$\begin{array}{rl}{\varphi }_{1,13}(x,t)=& \frac{(25c_4^4-256c_6^2)\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}{\sec }^2(x-\alpha t)}{8c_4\sqrt{c_6}(5c_4^2+16c_6)\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\\ & -\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}}{2\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\end{array}$$

(79)

under the situation of constraint

$$3{(\frac{25c_4^4}{256c_6^2}-1)}^2{c}_6-c_4^2{(-\frac{5c_4^2}{16c_6}-1)}^2(\frac{5c_4^2}{16c_6}-2)=0$$

(80)

if $p⟶1$ , then we have periodic wave solution

$$\begin{array}{rl}{\psi }_{1,14}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}{\cosh }^4(x-\alpha t)}{\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}(\frac{768c_4{c}_6^2}{256c_6^2-25c_4^4}-\frac{16c_4(5c_4^2-16c_6)c_6{\cosh }^2(x-\alpha t)}{25c_4^4-256c_6^2})}\\ & -\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}}{2\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\end{array}$$

(81)

$$\begin{array}{rl}{\varphi }_{1,14}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}{\cosh }^4(x-\alpha t)}{\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}(\frac{768c_4{c}_6^2}{256c_6^2-25c_4^4}-\frac{16c_4(5c_4^2-16c_6)c_6{\cosh }^2(x-\alpha t)}{25c_4^4-256c_6^2})}\\ & -\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}}{2\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\end{array}$$

(82)

under the situation of constraint

$$3{(\frac{25c_4^4}{256c_6^2}-1)}^2{c}_6-c_4^2{(1-\frac{5c_4^2}{16c_6})}^2(\frac{5c_4^2}{16c_6}+2)=0$$

(83)

Case $6$ : If $k_0=-1$ , $k_2=2-p^2$ , $k_4=-(1-p^2)$ then $\mathrm{\Delta }(\eta )=\text{nd}(\eta )$ and we have the Jacobi elliptic function solution

$$\begin{array}{rl}{\psi }_{1,15}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}\text{nd}(\eta {)}^4}{\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}(\frac{768c_4{c}_6^2}{256c_6^2(p^4-p^2+1)-25c_4^4}}\\ & -\frac{16c_4{c}_6(16c_6(p^2-2)+5c_4^2)\text{nd}(\eta {)}^2}{25c_4^4-256c_6^2(p^4-p^2+1)})-\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}}{2\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\end{array}$$

(84)

$$\begin{array}{rl}{\varphi }_{1,15}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}\text{nd}(\eta {)}^4}{\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}(\frac{768c_4{c}_6^2}{256c_6^2(p^4-p^2+1)-25c_4^4}}\\ & -\frac{16c_4{c}_6(16c_6(p^2-2)+5c_4^2)\text{nd}(\eta {)}^2}{25c_4^4-256c_6^2(p^4-p^2+1)})-\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}}{2\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\end{array}$$

(85)

where $e_1$ and $e_2$ are as follows:

$$e_1=-\frac{16c_4{c}_6(16c_6(p^2-2)+5c_4^2)}{25c_4^4-256c_6^2(p^4-p^2+1)}$$

(86)

and

$$e_2=\frac{768c_4{c}_6^2}{256c_6^2(p^4-p^2+1)-25c_4^4}$$

(87)

under the situation of constraint

$$\begin{array}{rl}& c_4^2(-\frac{5c_4^2}{16c_6}-p^2+2)(-(-\frac{5c_4^2}{16c_6}-p^2+2)(\frac{5c_4^2}{16c_6}+2(2-p^2))-9(p^2-1))\\ & +3c_6{(\frac{25c_4^4}{256c_6^2}-{(2-p^2)}^2-3(p^2-1))}^2=0\end{array}$$

(88)

if $p⟶1$ , then we have periodic wave solution

$$\begin{array}{rl}{\psi }_{1,16}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}{\text{sech}}^4(x-\alpha t)}{\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}(\frac{768c_4{c}_6^2}{256c_6^2-25c_4^4}-\frac{16c_4(5c_4^2-16c_6)c_6{\text{sech}}^2(x-\alpha t)}{25c_4^4-256c_6^2})}\\ & -\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}}{2\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\end{array}$$

(89)

$$\begin{array}{rl}{\varphi }_{1,16}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}{\text{sech}}^4(x-\alpha t)}{\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}(\frac{768c_4{c}_6^2}{256c_6^2-25c_4^4}-\frac{16c_4(5c_4^2-16c_6)c_6{\text{sech}}^2(x-\alpha t)}{25c_4^4-256c_6^2})}\\ & -\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}}{2\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\end{array}$$

(90)

under the situation of constraint

$$3{(\frac{25c_4^4}{256c_6^2}-1)}^2{c}_6-c_4^2{(1-\frac{5c_4^2}{16c_6})}^2(\frac{5c_4^2}{16c_6}+2)=0$$

(91)

Case $7$ : If $k_0=-1$ , $k_2=2-p^2$ , $k_4=1-p^2$ then $\mathrm{\Delta }(\eta )=\text{sc}(\eta )$ and we have the Jacobi elliptic function solution

$$\begin{array}{rl}{\psi }_{1,17}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}\text{sc}(\eta {)}^4}{\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}(\frac{768c_4{c}_6^2}{25c_4^4-256c_6^2(p^4-p^2+1)}}\\ & -\frac{16c_4{c}_6(16c_6(p^2-2)+5c_4^2)\text{sc}(\eta {)}^2}{25c_4^4-256c_6^2(p^4-p^2+1)})-\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}}{2\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\end{array}$$

(92)

$$\begin{array}{rl}{\varphi }_{1,17}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}\text{sc}(\eta {)}^4}{\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}(\frac{768c_4{c}_6^2}{25c_4^4-256c_6^2(p^4-p^2+1)}}\\ & -\frac{16c_4{c}_6(16c_6(p^2-2)+5c_4^2)\text{sc}(\eta {)}^2}{25c_4^4-256c_6^2(p^4-p^2+1)})\\ & -\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}}{2\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\end{array}$$

(93)

where $e_1$ and $e_2$ are as follows:

$$e_1=-\frac{16c_4{c}_6(16c_6(p^2-2)+5c_4^2)}{25c_4^4-256c_6^2(p^4-p^2+1)}$$

(94)

and

$$e_2=\frac{768c_4{c}_6^2}{25c_4^4-256c_6^2(p^4-p^2+1)}$$

(95)

under the situation of constraint

$$\begin{array}{rl}& c_4^2(-\frac{5c_4^2}{16c_6}-p^2+2)(9(1-p^2)-(-\frac{5c_4^2}{16c_6}-p^2+2)(\frac{5c_4^2}{16c_6}+2(2-p^2)))\\ & +3c_6{(\frac{25c_4^4}{256c_6^2}-{(2-p^2)}^2+3(1-p^2))}^2=0\end{array}$$

(96)

if $p⟶0$ , then we have periodic wave solution

$$\begin{array}{rl}{\psi }_{1,18}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}{\tan }^4(x-\alpha t)}{\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}(\frac{768c_4{c}_6^2}{25c_4^4-256c_6^2}-\frac{16c_4(5c_4^2-32c_6)c_6{\tan }^2(x-\alpha t)}{25c_4^4-256c_6^2})}\\ & -\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}}{2\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\end{array}$$

(97)

$$\begin{array}{rl}{\varphi }_{1,18}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}{\tan }^4(x-\alpha t)}{\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}(\frac{768c_4{c}_6^2}{25c_4^4-256c_6^2}-\frac{16c_4(5c_4^2-32c_6)c_6{\tan }^2(x-\alpha t)}{25c_4^4-256c_6^2})}\\ & -\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}}{2\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\end{array}$$

(98)

under the situation of constraint

$$(2-\frac{5c_4^2}{16c_6})(9-(2-\frac{5c_4^2}{16c_6})(\frac{5c_4^2}{16c_6}+4))c_4^2+3c_6{(\frac{25c_4^4}{256c_6^2}-1)}^2=0$$

(99)

if $p⟶1$ , then we have periodic wave solution

$$\begin{array}{rl}{\psi }_{1,19}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}{\sinh }^4(x-\alpha t)}{\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}(\frac{768c_4{c}_6^2}{25c_4^4-256c_6^2}-\frac{16c_4(5c_4^2-16c_6)c_6{\sinh }^2(x-\alpha t)}{25c_4^4-256c_6^2})}\\ & -\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}}{2\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\end{array}$$

(100)

$$\begin{array}{rl}{\varphi }_{1,19}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}{\sinh }^4(x-\alpha t)}{\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}(\frac{768c_4{c}_6^2}{25c_4^4-256c_6^2}-\frac{16c_4(5c_4^2-16c_6)c_6{\sinh }^2(x-\alpha t)}{25c_4^4-256c_6^2})}\\ & -\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}}{2\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\end{array}$$

(101)

under the situation of constraint

$$3{(\frac{25c_4^4}{256c_6^2}-1)}^2{c}_6-c_4^2{(1-\frac{5c_4^2}{16c_6})}^2(\frac{5c_4^2}{16c_6}+2)=0$$

(102)

Case $8$ : If $k_0=1$ , $k_2=2p^2-1$ , $k_4=-p^2(1-p^2)$ then $\mathrm{\Delta }(\eta )=\text{sd}(\eta )$ and we have the Jacobi elliptic function solution

$$\begin{array}{rl}{\psi }_{1,20}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}\text{sd}(\eta {)}^4}{\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}(\frac{768c_4{c}_6^2}{25c_4^4-256c_6^2(p^4-p^2+1)}}\\ & -\frac{16c_4{c}_6(16c_6(1-2p^2)+5c_4^2)\text{sd}(\eta {)}^2}{25c_4^4-256c_6^2(p^4-p^2+1)})-\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}}{2\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\end{array}$$

(103)

$$\begin{array}{rl}{\varphi }_{1,20}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}\text{sd}(\eta {)}^4}{\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}(\frac{768c_4{c}_6^2}{25c_4^4-256c_6^2(p^4-p^2+1)}}\\ & -\frac{16c_4{c}_6(16c_6(1-2p^2)+5c_4^2)\text{sd}(\eta {)}^2}{25c_4^4-256c_6^2(p^4-p^2+1)})-\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}}{2\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\end{array}$$

(104)

where $e_1$ and $e_2$ are as follows:

$$e_1=-\frac{16c_4{c}_6(16c_6(1-2p^2)+5c_4^2)}{25c_4^4-256c_6^2(p^4-p^2+1)}$$

(105)

and

$$e_2=\frac{768c_4{c}_6^2}{25c_4^4-256c_6^2(p^4-p^2+1)}$$

(106)

under the situation of constraint

$$\begin{array}{rl}& c_4^2(-\frac{5c_4^2}{16c_6}+2p^2-1)(-(-\frac{5c_4^2}{16c_6}+2p^2-1)(\frac{5c_4^2}{16c_6}+2(2p^2-1))-9(1-p^2)p^2)\\ & +3c_6{(\frac{25c_4^4}{256c_6^2}-{(2p^2-1)}^2-3p^2(1-p^2))}^2=0\end{array}$$

(107)

if $p⟶0$ , then we have periodic wave solution

$$\begin{array}{rl}{\psi }_{1,21}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}{\sin }^4(x-\alpha t)}{\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}(\frac{768c_4{c}_6^2}{25c_4^4-256c_6^2}-\frac{16c_4{c}_6(5c_4^2+16c_6){\sin }^2(x-\alpha t)}{25c_4^4-256c_6^2})}\\ & -\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}}{2\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\end{array}$$

(108)

$$\begin{array}{rl}{\varphi }_{1,21}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}{\sin }^4(x-\alpha t)}{\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}(\frac{768c_4{c}_6^2}{25c_4^4-256c_6^2}-\frac{16c_4{c}_6(5c_4^2+16c_6){\sin }^2(x-\alpha t)}{25c_4^4-256c_6^2})}\\ & -\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}}{2\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\end{array}$$

(109)

under the situation of constraint

$$3{(\frac{25c_4^4}{256c_6^2}-1)}^2{c}_6-c_4^2{(-\frac{5c_4^2}{16c_6}-1)}^2(\frac{5c_4^2}{16c_6}-2)=0$$

(110)

if $p⟶1$ , then we have periodic wave solution

$$\begin{array}{rl}{\psi }_{1,22}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}{\sinh }^4(x-\alpha t)}{\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}(\frac{768c_4{c}_6^2}{25c_4^4-256c_6^2}-\frac{16c_4(5c_4^2-16c_6)c_6{\sinh }^2(x-\alpha t)}{25c_4^4-256c_6^2})}\\ & -\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}}{2\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\end{array}$$

(111)

$$\begin{array}{rl}{\varphi }_{1,22}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}{\sinh }^4(x-\alpha t)}{\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}(\frac{768c_4{c}_6^2}{25c_4^4-256c_6^2}-\frac{16c_4(5c_4^2-16c_6)c_6{\sinh }^2(x-\alpha t)}{25c_4^4-256c_6^2})}\\ & -\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}}{2\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\end{array}$$

(112)

under the situation of constraint

$$3{(\frac{25c_4^4}{256c_6^2}-1)}^2{c}_6-c_4^2{(1-\frac{5c_4^2}{16c_6})}^2(\frac{5c_4^2}{16c_6}+2)=0$$

(113)

Case $9$ : If $k_0=1-p^2$ , $k_2=2-p^2$ , $k_4=1$ then $\mathrm{\Delta }(\eta )=\text{cs}(\eta )$ and we have the Jacobi elliptic function solution

$$\begin{array}{rl}{\psi }_{1,23}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\text{cs}(\eta {)}^4\sqrt{{\sigma }_5-{\delta }_2}}{\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}(\frac{768c_4{c}_6^2(p^2-1)}{256c_6^2(p^4-p^2+1)-25c_4^4}}\\ & -\frac{16c_4{c}_6(16c_6(p^2-2)+5c_4^2)\text{cs}(\eta {)}^2}{25c_4^4-256c_6^2(p^4-p^2+1)})-\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}}{2\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\end{array}$$

(114)

$$\begin{array}{rl}{\varphi }_{1,23}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\text{cs}(\eta {)}^4\sqrt{{\sigma }_2-{\delta }_5}}{\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}(\frac{768c_4{c}_6^2(p^2-1)}{256c_6^2(p^4-p^2+1)-25c_4^4}}\\ & -\frac{16c_4{c}_6(16c_6(p^2-2)+5c_4^2)\text{cs}(\eta {)}^2}{25c_4^4-256c_6^2(p^4-p^2+1)})-\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}}{2\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\end{array}$$

(115)

where $e_1$ and $e_2$ are as follows:

$$e_1=-\frac{16c_4{c}_6(16c_6(p^2-2)+5c_4^2)}{25c_4^4-256c_6^2(p^4-p^2+1)}$$

(116)

and

$$e_2=\frac{768c_4{c}_6^2(p^2-1)}{256c_6^2(p^4-p^2+1)-25c_4^4}$$

(117)

under the situation of constraint

$$\begin{array}{rl}& c_4^2(-\frac{5c_4^2}{16c_6}-p^2+2)(9(1-p^2)-(-\frac{5c_4^2}{16c_6}-p^2+2)(\frac{5c_4^2}{16c_6}+2(2-p^2)))\\ & +3c_6{(\frac{25c_4^4}{256c_6^2}-{(2-p^2)}^2+3(1-p^2))}^2=0\end{array}$$

(118)

if $p⟶0$ , then we have periodic wave solution

$$\begin{array}{rl}{\psi }_{1,24}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}{\cot }^4(x-\alpha t)}{\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}(-\frac{16c_4(5c_4^2-32c_6)c_6{\cot }^2(x-\alpha t)}{25c_4^4-256c_6^2}-\frac{768c_4{c}_6^2}{256c_6^2-25c_4^4})}\\ & -\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}}{2\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\end{array}$$

(119)

$$\begin{array}{rl}{\varphi }_{1,24}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}{\cot }^4(x-\alpha t)}{\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}(-\frac{16c_4(5c_4^2-32c_6)c_6{\cot }^2(x-\alpha t)}{25c_4^4-256c_6^2}-\frac{768c_4{c}_6^2}{256c_6^2-25c_4^4})}\\ & -\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}}{2\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\end{array}$$

(120)

under the situation of constraint

$$(2-\frac{5c_4^2}{16c_6})(9-(2-\frac{5c_4^2}{16c_6})(\frac{5c_4^2}{16c_6}+4))c_4^2+3c_6{(\frac{25c_4^4}{256c_6^2}-1)}^2=0$$

(121)

if $p⟶1$ , then we have periodic wave solution

$$\begin{array}{rl}{\psi }_{1,25}(x,t)=& \frac{(25c_4^4-256c_6^2)\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}{\text{csch}}^2(x-\alpha t)}{8c_4(5c_4^2-16c_6)\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\\ & -\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}}{2\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\end{array}$$

(122)

$$\begin{array}{rl}{\varphi }_{1,25}(x,t)=& \frac{(25c_4^4-256c_6^2)\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}{\text{csch}}^2(x-\alpha t)}{8c_4(5c_4^2-16c_6)\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\\ & -\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}}{2\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\end{array}$$

(123)

under the situation of constraint

$$3{(\frac{25c_4^4}{256c_6^2}-1)}^2{c}_6-c_4^2{(1-\frac{5c_4^2}{16c_6})}^2(\frac{5c_4^2}{16c_6}+2)=0$$

(124)

Case $10$ : If $k_0=-p^2(1-p^2)$ , $k_2=2p^2-1$ , $k_4=1$ then $\mathrm{\Delta }(\eta )=\text{ds}(\eta )$ and we have the Jacobi elliptic function solution

$$\begin{array}{rl}{\psi }_{1,26}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}\text{ds}(\eta {)}^4}{\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}(-\frac{16c_4{c}_6(16c_6(1-2p^2)+5c_4^2)\text{ds}(\eta {)}^2}{25c_4^4-256c_6^2(p^4-p^2+1)}}\\ & -\frac{768c_4{c}_6^2{p}^2(p^2-1)}{256c_6^2(p^4-p^2+1)-25c_4^4})-\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}}{2\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\end{array}$$

(125)

$$\begin{array}{rl}{\varphi }_{1,26}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}\text{ds}(\eta {)}^4}{\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}(-\frac{16c_4{c}_6(16c_6(1-2p^2)+5c_4^2)\text{ds}(\eta {)}^2}{25c_4^4-256c_6^2(p^4-p^2+1)}}\\ & -\frac{768c_4{c}_6^2{p}^2(p^2-1)}{256c_6^2(p^4-p^2+1)-25c_4^4})-\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}}{2\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\end{array}$$

(126)

where $e_1$ and $e_2$ are as follows:

$$e_1=-\frac{16c_4{c}_6(16c_6(1-2p^2)+5c_4^2)}{25c_4^4-256c_6^2(p^4-p^2+1)}$$

(127)

and

$$e_2=-\frac{768c_4{c}_6^2{p}^2(p^2-1)}{256c_6^2(p^4-p^2+1)-25c_4^4}$$

(128)

under the situation of constraint

$$\begin{array}{rl}& c_4^2(-\frac{5c_4^2}{16c_6}+2p^2-1)(-(-\frac{5c_4^2}{16c_6}+2p^2-1)(\frac{5c_4^2}{16c_6}+2(2p^2-1))-9(1-p^2)p^2)\\ & +3c_6{(\frac{25c_4^4}{256c_6^2}-{(2p^2-1)}^2-3p^2(1-p^2))}^2=0\end{array}$$

(129)

if $p⟶0$ , then we have periodic wave solution

$$\begin{array}{rl}{\psi }_{1,27}(x,t)=& \frac{(25c_4^4-256c_6^2)\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}{\csc }^2(x-\alpha t)}{8c_4\sqrt{c_6}(5c_4^2+16c_6)\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\\ & -\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}}{2\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\end{array}$$

(130)

$$\begin{array}{rl}{\varphi }_{1,27}(x,t)=& \frac{(25c_4^4-256c_6^2)\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}{\csc }^2(x-\alpha t)}{8c_4\sqrt{c_6}(5c_4^2+16c_6)\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\\ & -\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}}{2\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\end{array}$$

(131)

under the situation of constraint

$$3{(\frac{25c_4^4}{256c_6^2}-1)}^2{c}_6-c_4^2{(-\frac{5c_4^2}{16c_6}-1)}^2(\frac{5c_4^2}{16c_6}-2)=0$$

(132)

if $p⟶1$ , then we have periodic wave solution

$$\begin{array}{rl}{\psi }_{1,28}(x,t)=& \frac{(25c_4^4-256c_6^2)\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}{\text{csch}}^2(x-\alpha t)}{8c_4(5c_4^2-16c_6)\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\\ & -\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}}{2\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\end{array}$$

(133)

$$\begin{array}{rl}{\varphi }_{1,28}(x,t)=& \frac{(25c_4^4-256c_6^2)\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}{\text{csch}}^2(x-\alpha t)}{8c_4(5c_4^2-16c_6)\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\\ & -\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}}{2\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\end{array}$$

(134)

under the situation of constraint

$$3{(\frac{25c_4^4}{256c_6^2}-1)}^2{c}_6-c_4^2{(1-\frac{5c_4^2}{16c_6})}^2(\frac{5c_4^2}{16c_6}+2)=0$$

(135)

Case $11$ : If $k_0=\frac{1}{4}(1-p^2)$ , $k_2=\frac{1}{2}(p^2+1)$ , $k_4=\frac{1}{4}(1-p^2)$ then $\mathrm{\Delta }(\eta )=\frac{\text{cn}(\eta )}{\text{sn}(\eta )+1}$ then $\mathrm{\Delta }(\eta )=\text{ds}(\eta )$ and we have the Jacobi elliptic function solution

$$\begin{array}{rl}{\psi }_{1,29}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\text{cn}(\eta {)}^4\sqrt{{\sigma }_5-{\delta }_2}}{(\text{sn}(\eta )+1{)}^4\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}(\frac{192c_4{c}_6^2(p^2-1)}{16c_6^2(p^4+14p^2+1)-25c_4^4}-}\\ & \frac{16c_4{c}_6(5c_4^2-8c_6(p^2+1))\text{cn}(\eta {)}^2}{(25c_4^4-16c_6^2(p^4+14p^2+1))(\text{sn}(\eta )+1{)}^2})-\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}}{2\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\end{array}$$

(136)

$$\begin{array}{rl}{\varphi }_{1,29}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\text{cn}(\eta {)}^4\sqrt{{\sigma }_2-{\delta }_5}}{(\text{sn}(\eta )+1{)}^4\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}(\frac{192c_4{c}_6^2(p^2-1)}{16c_6^2(p^4+14p^2+1)-25c_4^4}}\\ & -\frac{16c_4{c}_6(5c_4^2-8c_6(p^2+1))\text{cn}(\eta {)}^2}{(25c_4^4-16c_6^2(p^4+14p^2+1))(\text{sn}(\eta )+1{)}^2})-\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}}{2\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\end{array}$$

(137)

where $e_1$ and $e_2$ are as follows:

$$e_1=-\frac{16c_4{c}_6(5c_4^2-8c_6(p^2+1))}{25c_4^4-16c_6^2(p^4+14p^2+1)}$$

(138)

and

$$e_2=\frac{192c_4{c}_6^2(p^2-1)}{16c_6^2(p^4+14p^2+1)-25c_4^4}$$

(139)

under the situation of constraint

$$\begin{array}{rl}& c_4^2(\frac{1}{2}(p^2+1)-\frac{5c_4^2}{16c_6})(\frac{9}{16}{(1-p^2)}^2-(\frac{1}{2}(p^2+1)-\frac{5c_4^2}{16c_6})(\frac{5c_4^2}{16c_6}+p^2+1))\\ & +3c_6{(\frac{25c_4^4}{256c_6^2}+\frac{3}{16}{(1-p^2)}^2-\frac{1}{4}{(p^2+1)}^2)}^2=0\end{array}$$

(140)

if $p⟶0$ , then we have periodic wave solution

$$\begin{array}{rl}{\psi }_{1,30}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}{\cos }^4(x-\alpha t)}{\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}(\sin (x-\alpha t)+1{)}^4(-\frac{16c_4(5c_4^2-8c_6)c_6{\cos }^2(x-\alpha t)}{(25c_4^4-16c_6^2)(\sin (x-\alpha t)+1{)}^2}}\\ & -\frac{192c_4{c}_6^2}{16c_6^2-25c_4^4})-\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}}{2\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\end{array}$$

(141)

$$\begin{array}{rl}{\varphi }_{1,30}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}{\cos }^4(x-\alpha t)}{\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}(\sin (x-\alpha t)+1{)}^4(-\frac{16c_4(5c_4^2-8c_6)c_6{\cos }^2(x-\alpha t)}{(25c_4^4-16c_6^2)(\sin (x-\alpha t)+1{)}^2}}\\ & -\frac{192c_4{c}_6^2}{16c_6^2-25c_4^4})-\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}}{2\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\end{array}$$

(142)

under the situation of constraint

$$\begin{array}{rl}& (\frac{1}{2}-\frac{5c_4^2}{16c_6})(\frac{9}{16}-(\frac{1}{2}-\frac{5c_4^2}{16c_6})(\frac{5c_4^2}{16c_6}+1))c_4^2\\ & +3c_6{(\frac{25c_4^4}{256c_6^2}-\frac{1}{16})}^2=0\end{array}$$

(143)

if $p⟶1$ , then we have periodic wave solution

$$\begin{array}{rl}{\psi }_{1,31}(x,t)=& \frac{(25c_4^4-256c_6^2)\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}{\text{sech}}^2(x-\alpha t)}{8c_4(5c_4^2-16c_6)\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}(\tanh (x-\alpha t)+1{)}^2}\\ & -\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}}{2\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\end{array}$$

(144)

$$\begin{array}{rl}{\varphi }_{1,31}(x,t)=& \frac{(25c_4^4-256c_6^2)\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}{\text{sech}}^2(x-\alpha t)}{8c_4(5c_4^2-16c_6)\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}(\tanh (x-\alpha t)+1{)}^2}\\ & -\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}}{2\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\end{array}$$

(145)

under the situation of constraint

$$3{(\frac{25c_4^4}{256c_6^2}-1)}^2{c}_6-c_4^2{(1-\frac{5c_4^2}{16c_6})}^2(\frac{5c_4^2}{16c_6}+2)=0$$

(146)

Case $12$ : If $k_0=-\frac{1}{4}(1-p^2{)}^2$ , $k_2=\frac{1}{2}(p^2+1)$ , $k_4=-\frac{1}{4}$ then $\mathrm{\Delta }(\eta )=p\text{cn}(\eta )+\text{dn}(\eta )$ and we have the Jacobi elliptic function solution

$$\begin{array}{rl}{\psi }_{1,32}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}(p\text{cn}(\eta )+\text{dn}(\eta ){)}^4}{\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}(\frac{192c_4{c}_6^2{(p^2-1)}^2}{16c_6^2(p^4+14p^2+1)-25c_4^4}}\\ & -\frac{16c_4{c}_6(5c_4^2-8c_6(p^2+1))(p\text{cn}(\eta )+\text{dn}(\eta ){)}^2}{25c_4^4-16c_6^2(p^4+14p^2+1)})-\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}}{2\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\end{array}$$

(147)

$$\begin{array}{rl}{\varphi }_{1,32}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}(p\text{cn}(\eta )+\text{dn}(\eta ){)}^4}{\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}(\frac{192c_4{c}_6^2{(p^2-1)}^2}{16c_6^2(p^4+14p^2+1)-25c_4^4}}\\ & -\frac{16c_4{c}_6(5c_4^2-8c_6(p^2+1))(p\text{cn}(\eta )+\text{dn}(\eta ){)}^2}{25c_4^4-16c_6^2(p^4+14p^2+1)})-\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}}{2\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\end{array}$$

(148)

where $e_1$ and $e_2$ are as follows:

$$e_1=-\frac{16c_4{c}_6(5c_4^2-8c_6(p^2+1))}{25c_4^4-16c_6^2(p^4+14p^2+1)}$$

(149)

and

$$e_2=\frac{192c_4{c}_6^2{(p^2-1)}^2}{16c_6^2(p^4+14p^2+1)-25c_4^4}$$

(150)

under the situation of constraint

$$\begin{array}{rl}& c_4^2(\frac{1}{2}(p^2+1)-\frac{5c_4^2}{16c_6})(\frac{9}{16}{(1-p^2)}^2-(\frac{1}{2}(p^2+1)-\frac{5c_4^2}{16c_6})(\frac{5c_4^2}{16c_6}+p^2+1))\\ & +3c_6{(\frac{25c_4^4}{256c_6^2}+\frac{3}{16}{(1-p^2)}^2-\frac{1}{4}{(p^2+1)}^2)}^2=0\end{array}$$

(151)

if $p⟶1$ , then we have periodic wave solution

$$\begin{array}{rl}{\psi }_{1,33}(x,t)=& \frac{(25c_4^4-256c_6^2)\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}{\text{sech}}^2(x-\alpha t)}{2c_4(5c_4^2-16c_6)\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\\ & -\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}}{2\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\end{array}$$

(152)

$$\begin{array}{rl}{\varphi }_{1,33}(x,t)=& \frac{(25c_4^4-256c_6^2)\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}{\text{sech}}^2(x-\alpha t)}{2c_4(5c_4^2-16c_6)\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\\ & -\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}}{2\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\end{array}$$

(153)

under the situation of constraint

$$3{(\frac{25c_4^4}{256c_6^2}-1)}^2{c}_6-c_4^2{(1-\frac{5c_4^2}{16c_6})}^2(\frac{5c_4^2}{16c_6}+2)=0.$$

(154)

Case $13$ : If $k_0=\frac{1}{4}$ , $k_2=\frac{1}{2}(1-2p^2)$ , $k_4=\frac{1}{4}$ then $\mathrm{\Delta }(\eta )=\frac{\text{sn}(\eta )}{\text{cn}(\eta )+1}$ and we have the Jacobi elliptic function solution

$$\begin{array}{rl}{\psi }_{1,35}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}\text{sn}(\eta {)}^4}{(\text{cn}(\eta )+1{)}^4\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}(\frac{192c_4{c}_6^2}{25c_4^4-16c_6^2(16p^4-16p^2+1)}}\\ & -\frac{16c_4{c}_6(8c_6(2p^2-1)+5c_4^2)\text{sn}(\eta {)}^2}{(25c_4^4-16c_6^2(16p^4-16p^2+1))(\text{cn}(\eta )+1{)}^2})-\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}}{2\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\end{array}$$

(155)

$$\begin{array}{rl}{\varphi }_{1,35}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}\text{sn}(\eta {)}^4}{(\text{cn}(\eta )+1{)}^4\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}(\frac{192c_4{c}_6^2}{25c_4^4-16c_6^2(16p^4-16p^2+1)}}\\ & -\frac{16c_4{c}_6(8c_6(2p^2-1)+5c_4^2)\text{sn}(\eta {)}^2}{(25c_4^4-16c_6^2(16p^4-16p^2+1))(\text{cn}(\eta )+1{)}^2})-\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}}{2\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\end{array}$$

(156)

where $e_1$ and $e_2$ are as follows:

$$e_1=-\frac{16c_4{c}_6(8c_6(2p^2-1)+5c_4^2)}{25c_4^4-16c_6^2(16p^4-16p^2+1)}$$

(157)

and

$$e_2=\frac{192c_4{c}_6^2}{25c_4^4-16c_6^2(16p^4-16p^2+1)}$$

(158)

under the situation of constraint

$$\begin{array}{rl}& c_4^2(\frac{1}{2}(1-2p^2)-\frac{5c_4^2}{16c_6})(\frac{9}{16}-(\frac{1}{2}(1-2p^2)-\frac{5c_4^2}{16c_6})(\frac{5c_4^2}{16c_6}-2p^2+1))\\ & +3c_6{(\frac{25c_4^4}{256c_6^2}-\frac{1}{4}{(1-2p^2)}^2+\frac{3}{16})}^2=0\end{array}$$

(159)

if $p⟶0$ , then we have periodic wave solution

$$\begin{array}{rl}{\psi }_{1,36}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}{\sin }^4(x-\alpha t)}{\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}(\cos (x-\alpha t)+1{)}^4(\frac{192c_4{c}_6^2}{25c_4^4-16c_6^2}}\\ & -\frac{16c_4(5c_4^2-8c_6)c_6{\sin }^2(x-\alpha t)}{(25c_4^4-16c_6^2)(\cos (x-\alpha t)+1{)}^2})\\ & -\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}}{2\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\end{array}$$

(160)

$$\begin{array}{rl}{\varphi }_{1,36}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}{\sin }^4(x-\alpha t)}{\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}(\cos (x-\alpha t)+1{)}^4(\frac{192c_4{c}_6^2}{25c_4^4-16c_6^2}}\\ & -\frac{16c_4(5c_4^2-8c_6)c_6{\sin }^2(x-\alpha t)}{(25c_4^4-16c_6^2)(\cos (x-\alpha t)+1{)}^2})-\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}}{2\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\end{array}$$

(161)

under the situation of constraint

$$\begin{array}{rl}& (\frac{1}{2}-\frac{5c_4^2}{16c_6})(\frac{9}{16}-(\frac{1}{2}-\frac{5c_4^2}{16c_6})(\frac{5c_4^2}{16c_6}+1))c_4^2\\ & +3c_6{(\frac{25c_4^4}{256c_6^2}-\frac{1}{16})}^2=0\end{array}$$

(162)

if $p⟶1$ , then we have periodic wave solution

$$\begin{array}{rl}{\psi }_{1,37}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}{\tanh }^4(x-\alpha t)}{\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}(\text{sech}(x-\alpha t)+1{)}^4(\frac{192c_4{c}_6^2}{25c_4^4-16c_6^2}}\\ & -\frac{16c_4{c}_6(5c_4^2+8c_6){\tanh }^2(x-\alpha t)}{(25c_4^4-16c_6^2)(\text{sech}(x-\alpha t)+1{)}^2})-\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}}{2\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\end{array}$$

(163)

$$\begin{array}{rl}{\varphi }_{1,37}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}{\tanh }^4(x-\alpha t)}{\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}(\text{sech}(x-\alpha t)+1{)}^4(\frac{192c_4{c}_6^2}{25c_4^4-16c_6^2}}\\ & -\frac{16c_4{c}_6(5c_4^2+8c_6){\tanh }^2(x-\alpha t)}{(25c_4^4-16c_6^2)(\text{sech}(x-\alpha t)+1{)}^2})-\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}}{2\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\end{array}$$

(164)

under the situation of constraint

$$\begin{array}{rl}& (-\frac{5c_4^2}{16c_6}-\frac{1}{2})(\frac{9}{16}-(-\frac{5c_4^2}{16c_6}-\frac{1}{2})(\frac{5c_4^2}{16c_6}-1))c_4^2\\ & +3c_6{(\frac{25c_4^4}{256c_6^2}-\frac{1}{16})}^2=0\end{array}$$

(165)

Case $14$ : If $k_0=\frac{1}{4}$ , $k_2=\frac{1}{2}(p^2+1)$ , $k_4=\frac{1}{4}(1-p^2{)}^2$ then $\mathrm{\Delta }(\eta )=\frac{\text{sn}(\eta )}{\text{cn}(\eta )+\text{dn}(\eta )}$ and we have the Jacobi elliptic function solution

$$\begin{array}{rl}{\psi }_{1,38}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}\text{sn}(\eta {)}^4}{(\text{cn}(\eta )+\text{dn}(\eta ){)}^4\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}(\frac{192c_4{c}_6^2}{25c_4^4-16c_6^2(p^4+14p^2+1)}}\\ & -\frac{16c_4{c}_6(5c_4^2-8c_6(p^2+1))\text{sn}(\eta {)}^2}{(25c_4^4-16c_6^2(p^4+14p^2+1))(\text{cn}(\eta )+\text{dn}(\eta ){)}^2})-\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}}{2\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\end{array}$$

(166)

$$\begin{array}{rl}{\varphi }_{1,38}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}\text{sn}(\eta {)}^4}{(\text{cn}(\eta )+\text{dn}(\eta ){)}^4\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}(\frac{192c_4{c}_6^2}{25c_4^4-16c_6^2(p^4+14p^2+1)}}\\ & -\frac{16c_4{c}_6(5c_4^2-8c_6(p^2+1))\text{sn}(\eta {)}^2}{(25c_4^4-16c_6^2(p^4+14p^2+1))(\text{cn}(\eta )+\text{dn}(\eta ){)}^2})-\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}}{2\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\end{array}$$

(167)

where $e_1$ and $e_2$ are as follows:

$$e_1=-\frac{16c_4{c}_6(5c_4^2-8c_6(p^2+1))}{25c_4^4-16c_6^2(p^4+14p^2+1)}$$

(168)

and

$$e_2=\frac{192c_4{c}_6^2}{25c_4^4-16c_6^2(p^4+14p^2+1)}$$

(169)

under the situation of constraint

$$\begin{array}{rl}& c_4^2(\frac{1}{2}(p^2+1)-\frac{5c_4^2}{16c_6})(\frac{9}{16}{(1-p^2)}^2-(\frac{1}{2}(p^2+1)-\frac{5c_4^2}{16c_6})(\frac{5c_4^2}{16c_6}+p^2+1))\\ & +3c_6{(\frac{25c_4^4}{256c_6^2}+\frac{3}{16}{(1-p^2)}^2-\frac{1}{4}{(p^2+1)}^2)}^2=0\end{array}$$

(170)

if $p⟶0$ , then we have periodic wave solution

$$\begin{array}{rl}{\psi }_{1,39}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}{\sin }^4(x-\alpha t)}{\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}(\cos (x-\alpha t)+1{)}^4(\frac{192c_4{c}_6^2}{25c_4^4-16c_6^2}}\\ & -\frac{16c_4(5c_4^2-8c_6)c_6{\sin }^2(x-\alpha t)}{(25c_4^4-16c_6^2)(\cos (x-\alpha t)+1{)}^2})-\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}}{2\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\end{array}$$

(171)

$$\begin{array}{rl}{\varphi }_{1,39}(x,t)=& -\frac{2\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}{\sin }^4(x-\alpha t)}{\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}(\cos (x-\alpha t)+1{)}^4(\frac{192c_4{c}_6^2}{25c_4^4-16c_6^2}}\\ & -\frac{16c_4(5c_4^2-8c_6)c_6{\sin }^2(x-\alpha t)}{(25c_4^4-16c_6^2)(\cos (x-\alpha t)+1{)}^2})-\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}}{2\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\end{array}$$

(172)

under the situation of constraint

$$\begin{array}{rl}& (\frac{1}{2}-\frac{5c_4^2}{16c_6})(\frac{9}{16}-(\frac{1}{2}-\frac{5c_4^2}{16c_6})(\frac{5c_4^2}{16c_6}+1))c_4^2\\ & +3c_6{(\frac{25c_4^4}{256c_6^2}-\frac{1}{16})}^2=0\end{array}$$

(173)

if $p⟶1$ , then we have periodic wave solution

$$\begin{array}{rl}{\psi }_{1,40}(x,t)=& -\frac{\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}{\sinh }^4(x-\alpha t)}{8\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}(\frac{192c_4{c}_6^2}{25c_4^4-256c_6^2}-\frac{4c_4(5c_4^2-16c_6)c_6{\sinh }^2(x-\alpha t)}{25c_4^4-256c_6^2})}\\ & -\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_5-{\delta }_2}}{2\sqrt{c_6}\sqrt{n({\delta }_2{\sigma }_2-{\delta }_5{\sigma }_5)}}\end{array}$$

(174)

$$\begin{array}{rl}{\varphi }_{1,40}(x,t)=& -\frac{\sqrt{c_6}\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}{\sinh }^4(x-\alpha t)}{8\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}(\frac{192c_4{c}_6^2}{25c_4^4-256c_6^2}-\frac{4c_4(5c_4^2-16c_6)c_6{\sinh }^2(x-\alpha t)}{25c_4^4-256c_6^2})}\\ & -\frac{c_4\sqrt{M+2}\sqrt{u}\sqrt{{\sigma }_2-{\delta }_5}}{2\sqrt{c_6}\sqrt{{\delta }_{5}n{\sigma }_5-{\delta }_{2}n{\sigma }_2}}\end{array}$$

(175)

under the situation of constraint

$$3{(\frac{25c_4^4}{256c_6^2}-1)}^2{c}_6-c_4^2{(1-\frac{5c_4^2}{16c_6})}^2(\frac{5c_4^2}{16c_6}+2)=0$$

(176)

Graphical discussion

In this section, we use contour charts and 3D to demonstrate a few solutions. All of the graphs are created using Mathematica. The ${\varphi }^6$ approach is an effective tool for magneto-optic waveguide modeling since it provides robust answers to a variety of soliton topologies and is used to generate accurate solutions for NLPDEs. In this section, we display the 3D and contour graphs of some of our responses to show their physical relevance. The magneto-optic waveguide model offers lumps, solitons, kink waves, periodic-wave solutions, and combinations in addition to other wave solutions in exponential, trigonometric, and hyperbolic functions. The magneto-optic waveguide model is a temporal derivative of second order. Negative time values are plotted to show the pre-evolution (initial) state of the wave and to present a complete symmetric picture of the solution over the entire time domain. The solution ${\psi }_{1,2}$ and ${\varphi }_{1,2}$ that yields the solitary wave solution in the form of 3D, line and contour graphs are depicted in Figure 1 by selecting the parameters, such as $\alpha =3.9$ , $b=0.2$ , $c_4=0.1$ , $c_6=-0.1$ , ${\delta }_2=3.1$ , ${\delta }_5=1.02$ , $n=1$ , ${\sigma }_5=0.3$ , ${\sigma }_2=2.01$ , $u=2.2$ , and $M=\frac{1}{1+n}=\frac{1}{2}$ . The solution ${\psi }_{1,3}$ and ${\varphi }_{1,3}$ that yields the dark soliton in the form of 3D, line and contour graphs are depicted in Figure 2 by selecting the parameters, such as $\alpha =3.9$ , $b=0.2$ , $c_4=0.1$ , $c_6=-0.1$ , ${\delta }_2=3.1$ , ${\delta }_5=1.02$ , $n=1$ , ${\sigma }_5=0.3$ , ${\sigma }_2=2.01$ , $u=2.2$ , and $M=\frac{1}{1+n}=\frac{1}{2}$ . The solution ${\psi }_{1,5}$ and ${\varphi }_{1,5}$ that provides a single wave solution in 3D, line and contour graphs are depicted in Figure 3 by selecting the parameters, such as $\alpha =2.9$ , $b=0.2$ , $c_4=0.1$ , $c_6=0.1$ , ${\delta }_2=2.1$ , ${\delta }_5=1.01$ , $n=2$ , ${\sigma }_5=0.4$ , ${\sigma }_2=2.02$ , $u=2.1$ , and $M=\frac{1}{1+n}=\frac{1}{3}$ . The solution ${\psi }_{1,6}$ and ${\varphi }_{1,6}$ that yields a bright solution is also depicted in Figure 4 by selecting the parameters, such as $\alpha =2.9$ , $b=0.2$ , $c_4=0.1$ , $c_6=0.1$ , ${\delta }_2=2.1$ , ${\delta }_5=2.01$ , $n=2$ , ${\sigma }_5=1.4$ , ${\sigma }_2=1.02$ , $u=2.1$ , and $M=\frac{1}{1+n}=\frac{1}{3}$ . The solution ${\psi }_{1,10}$ and ${\varphi }_{1,10}$ that yields a solitary wave solution is also depicted in Figure 5 by selecting the parameters, such as $\alpha =3.9$ , $b=0.2$ , $c_4=0.1$ , $c_6=0.1$ , ${\delta }_2=3.1$ , ${\delta }_5=0.03$ , $n=1$ , ${\sigma }_5=0.5$ , ${\sigma }_2=0.01$ , $u=2.1$ , and $M=\frac{1}{1+n}=\frac{1}{2}$ . The solution ${\psi }_{1,13}$ and ${\varphi }_{1,13}$ that yields a solitary wave solution is also depicted in Figure 6 by selecting the parameters, such as $\alpha =3.9$ , $b=0.2$ , $c_4=0.1$ , $c_6=0.1$ , ${\delta }_2=3.1$ , ${\delta }_5=0.03$ , $n=1$ , ${\sigma }_5=0.5$ , ${\sigma }_2=0.01$ , $u=2.1$ , and $M=\frac{1}{1+n}=\frac{1}{2}$ . The findings made in this article, which apply contour charts and 3D graphs, provide important information on how nonlinear waves can be used in magneto-optic waveguides, and the implications directly relate to practical use. The solitons, kink wave modeling, periodic-wave solutions, and hybrid waveforms modeling proves how these phenomena can be exploited in optical communication systems. An example is solitons, which preserve their shape and energy throughout long distances, and, therefore, solitons are the best in transmitting data across long distances without losing its strength. Kink waves and dark solitons, which are a sudden jumps and a localized depression, can be applied to optical switching and signal modulation. Changing different parameters (strength of magnetic field and waveguide characteristics) demonstrated in the graphs indicates the effect of these factors on the propagation of waves. The use of this work is essential in the development of optical equipment, such as modulators, isolators, and sensors, which depends on the accurate light control, the future of telecommunications and optical computing, and other magneto-optic systems. The stability of the obtained solutions under small perturbations is a critical aspect of their physical applicability, especially for real-world systems like optical communication devices or sensors. In nonlinear wave systems, such as the ones described in this study, it is essential to assess whether the solitons, kink waves, or other solutions remain robust when subjected to small disturbances, as this determines their feasibility for practical use. For example, in the case of solitons, their inherent stability against small perturbations is one of the key features that makes them attractive for optical applications like data transmission. However, this stability must be carefully examined, particularly for complex solutions like kink or dark solitons, which may exhibit different stability characteristics depending on the specific parameters of the waveguide and the strength of external fields.

Figure 1.

Physical representation for ${\psi }_{1,2}(x,t)$ and ${\varphi }_{1,2}(x,t)$ by selecting the parameters such as $\alpha =3.9,b=0.2,c_4=0.1,c_6=-0.1,{\delta }_2=3.1,{\delta }_5=1.02,n=1,{\sigma }_5=0.3,{\sigma }_2=2.01,$ $u=2.2,\mathrm{and}M=\frac{1}{1+n}=\frac{1}{2}$ .

Figure 2.

Physical representation for ${\psi }_{1,3}(x,t)$ and ${\varphi }_{1,3}(x,t)$ by selecting the parameters such as $\alpha =3.9,b=0.2,c_4=0.1,c_6=-0.1,{\delta }_2=3.1,{\delta }_5=1.02,n=1,{\sigma }_5=0.3,{\sigma }_2=2.01,$ $u=2.2,\mathrm{and}M=\frac{1}{1+n}=\frac{1}{2}$ .

Figure 3.

Physical representation for ${\psi }_{1,5}(x,t)$ and ${\varphi }_{1,5}(x,t)$ by selecting the parameters such as $\alpha =2.9,b=0.2,c_4=0.1,c_6=0.1,{\delta }_2=2.1,{\delta }_5=1.01,n=2,{\sigma }_5=0.4,{\sigma }_2=2.02,$ $u=2.1,\mathrm{and}M=\frac{1}{1+n}=\frac{1}{3}$ .

Figure 4.

Physical representation for ${\psi }_{1,6}(x,t)$ and ${\varphi }_{1,6}(x,t)$ by selecting the parameters such as $\alpha =2.9,b=0.2,c_4=0.1,c_6=0.1,{\delta }_2=2.1,{\delta }_5=2.01,n=2,{\sigma }_5=1.4,{\sigma }_2=1.02,$ $u=2.1,\mathrm{and}M=\frac{1}{1+n}=\frac{1}{3}$ .

Figure 5.

Physical representation for ${\psi }_{1,10}(x,t)$ and ${\varphi }_{1,10}(x,t)$ by selecting the parameters such as $\alpha =3.9,b=0.2,c_4=0.1,c_6=0.1,{\delta }_2=3.1,{\delta }_5=0.03,n=1,{\sigma }_5=0.5,{\sigma }_2=0.01,$ $u=2.1,\mathrm{and}M=\frac{1}{1+n}=\frac{1}{2}$ .

Figure 6.

Physical representation for ${\psi }_{1,13}(x,t)$ and ${\varphi }_{1,13}(x,t)$ by selecting the parameters such as $\alpha =3.9,b=0.2,c_4=0.1,c_6=0.1,{\delta }_2=3.1,{\delta }_5=0.03,n=1,{\sigma }_5=0.5,{\sigma }_2=0.01,$ $u=2.1,\mathrm{and}M=\frac{1}{1+n}=\frac{1}{2}$ .

Conclusion

This work preserves the GAC form of the nonlinear self-phase modulation structure while recovering optical solitons in magneto-optic waveguides. A novel ${\varphi }^6$ -model expansion method is presented, and its application in magneto-optic waveguides is explored. As an advanced extension of traditional perturbation techniques, the ${\varphi }^6$ method provides a precise and efficient approach for solving wave propagation problems in magneto-optic media. Comparing the proposed ${\varphi }^6$ -model expansion strategy to existing approaches, such as the ${\varphi }^4$ model and the standard coupled-mode approach, reveals that it is more effective. In particular, in the presence of strong nonlinear and magneto-optic interactions, it offers a more precise description of magneto-optic waveguide propagation characteristics. In practical applications, magneto-optic waveguides require reduced optical losses, integration with photonic platforms, and enhanced performance. By integrating the effects of electric and magnetic fields into a unified framework, this method enables more accurate simulation of light behavior in magneto-optic waveguides. The study provides exact solutions for dispersion relations, field distributions, and transmission characteristics, addressing the challenges posed by nonlinear magneto-optic effects on waveguide performance. Comprehensive numerical simulations validate the effectiveness of the ${\varphi }^6$ -model expansion method, demonstrating its capability to predict field intensities and propagation constants over a wide range of parameters. This approach facilitates the design of magneto-optic waveguides with tailored properties for next-generation photonic devices. By providing a more accurate and reliable modeling framework, the ${\varphi }^6$ approach promotes photonic integration, communication, and sensing technologies, leading to the development of more efficient magneto-optic devices. Furthermore, the ${\varphi }^6$ method exhibits faster convergence in some computational steps while maintaining numerical stability over a larger range of parameters. Thus, the proposed method provides a more precise and efficient analytical tool for studying magneto-optic waveguide phenomena.