Mathematical and Physical Analysis of Fractional Estevez–Mansfield–Clarkson Equation

Abstract

This paper presents the mathematical and physical analysis, as well as distinct types of exact wave solutions, of an important fluid flow dynamics model called the truncated M-fractional (1+1)-dimensional nonlinear Estevez–Mansfield–Clarkson (EMC) equation. This model is used to explain waves in shallow water, fluid dynamics, and other areas. We obtain kink, bright, singular, and other types of exact wave solutions using the modified extended direct algebraic method and the improved $(G^{\prime }/G)$-expansion method. Some solutions do not exist. These solutions may be useful in different areas of science and engineering. The results are represented as three-dimensional, contour, and two-dimensional graphs. Stability analysis is also performed to check the stability of the corresponding model. Furthermore, modulation instability analysis is performed to study the stationary solutions of the corresponding model. The results will be helpful for future studies of the corresponding system. The methods used are easy and useful.

1. Introduction

Mathematical modeling is important in applied sciences and engineering. Many models in different branches of science and engineering have been developed, such as the Biswas–Arshed model [1], the single-joint robot arm model [2], the Kundu–Mukherjee–Naskar model [3], the generalized Rosenau–Korteweg–de Vries regularized long-wave equation [4], etc. Only in the last few years has the use of fractional derivatives in mathematical modeling played a significant role. Nonlinear fractional partial differential equations are a more prominent way to represent any naturally occurring phenomenon. Various mathematical models are represented in a form involving different fractional derivatives, such as Caputo fractional diffusion models [5], Atangana–Baleanu fractional Newton’s law of cooling [6], conformable fractional Gerdjikov–Ivanov model [7], beta fractional modified Benjamin–Bona–Mahony model [8], truncated M-fractional Westervelt model [9], and many more.

There are different methods that have been developed to obtain exact soliton solutions. Recently, the generalized double auxiliary equation technique [10], the new extended direct algebraic scheme [11], the Ricatti equation mapping technique [12], the Paul–Painleve technique [13], the improved Fan sub-equation approach [14], the enhanced algebraic method [15], the exponential rational function technique [16], and the multiple exp-function scheme [17] have been used.

In our study, we utilized straightforward and powerful methods: the modified extended direct algebraic (MEDA) method and the improved $(G^{\prime }/G)$-expansion method. These methods have been used to discuss various models in the literature. For instance, the modified extended direct algebraic (MEDA) method is used for the nonlinear Schrödinger equation [18], the coupled Higgs system [19], the Lakshmanan–Porseizian–Daniel model [20], the Gerdjikov–Ivanov model [21], the highly dispersive perturbed Schrödinger model [22], and the extended (2+1)-dimensional perturbed nonlinear Schrödinger equation [23]. Similarly, the improved $(G^{\prime }/G)$-expansion method is utilized for the extended shallow-water wave models [24], the Calogero–Bogoyavlenskii–Schiff model [25], etc.

The motivation of this paper is to obtain distinct kinds of exact soliton solutions to the (1+1)-dimensional nonlinear Estevez–Mansfield–Clarkson equation, along with a novel definition of the derivative by employing the modified extended direct algebraic method and the improved $(G^{\prime }/G)$-expansion method. The novel definition of the fractional derivative, known as the truncated M-fractional derivative, fulfills the properties of both integer-order and fractional-order derivatives. A truncated M-fractional derivative provides results close to the numerical results. The modified extended direct algebraic method provides us with singular soliton, dark wave, bright wave, dark–bright, and rational wave solutions, while the improved $(G^{\prime }/G)$-expansion method provides kink, periodic soliton, and many more solutions. Our obtained results are newer than the existing results of the model under study. Both techniques are useful and reliable. These techniques can be used for other nonlinear, fractional partial differential equations. The advantage of both techniques is that there are no limitations or assumptions for their use. We also perform a physical analysis through graphical representation, stability analysis, and modulation instability.

This paper is organized as follows. The corresponding model and its mathematical analysis are explained in Section 2. The MEDA method and its application are given in Section 3. The improved $(G^{\prime }/G)$-expansion method and its application are shown in Section 4. Some results are shown graphically and physically in Section 5. Stability analysis is given in Section 6. Modulation instability is explained in Section 7. The conclusions are presented in Section 8.

2. Model Description and Mathematical Analysis

In this research, the model under study is the nonlinear (1+1)-dimensional Estevez–Mansfield–Clarkson (EMC) equation given as [26]

$$\Omega g_t{g}_{yy}+g_{tt}+\Omega g_{ty}{g}_y+g_{tyyy}=0,$$

(1)

where $g=g(y,t)$ is a wave function. The parameter $\Omega$ represents a non-zero constant.

The truncated M-fractional nonlinear (1+1)-dimensional Estevez–Mansfield–Clarkson equation is given as

$$\Omega D_{M,t}^{ϵ,\varrho }g{D}_{M,y}^{2ϵ,\varrho }g+D_{M,t}^{2ϵ,\varrho }g+\Omega D_{M,t}^{ϵ,\varrho }\left(D_{M,y}^{ϵ,\varrho }g\right)D_{M,y}^{ϵ,\varrho }g+D_{M,t}^{ϵ,\varrho }\left(D_{M,y}^{3ϵ,\varrho }g\right)=0,$$

(2)

where

$$D_{M,y}^{ϵ,\varrho }g=\underset{\tau \to 0}{lim}{\displaystyle \frac{g\left(y{E}_{\varrho }\left(\tau y^{1-ϵ}\right)\right)-g\left(y\right)}{\tau }},ϵ\in (0,1],\varrho >0,$$

(3)

and $E_{\varrho }(.)$ represents a truncated Mittag–Leffler function [27,28].

In 1997, Mansfield and Clarkson’s research on the pattern dispersion in liquid drops introduced the fourth-order nonlinear evolution equation. The Estevez–Mansfield–Clarkson equation is used in various areas of nonlinear science, such as plasma physics, optics, fluid dynamics, image processing, and many others. The EMC equation contains a difficult set of complications and is usually utilized to explain wave behavior in shallow water. In the literature, different methods are applied to solve Equation (1). For instance, some solitary wave solutions are obtained using the Riccati Bernoulli sub-ODE method [26], kink wave solutions are obtained using the simple equation method [29], and bell-shaped, peakon, kink, and compact soliton solutions are obtained utilizing the generalized Kudryashov method [30].

By applying the following wave transformation to Equation (2):

$$g(y,t)=G\left(\zeta \right)\times exp\left(\iota \zeta \right);\zeta ={\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}-\lambda t^{ϵ}).$$

(4)

we obtain

$${\delta }^3{G}^{‴}+2{\delta }^2\Omega {\left(G^{\prime }\right)}^2-\lambda G^{\prime }=0.$$

(5)

We obtain the value of m by utilizing the homogeneous balance scheme in Equation (5) as follows.

Balancing the terms $G^{‴}$ and ${\left(G^{\prime }\right)}^2$, we obtain

$3m=m+2$, so $m=1$.

In the next section, exact wave solitons of Equation (5) are found using two methods.

3. Modified Extended Direct Algebraic Method and Its Application

Some of the main steps are given below [19].

Step 1: Assume a nonlinear PDE

$$F(g,g^2,g^2{g}_y,g_{yy},g_{yt},\dots )=0,$$

(6)

where $g=g(y,t)$ denotes a wave function. Consider the following transformation:

$$g(y,t)=G\left(\xi \right),\xi =y+\lambda t.$$

(7)

Substituting Equation (7) into Equation (6) yields

$$H(G,G^2{G}^{\prime },G^{″},\dots )=0.$$

(8)

Step 2: Suppose the results of Equation (8) are given as

$$G\left(\xi \right)=\sum _{s=-m}^m{\alpha }_s{\psi }^s\left(\xi \right),$$

(9)

where ${\alpha }_s(s=-m,\dots ,0,1,2,3,\dots ,m)$ are undetermined. The function $\psi \left(\xi \right)$ satisfies the equation

$${\psi }^{\prime }\left(\xi \right)=log\left(d\right)\left(p+q\psi \left(\xi \right)+r\psi {\left(\xi \right)}^2\right),$$

(10)

where p, q, and r are constants and $d\ne 0.1$. Consider the solutions of Equation (10) in the following cases:

Case 1: When $\Theta =q^2-4pr<0$ and $r\ne 0$, then

$${\psi }_1\left(\xi \right)=-{\displaystyle \frac{q}{2r}}+{\displaystyle \frac{\sqrt{-\Theta }{tan}_B\left(\frac{1}{2}\sqrt{-\Theta }\xi \right)}{2r}}.$$

(11)

$${\psi }_2\left(\xi \right)=-{\displaystyle \frac{q}{2r}}-{\displaystyle \frac{\sqrt{-\Theta }{cot}_B\left(\frac{1}{2}\sqrt{-\Theta }\xi \right)}{2r}}.$$

(12)

$${\psi }_3\left(\xi \right)=-{\displaystyle \frac{q}{2r}}+{\displaystyle \frac{\sqrt{-\Theta }({tan}_B\left(\sqrt{-\Theta }\xi \right)\pm \left(\sqrt{cf}{sec}_B\left(\sqrt{-\Theta }\xi \right)\right))}{2r}}.$$

(13)

$${\psi }_4\left(\xi \right)=-{\displaystyle \frac{q}{2r}}-{\displaystyle \frac{\sqrt{-\Theta }({cot}_B\left(\sqrt{-\Theta }\xi \right)\pm \left(\sqrt{cf}{csc}_B\left(\sqrt{-\Theta }\xi \right)\right))}{2r}}.$$

(14)

$${\psi }_5\left(\xi \right)=-{\displaystyle \frac{q}{2r}}+{\displaystyle \frac{\sqrt{-\Theta }({tan}_B\left(\frac{1}{4}\sqrt{-\Theta }\xi \right)-\left({cot}_B\left(\frac{1}{4}\sqrt{-\Theta }\xi \right)\right))}{2r}}.$$

(15)

Case 2: When $\Theta =q^2-4pr>0$ and $r\ne 0$, then

$${\psi }_6\left(\xi \right)=-{\displaystyle \frac{q}{2r}}-{\displaystyle \frac{\sqrt{\Theta }{tanh}_B\left(\frac{1}{2}\sqrt{\Theta }\xi \right)}{2r}}.$$

(16)

$${\psi }_7\left(\xi \right)=-{\displaystyle \frac{q}{2r}}-{\displaystyle \frac{\sqrt{\Theta }{coth}_B\left(\frac{1}{2}\sqrt{\Theta }\xi \right)}{2r}}.$$

(17)

$${\psi }_8\left(\xi \right)=-{\displaystyle \frac{q}{2r}}-{\displaystyle \frac{\sqrt{\Theta }({tanh}_B\left(\sqrt{\Theta }\xi \right)\pm \left(\sqrt{cf}sec{h}_B\left(\sqrt{\Theta }\xi \right)\right))}{2r}}.$$

(18)

$${\psi }_9\left(\xi \right)=-{\displaystyle \frac{q}{2r}}-{\displaystyle \frac{\sqrt{\Theta }({coth}_B\left(\sqrt{\Theta }\xi \right)\pm \left(\sqrt{cf}csc{h}_B\left(\sqrt{\Theta }\xi \right)\right))}{2r}}.$$

(19)

$${\psi }_{10}\left(\xi \right)=-{\displaystyle \frac{q}{2r}}-{\displaystyle \frac{\sqrt{\Theta }({tanh}_B\left(\frac{1}{4}\sqrt{\Theta }\xi \right)-\left({coth}_B\left(\frac{1}{4}\sqrt{\Theta }\xi \right)\right))}{2r}}.$$

(20)

Case 3: When $pr>0$ and q is zero, then

$${\psi }_{11}\left(\xi \right)=\sqrt{{\displaystyle \frac{p}{r}}}{tan}_B\left(\sqrt{pr}\xi \right).$$

(21)

$${\psi }_{12}\left(\xi \right)=-\sqrt{{\displaystyle \frac{p}{r}}}{cot}_B\left(\sqrt{pr}\xi \right).$$

(22)

$${\psi }_{13}\left(\xi \right)=\sqrt{{\displaystyle \frac{p}{r}}}({tan}_B\left(2\sqrt{rp}\xi \right)\pm \left(\sqrt{cf}{sec}_B\left(2\sqrt{rp}\xi \right)\right)).$$

(23)

$${\psi }_{14}\left(\xi \right)=-\sqrt{{\displaystyle \frac{p}{r}}}({cot}_B\left(2\sqrt{rp}\xi \right)\pm \left(\sqrt{cf}{csc}_B\left(2\sqrt{rp}\xi \right)\right)).$$

(24)

$${\psi }_{15}\left(\xi \right)={\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{p}{r}}}({tan}_B\left({\displaystyle \frac{1}{2}}\sqrt{rp}\xi \right)-{cot}_B\left({\displaystyle \frac{1}{2}}\sqrt{pr}\xi \right)).$$

(25)

Case 4: When $pr<0$ and q is zero, then

$${\psi }_{16}\left(\xi \right)=-\sqrt{-{\displaystyle \frac{p}{r}}}{tanh}_B\left(\sqrt{-rp}\xi \right).$$

(26)

$${\psi }_{17}\left(\xi \right)=-\sqrt{-{\displaystyle \frac{p}{r}}}{coth}_B\left(\sqrt{-rp}\xi \right).$$

(27)

$${\psi }_{18}\left(\xi \right)=-\sqrt{-{\displaystyle \frac{p}{r}}}({tanh}_B\left(\sqrt{-rp}2\xi \right)\pm \left(\iota \sqrt{cf}sec{h}_B\left(\sqrt{-rp}2\xi \right)\right)).$$

(28)

$${\psi }_{19}\left(\xi \right)=-\sqrt{-{\displaystyle \frac{p}{r}}}({coth}_B\left(\sqrt{-rp}2\xi \right)\pm \left(\sqrt{cf}csc{h}_B\left(\sqrt{-rp}2\xi \right)\right)).$$

(29)

$${\psi }_{20}\left(\xi \right)=-{\displaystyle \frac{1}{2}}\sqrt{-{\displaystyle \frac{p}{r}}}({tanh}_B\left({\displaystyle \frac{1}{2}}\sqrt{-rp}\xi \right)+{coth}_B\left({\displaystyle \frac{1}{2}}\sqrt{-rp}\xi \right)).$$

(30)

Case 5: When $r=p$ and q is zero, then

$${\psi }_{21}\left(\xi \right)={tan}_B\left(p\xi \right).$$

(31)

$${\psi }_{22}\left(\xi \right)=-{cot}_B\left(p\xi \right).$$

(32)

$${\psi }_{23}\left(\xi \right)={tan}_B\left(p2\xi \right)\pm (\sqrt{cf}{sec}_B\left(p2\xi \right).$$

(33)

$${\psi }_{24}\left(\xi \right)=-{cot}_B\left(p2\xi \right)\pm (\sqrt{cf}{csc}_B\left(p2\xi \right).$$

(34)

$${\psi }_{25}\left(\xi \right)={\displaystyle \frac{1}{2}}{tan}_B\left({\displaystyle \frac{1}{2}}p\xi \right)-{\displaystyle \frac{1}{2}}{cot}_B\left({\displaystyle \frac{1}{2}}p\xi \right).$$

(35)

Case 6: When $r=-p$ and q is zero, then

$${\psi }_{26}\left(\xi \right)=-{tanh}_B\left(p\xi \right).$$

(36)

$${\psi }_{27}\left(\xi \right)=-{coth}_B\left(p\xi \right).$$

(37)

$${\psi }_{28}\left(\xi \right)=-{tanh}_B\left(p2\xi \right)\pm (\iota \sqrt{cf}sec{h}_B\left(p2\xi \right).$$

(38)

$${\psi }_{29}\left(\xi \right)=-{coth}_B\left(p2\xi \right)\pm (\sqrt{cf}csc{h}_B\left(p2\xi \right).$$

(39)

$${\psi }_{30}\left(\xi \right)=-{\displaystyle \frac{1}{2}}{tanh}_B\left({\displaystyle \frac{1}{2}}p\xi \right)-{\displaystyle \frac{1}{2}}{coth}_B\left({\displaystyle \frac{1}{2}}p\xi \right).$$

(40)

Case 7: When $q^2-4pr=0$, then

$${\psi }_{31}\left(\xi \right)=-2{\displaystyle \frac{p(q\xi log(d)+2)}{q^2\xi log\left(d\right)}}.$$

(41)

Case 8: When $q=\rho$, $p=\rho m$$(m\ne 0)$, and r is zero, then

$${\psi }_{32}\left(\xi \right)=d^{\rho \xi }-m.$$

(42)

Case 9: When q and r are zero, then

$${\psi }_{33}\left(\xi \right)=p\xi log\left(d\right).$$

(43)

Case 10: When p and q are zero, then

$${\psi }_{34}\left(\xi \right)=-{\displaystyle \frac{1}{r\xi log\left(d\right)}}.$$

(44)

Case 11: When p is zero, $q\ne 0$, and $r\ne 0$, then

$${\psi }_{35}\left(\xi \right)=-{\displaystyle \frac{cq}{r({cosh}_B\left(q\xi \right)-{sinh}_B\left(q\xi \right)+c)}}.$$

(45)

$${\psi }_{36}\left(\xi \right)=-{\displaystyle \frac{q({cosh}_B\left(q\xi \right)+{sinh}_B\left(q\xi \right))}{r({cosh}_B\left(q\xi \right)+{sinh}_B\left(q\xi \right)+f)}}.$$

(46)

Case 12: When $q=\rho$$r=m\rho$$(m\ne 0)$ and p is zero, then

$${\psi }_{37}\left(\xi \right)={\displaystyle \frac{c{d}^{\rho \xi }}{c-mf{d}^{\rho \xi }}},$$

(47)

where c and f are deformation positive constants.

Step 3: Substitute Equations (9) and (10) into Equation (8). Collect the coefficients of every power of $\psi$ and set them equal to zero, yielding a system of equations. By solving this system, we obtain results for the unknowns.

Step 4: Substitute Equation (9) into Equation (8) to find the solutions to Equation (6).

Application

Equation (9) takes the given form for $m=1$:

$$G\left(\xi \right)={\alpha }_{-1}{\left(\psi \left(\xi \right)\right)}^{-1}+{\alpha }_0+{\alpha }_1\psi \left(\xi \right).$$

(48)

Substituting Equations (48) and (10) into Equation (5) yields the following solution sets:

Set 1:

$$\left\{{\alpha }_{-1}={\displaystyle \frac{3\delta plog\left(d\right)}{\Omega }},{\alpha }_0={\alpha }_0,{\alpha }_1=0,\lambda ={\delta }^3{log}^2\left(d\right)\left(q^2-4pr\right)\right\}.$$

(49)

Case 1:

$$\begin{array}{c}g(y,t)=[{\alpha }_0+{\displaystyle \frac{3\delta plog\left(d\right)}{\Omega }}(-{\displaystyle \frac{q}{2r}}+\left(\sqrt{-(q^2-4pr)}{tan}_B\right({\displaystyle \frac{1}{2}}\sqrt{-(q^2-4pr)}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}\hfill \\ (\delta y^{ϵ}-{\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ})\left)\right)/\left(2r\right){)}^{-1}]\times exp(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}-\\ \hfill {\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ}\left)\right).\end{array}$$

(50)

$$\begin{array}{c}g(y,t)=[{\alpha }_0+{\displaystyle \frac{3\delta plog\left(d\right)}{\Omega }}(-{\displaystyle \frac{q}{2r}}-\left(\sqrt{-(q^2-4pr)}{cot}_B\right({\displaystyle \frac{1}{2}}\sqrt{-(q^2-4pr)}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}\hfill \\ (\delta y^{ϵ}-{\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ})\left)\right)/\left(2r\right){)}^{-1}]\times exp(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}-\\ \hfill {\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ}\left)\right).\end{array}$$

(51)

$$\begin{array}{c}g(y,t)=[{\alpha }_0+{\displaystyle \frac{3\delta plog\left(d\right)}{\Omega }}(-{\displaystyle \frac{q}{2r}}+(\sqrt{-(q^2-4pr)}\left({tan}_B\right(\sqrt{-(q^2-4pr)}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}\hfill \\ (\delta y^{ϵ}-{\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ}))\pm (\sqrt{cf}{sec}_B\left(\sqrt{-(q^2-4pr)}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}\right(\delta y^{ϵ}-\\ \hfill {\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ}\left)\right)\left)\right))/\left(2r\right){)}^{-1}]\times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}-{\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ})\right).\end{array}$$

(52)

$$\begin{array}{c}g(y,t)=[{\alpha }_0+{\displaystyle \frac{3\delta plog\left(d\right)}{\Omega }}(-{\displaystyle \frac{q}{2r}}-(\sqrt{-(q^2-4pr)}\left({cot}_B\right(\sqrt{-(q^2-4pr)}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}\hfill \\ (\delta y^{ϵ}-{\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ}))\pm (\sqrt{cf}{csc}_B\left(\sqrt{-(q^2-4pr)}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}\right(\delta y^{ϵ}-\\ \hfill {\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ}\left)\right)\left)\right))/\left(2r\right){)}^{-1}]\times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}-{\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ})\right).\end{array}$$

(53)

$$\begin{array}{c}g(y,t)=[{\alpha }_0+{\displaystyle \frac{3\delta plog\left(d\right)}{\Omega }}(-{\displaystyle \frac{q}{2r}}+(\sqrt{-(q^2-4pr)}\left({tan}_B\right({\displaystyle \frac{1}{4}}\sqrt{-(q^2-4pr)}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}\hfill \\ (\delta y^{ϵ}-{\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ}))-({cot}_B\left({\displaystyle \frac{1}{4}}\sqrt{-(q^2-4pr)}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}\right(\delta y^{ϵ}-\\ \hfill {\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ}\left)\right)\left)\right))/\left(2r\right){)}^{-1}]\times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}-{\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ})\right).\end{array}$$

(54)

Case 2:

$$\begin{array}{c}g(y,t)=[{\alpha }_0+{\displaystyle \frac{3\delta plog\left(d\right)}{\Omega }}(-{\displaystyle \frac{q}{2r}}-\left(\sqrt{(q^2-4pr)}{tanh}_B\right({\displaystyle \frac{1}{2}}\sqrt{(q^2-4pr)}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}\hfill \\ (\delta y^{ϵ}-{\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ})\left)\right)/\left(2r\right){)}^{-1}]\times exp(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}-\\ \hfill {\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ}\left)\right).\end{array}$$

(55)

$$\begin{array}{c}g(y,t)=[{\alpha }_0+{\displaystyle \frac{3\delta plog\left(d\right)}{\Omega }}(-{\displaystyle \frac{q}{2r}}-\left(\sqrt{(q^2-4pr)}{coth}_B\right({\displaystyle \frac{1}{2}}\sqrt{(q^2-4pr)}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}\hfill \\ (\delta y^{ϵ}-{\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ})\left)\right)/\left(2r\right){)}^{-1}]\times exp(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}-\\ \hfill {\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ}\left)\right).\end{array}$$

(56)

$$\begin{array}{c}g(y,t)=[{\alpha }_0+{\displaystyle \frac{3\delta plog\left(d\right)}{\Omega }}(-{\displaystyle \frac{q}{2r}}-(\sqrt{(q^2-4pr)}\left({tanh}_B\right(\sqrt{-(q^2-4pr)}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}\hfill \\ (\delta y^{ϵ}-{\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ}))\pm (\sqrt{cf}sec{h}_B\left(\sqrt{(q^2-4pr)}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}\right(\delta y^{ϵ}-\\ \hfill {\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ}\left)\right)\left)\right))/\left(2r\right){)}^{-1}]\times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}-{\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ})\right).\end{array}$$

(57)

$$\begin{array}{c}g(y,t)=[{\alpha }_0+{\displaystyle \frac{3\delta plog\left(d\right)}{\Omega }}(-{\displaystyle \frac{q}{2r}}-(\sqrt{(q^2-4pr)}\left({coth}_B\right(\sqrt{(q^2-4pr)}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}\hfill \\ (\delta y^{ϵ}-{\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ}))\pm (\sqrt{cf}csc{h}_B\left(\sqrt{(q^2-4pr)}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}\right(\delta y^{ϵ}-\\ \hfill {\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ}\left)\right)\left)\right))/\left(2r\right){)}^{-1}]\times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}-{\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ})\right).\end{array}$$

(58)

$$\begin{array}{c}g(y,t)=[{\alpha }_0+{\displaystyle \frac{3\delta plog\left(d\right)}{\Omega }}(-{\displaystyle \frac{q}{2r}}-(\sqrt{(q^2-4pr)}\left({tanh}_B\right({\displaystyle \frac{1}{4}}\sqrt{(q^2-4pr)}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}\hfill \\ (\delta y^{ϵ}-{\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ}))-({coth}_B\left({\displaystyle \frac{1}{4}}\sqrt{(q^2-4pr)}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}\right(\delta y^{ϵ}-\\ \hfill {\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ}\left)\right)\left)\right))/\left(2r\right){)}^{-1}]\times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}-{\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ})\right).\end{array}$$

(59)

Case 3:

$$\begin{array}{c}g(y,t)=[{\alpha }_0+{\displaystyle \frac{3\delta plog\left(d\right)}{\Omega }}{\left(\sqrt{{\displaystyle \frac{p}{r}}}{tan}_B\left(\sqrt{pr}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right)\right)}^{-1}]\hfill \\ \hfill \times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right).\end{array}$$

(60)

$$\begin{array}{c}g(y,t)=[{\alpha }_0+{\displaystyle \frac{3\delta plog\left(d\right)}{\Omega }}{(-\sqrt{{\displaystyle \frac{p}{r}}}{cot}_B\left(\sqrt{pr}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right))}^{-1}]\hfill \\ \hfill \times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right).\end{array}$$

(61)

$$\begin{array}{c}g(y,t)=[{\alpha }_0+{\displaystyle \frac{3\delta plog\left(d\right)}{\Omega }}\left(\sqrt{{\displaystyle \frac{p}{r}}}\right({tan}_B\left(2\sqrt{pr}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right)\pm \hfill \\ \left(\sqrt{cf}{sec}_B\left(2\sqrt{pr}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right)\right){\left)\right)}^{-1}]\\ \hfill \times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right).\end{array}$$

(62)

$$\begin{array}{c}g(y,t)=[{\alpha }_0+{\displaystyle \frac{3\delta plog\left(d\right)}{\Omega }}(-\sqrt{{\displaystyle \frac{p}{r}}}({cot}_B\left(2\sqrt{pr}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right)\pm \hfill \\ \left(\sqrt{cf}{csc}_B\left(2\sqrt{pr}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right)\right){\left)\right)}^{-1}]\\ \hfill \times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right).\end{array}$$

(63)

$$\begin{array}{c}g(y,t)=[{\alpha }_0+{\displaystyle \frac{3\delta plog\left(d\right)}{\Omega }}\left({\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{p}{r}}}\right({tan}_B\left({\displaystyle \frac{1}{2}}\sqrt{pr}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right)-\hfill \\ {cot}_B\left({\displaystyle \frac{1}{2}}\sqrt{pr}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right){\left)\right)}^{-1}]\\ \hfill \times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right).\end{array}$$

(64)

Case 4:

$$\begin{array}{c}g(y,t)=[{\alpha }_0+{\displaystyle \frac{3\delta plog\left(d\right)}{\Omega }}{(-\sqrt{-{\displaystyle \frac{p}{r}}}{tanh}_B\left(\sqrt{-pr}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right))}^{-1}]\hfill \\ \hfill \times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right).\end{array}$$

(65)

$$\begin{array}{c}g(y,t)=[{\alpha }_0+{\displaystyle \frac{3\delta plog\left(d\right)}{\Omega }}{(-\sqrt{-{\displaystyle \frac{p}{r}}}{coth}_B\left(\sqrt{-pr}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right))}^{-1}]\hfill \\ \hfill \times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right).\end{array}$$

(66)

$$\begin{array}{c}g(y,t)=[{\alpha }_0+{\displaystyle \frac{3\delta plog\left(d\right)}{\Omega }}(-\sqrt{-{\displaystyle \frac{p}{r}}}({tanh}_B\left(2\sqrt{-pr}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right)\pm \hfill \\ \left(\iota \sqrt{cf}sec{h}_B\left(2\sqrt{-pr}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right)\right){\left)\right)}^{-1}]\\ \hfill \times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right).\end{array}$$

(67)

$$\begin{array}{c}g(y,t)=[{\alpha }_0+{\displaystyle \frac{3\delta plog\left(d\right)}{\Omega }}(-\sqrt{-{\displaystyle \frac{p}{r}}}({coth}_B\left(2\sqrt{-pr}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right)\pm \hfill \\ \left(\sqrt{cf}csc{h}_B\left(2\sqrt{-pr}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right)\right){\left)\right)}^{-1}]\\ \hfill \times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right).\end{array}$$

(68)

$$\begin{array}{c}g(y,t)=[{\alpha }_0+{\displaystyle \frac{3\delta plog\left(d\right)}{\Omega }}(-{\displaystyle \frac{1}{2}}\sqrt{-{\displaystyle \frac{p}{r}}}({tanh}_B\left({\displaystyle \frac{1}{2}}\sqrt{-pr}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right)+\hfill \\ {coth}_B\left({\displaystyle \frac{1}{2}}\sqrt{-pr}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right){\left)\right)}^{-1}]\\ \hfill \times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right).\end{array}$$

(69)

Case 5:

$$\begin{array}{c}g(y,t)=[{\alpha }_0+{\displaystyle \frac{3\delta plog\left(d\right)}{\Omega }}{\left({tan}_B\left(p{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4p^2{\delta }^3{log}^2\left(d\right)t^{ϵ})\right)\right)}^{-1}]\hfill \\ \hfill \times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4p^2{\delta }^3{log}^2\left(d\right)t^{ϵ})\right).\end{array}$$

(70)

$$\begin{array}{c}g(y,t)=[{\alpha }_0+{\displaystyle \frac{3\delta plog\left(d\right)}{\Omega }}{(-{cot}_B\left(p{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4p^2{\delta }^3{log}^2\left(d\right)t^{ϵ})\right))}^{-1}]\hfill \\ \hfill \times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4p^2{\delta }^3{log}^2\left(d\right)t^{ϵ})\right).\end{array}$$

(71)

$$\begin{array}{c}g(y,t)=[{\alpha }_0+{\displaystyle \frac{3\delta plog\left(d\right)}{\Omega }}({tan}_B\left(2p{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4p^2{\delta }^3{log}^2\left(d\right)t^{ϵ})\right)\pm \hfill \\ \hfill {\left(\sqrt{cf}{sec}_B\left(2p{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4p^2{\delta }^3{log}^2\left(d\right)t^{ϵ})\right)\right)}^{-1}]\times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4p^2{\delta }^3{log}^2\left(d\right)t^{ϵ})\right).\end{array}$$

(72)

$$\begin{array}{c}g(y,t)=[{\alpha }_0+{\displaystyle \frac{3\delta plog\left(d\right)}{\Omega }}(-{cot}_B\left(2p{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4p^2{\delta }^3{log}^2\left(d\right)t^{ϵ})\right)\pm \hfill \\ \hfill {\left(\sqrt{cf}{csc}_B\left(2p{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4p^2{\delta }^3{log}^2\left(d\right)t^{ϵ})\right)\right)}^{-1}]\times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4p^2{\delta }^3{log}^2\left(d\right)t^{ϵ})\right).\end{array}$$

(73)

$$\begin{array}{c}g(y,t)=[{\alpha }_0+{\displaystyle \frac{3\delta plog\left(d\right)}{\Omega }}({\displaystyle \frac{1}{2}}{tan}_B\left({\displaystyle \frac{1}{2}}p{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4p^2{\delta }^3{log}^2\left(d\right)t^{ϵ})\right)-\hfill \\ \hfill {\displaystyle \frac{1}{2}}{cot}_B\left({\displaystyle \frac{1}{2}}p{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4p^2{\delta }^3{log}^2\left(d\right)t^{ϵ})\right){)}^{-1}]\times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4p^2{\delta }^3{log}^2\left(d\right)t^{ϵ})\right).\end{array}$$

(74)

Case 6:

$$\begin{array}{c}g(y,t)=[{\alpha }_0+{\displaystyle \frac{3\delta plog\left(d\right)}{\Omega }}{(-{tanh}_B\left(p{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}-4p^2{\delta }^3{log}^2\left(d\right)t^{ϵ})\right))}^{-1}]\times \hfill \\ \hfill exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}-4p^2{\delta }^3{log}^2\left(d\right)t^{ϵ})\right).\end{array}$$

(75)

$$\begin{array}{c}g(y,t)=[{\alpha }_0+{\displaystyle \frac{3\delta plog\left(d\right)}{\Omega }}{(-{coth}_B\left(p{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}-4p^2{\delta }^3{log}^2\left(d\right)t^{ϵ})\right))}^{-1}]\times \hfill \\ \hfill exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}-4p^2{\delta }^3{log}^2\left(d\right)t^{ϵ})\right).\end{array}$$

(76)

$$\begin{array}{c}g(y,t)=[{\alpha }_0+{\displaystyle \frac{3\delta plog\left(d\right)}{\Omega }}(-{tanh}_B\left(2p{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}-4p^2{\delta }^3{log}^2\left(d\right)t^{ϵ})\right)\pm \hfill \\ \hfill {\left(\iota \sqrt{cf}sec{h}_B\left(2p{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}-4p^2{\delta }^3{log}^2\left(d\right)t^{ϵ})\right)\right)}^{-1}]\times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}-4p^2{\delta }^3{log}^2\left(d\right)t^{ϵ})\right).\end{array}$$

(77)

$$\begin{array}{c}g(y,t)=[{\alpha }_0+{\displaystyle \frac{3\delta plog\left(d\right)}{\Omega }}(-{coth}_B\left(2p{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}-4p^2{\delta }^3{log}^2\left(d\right)t^{ϵ})\right)\pm \hfill \\ \hfill {\left(\sqrt{cf}csc{h}_B\left(2p{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}-4p^2{\delta }^3{log}^2\left(d\right)t^{ϵ})\right)\right)}^{-1}]\times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}-4p^2{\delta }^3{log}^2\left(d\right)t^{ϵ})\right).\end{array}$$

(78)

$$\begin{array}{c}g(y,t)=[{\alpha }_0+{\displaystyle \frac{3\delta plog\left(d\right)}{\Omega }}(-{\displaystyle \frac{1}{2}}{tanh}_B\left({\displaystyle \frac{1}{2}}p{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}-4p^2{\delta }^3{log}^2\left(d\right)t^{ϵ})\right)-\hfill \\ \hfill {\displaystyle \frac{1}{2}}{coth}_B\left({\displaystyle \frac{1}{2}}p{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}-4p^2{\delta }^3{log}^2\left(d\right)t^{ϵ})\right){)}^{-1}]\times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}-4p^2{\delta }^3{log}^2\left(d\right)t^{ϵ})\right).\end{array}$$

(79)

Case 8:

$$\begin{array}{c}g(y,t)=[{\alpha }_0+{\displaystyle \frac{3\delta m\rho log\left(d\right)}{\Omega }}{(d^{\rho {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}-{\rho }^2{\delta }^3{log}^2\left(d\right)t^{ϵ})}-m)}^{-1}]\times \hfill \\ \hfill exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}-{\rho }^2{\delta }^3{log}^2\left(d\right)t^{ϵ})\right).\end{array}$$

(80)

Note: Cases 7, 9, 10, 11, and 12 result in zero solutions.

Set 2:

$$\left\{{\alpha }_{-1}=0,{\alpha }_1=-{\displaystyle \frac{3\delta rlog\left(d\right)}{\Omega }},\lambda ={\delta }^3{log}^2\left(d\right)\left(q^2-4pr\right)\right\}.$$

(81)

Case 1:

$$\begin{array}{c}g(y,t)=[{\alpha }_0-{\displaystyle \frac{3\delta rlog\left(d\right)}{\Omega }}(-{\displaystyle \frac{q}{2r}}+\left(\sqrt{-(q^2-4pr)}{tan}_B\right({\displaystyle \frac{1}{2}}\sqrt{-(q^2-4pr)}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}\hfill \\ (\delta y^{ϵ}-{\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ})\left)\right)/\left(2r\right)\left)\right]\times exp(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}-\\ \hfill {\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ}\left)\right).\end{array}$$

(82)

$$\begin{array}{c}g(y,t)=[{\alpha }_0-{\displaystyle \frac{3\delta rlog\left(d\right)}{\Omega }}(-{\displaystyle \frac{q}{2r}}-\left(\sqrt{-(q^2-4pr)}{cot}_B\right({\displaystyle \frac{1}{2}}\sqrt{-(q^2-4pr)}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}\hfill \\ (\delta y^{ϵ}-{\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ})\left)\right)/\left(2r\right)\left)\right]\times exp(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}\\ \hfill -{\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ}\left)\right).\end{array}$$

(83)

$$\begin{array}{c}g(y,t)=[{\alpha }_0-{\displaystyle \frac{3\delta rlog\left(d\right)}{\Omega }}(-{\displaystyle \frac{q}{2r}}+(\sqrt{-(q^2-4pr)}\left({tan}_B\right(\sqrt{-(q^2-4pr)}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}\hfill \\ (\delta y^{ϵ}-{\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ}))\pm (\sqrt{cf}{sec}_B\left(\sqrt{-(q^2-4pr)}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}\right(\delta y^{ϵ}-\\ \hfill {\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ}\left)\right)\left)\right))/\left(2r\right)\left)\right]\times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}-{\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ})\right).\end{array}$$

(84)

$$\begin{array}{c}g(y,t)=[{\alpha }_0-{\displaystyle \frac{3\delta rlog\left(d\right)}{\Omega }}(-{\displaystyle \frac{q}{2r}}-(\sqrt{-(q^2-4pr)}\left({cot}_B\right(\sqrt{-(q^2-4pr)}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}\hfill \\ (\delta y^{ϵ}-{\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ}))\pm (\sqrt{cf}{csc}_B\left(\sqrt{-(q^2-4pr)}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}\right(\delta y^{ϵ}-\\ \hfill {\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ}\left)\right)\left)\right))/\left(2r\right)\left)\right]\times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}-{\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ})\right).\end{array}$$

(85)

$$\begin{array}{c}g(y,t)=[{\alpha }_0-{\displaystyle \frac{3\delta rlog\left(d\right)}{\Omega }}(-{\displaystyle \frac{q}{2r}}+(\sqrt{-(q^2-4pr)}\left({tan}_B\right({\displaystyle \frac{1}{4}}\sqrt{-(q^2-4pr)}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}\hfill \\ (\delta y^{ϵ}-{\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ}))-({cot}_B\left({\displaystyle \frac{1}{4}}\sqrt{-(q^2-4pr)}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}\right(\delta y^{ϵ}\\ \hfill -{\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ}\left)\right)\left)\right))/\left(2r\right)\left)\right]\times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}-{\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ})\right).\end{array}$$

(86)

Case 2:

$$\begin{array}{c}g(y,t)=[{\alpha }_0-{\displaystyle \frac{3\delta rlog\left(d\right)}{\Omega }}(-{\displaystyle \frac{q}{2r}}-\left(\sqrt{(q^2-4pr)}{tanh}_B\right({\displaystyle \frac{1}{2}}\sqrt{(q^2-4pr)}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}\hfill \\ (\delta y^{ϵ}-{\delta }^3{log}^2\left(d\right)\left(q^2-4pr\right)t^{ϵ})\left)\right)/\left(2r\right)\left)\right]\times exp(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}-\\ \hfill {\delta }^3{log}^2\left(d\right)\left(q^2-4pr\right)t^{ϵ}\left)\right).\end{array}$$

(87)

$$\begin{array}{c}g(y,t)=[{\alpha }_0-{\displaystyle \frac{3\delta rlog\left(d\right)}{\Omega }}(-{\displaystyle \frac{q}{2r}}-\left(\sqrt{(q^2-4pr)}{coth}_B\right({\displaystyle \frac{1}{2}}\sqrt{(q^2-4pr)}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}\hfill \\ (\delta y^{ϵ}-{\delta }^3{log}^2\left(d\right)\left(q^2-4pr\right)t^{ϵ})\left)\right)/\left(2r\right)\left)\right]\times exp(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}\\ \hfill -{\delta }^3{log}^2\left(d\right)\left(q^2-4pr\right)t^{ϵ}\left)\right).\end{array}$$

(88)

$$\begin{array}{c}g(y,t)=[{\alpha }_0-{\displaystyle \frac{3\delta rlog\left(d\right)}{\Omega }}(-{\displaystyle \frac{q}{2r}}-(\sqrt{(q^2-4pr)}\left({tanh}_B\right(\sqrt{-(q^2-4pr)}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}\hfill \\ (\delta y^{ϵ}-{\delta }^3{log}^2\left(d\right)\left(q^2-4pr\right)t^{ϵ}))\pm (\sqrt{cf}sec{h}_B\left(\sqrt{(q^2-4pr)}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}\right(\delta y^{ϵ}\\ \hfill -{\delta }^3{log}^2\left(d\right)\left(q^2-4pr\right)t^{ϵ}\left)\right)\left)\right))/\left(2r\right)\left)\right]\times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}-{\delta }^3{log}^2\left(d\right)\left(q^2-4pr\right)t^{ϵ})\right).\end{array}$$

(89)

$$\begin{array}{c}g(y,t)=[{\alpha }_0-{\displaystyle \frac{3\delta rlog\left(d\right)}{\Omega }}(-{\displaystyle \frac{q}{2r}}-(\sqrt{(q^2-4pr)}\left({coth}_B\right(\sqrt{(q^2-4pr)}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}\hfill \\ (\delta y^{ϵ}-{\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ}))\pm (\sqrt{cf}csc{h}_B\left(\sqrt{(q^2-4pr)}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}\right(\delta y^{ϵ}\\ \hfill -{\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ}\left)\right)\left)\right))/\left(2r\right)\left)\right]\times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}-{\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ})\right).\end{array}$$

(90)

$$\begin{array}{c}g(y,t)=[{\alpha }_0-{\displaystyle \frac{3\delta rlog\left(d\right)}{\Omega }}(-{\displaystyle \frac{q}{2r}}-(\sqrt{(q^2-4pr)}\left({tanh}_B\right({\displaystyle \frac{1}{4}}\sqrt{(q^2-4pr)}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}\hfill \\ (\delta y^{ϵ}-{\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ}))-({coth}_B\left({\displaystyle \frac{1}{4}}\sqrt{(q^2-4pr)}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}\right(\delta y^{ϵ}\\ \hfill -{\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ}\left)\right)\left)\right))/\left(2r\right)\left)\right]\times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}-{\delta }^3{log}^2\left(d\right)(q^2-4pr)t^{ϵ})\right).\end{array}$$

(91)

Case 3:

$$\begin{array}{c}g(y,t)=[{\alpha }_0-{\displaystyle \frac{3\delta rlog\left(d\right)}{\Omega }}\left(\sqrt{{\displaystyle \frac{p}{r}}}{tan}_B\left(\sqrt{pr}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right)\right)]\times \hfill \\ \hfill exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right).\end{array}$$

(92)

$$\begin{array}{c}g(y,t)=[{\alpha }_0-{\displaystyle \frac{3\delta rlog\left(d\right)}{\Omega }}(-\sqrt{{\displaystyle \frac{p}{r}}}{cot}_B\left(\sqrt{pr}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right))]\times \hfill \\ \hfill exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right).\end{array}$$

(93)

$$\begin{array}{c}g(y,t)=[{\alpha }_0-{\displaystyle \frac{3\delta rlog\left(d\right)}{\Omega }}\left(\sqrt{{\displaystyle \frac{p}{r}}}\right({tan}_B\left(2\sqrt{pr}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right)\pm \hfill \\ \left(\sqrt{cf}{sec}_B\left(2\sqrt{pr}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right)\right)\left)\right)]\times exp\left(\iota \zeta \right)]\times \\ \hfill exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right).\end{array}$$

(94)

$$\begin{array}{c}g(y,t)=[{\alpha }_0-{\displaystyle \frac{3\delta rlog\left(d\right)}{\Omega }}(-\sqrt{{\displaystyle \frac{p}{r}}}({cot}_B\left(2\sqrt{pr}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right)\pm \hfill \\ \hfill \left(\sqrt{cf}{csc}_B\left(2\sqrt{pr}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right)\right)\left)\right)]\times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right).\end{array}$$

(95)

$$\begin{array}{c}g(y,t)=[{\alpha }_0-{\displaystyle \frac{3\delta rlog\left(d\right)}{\Omega }}\left({\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{p}{r}}}\right({tan}_B\left({\displaystyle \frac{1}{2}}\sqrt{pr}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right)-\hfill \\ \hfill {cot}_B\left({\displaystyle \frac{1}{2}}\sqrt{pr}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right)\left)\right)]\times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right).\end{array}$$

(96)

Case 4:

$$\begin{array}{c}g(y,t)=[{\alpha }_0-{\displaystyle \frac{3\delta rlog\left(d\right)}{\Omega }}(-\sqrt{-{\displaystyle \frac{p}{r}}}{tanh}_B\left(\sqrt{-pr}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right))]\hfill \\ \hfill \times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right).\end{array}$$

(97)

$$\begin{array}{c}g(y,t)=[{\alpha }_0-{\displaystyle \frac{3\delta plog\left(d\right)}{\Omega }}(-\sqrt{-{\displaystyle \frac{p}{r}}}{coth}_B\left(\sqrt{-pr}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right))]\hfill \\ \hfill \times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right).\end{array}$$

(98)

$$\begin{array}{c}g(y,t)=[{\alpha }_0-{\displaystyle \frac{3\delta rlog\left(d\right)}{\Omega }}(-\sqrt{-{\displaystyle \frac{p}{r}}}({tanh}_B\left(2\sqrt{-pr}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right)\pm \hfill \\ \left(\iota \sqrt{cf}sec{h}_B\left(2\sqrt{-pr}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right)\right)\left)\right)]\times \\ \hfill exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right).\end{array}$$

(99)

$$\begin{array}{c}g(y,t)=[{\alpha }_0-{\displaystyle \frac{3\delta rlog\left(d\right)}{\Omega }}(-\sqrt{-{\displaystyle \frac{p}{r}}}({coth}_B\left(2\sqrt{-pr}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right)\pm \hfill \\ \left(\sqrt{cf}csc{h}_B\left(2\sqrt{-pr}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right)\right)\left)\right)]\times \\ \hfill exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right).\end{array}$$

(100)

$$\begin{array}{c}g(y,t)=[{\alpha }_0-{\displaystyle \frac{3\delta rlog\left(d\right)}{\Omega }}(-{\displaystyle \frac{1}{2}}\sqrt{-{\displaystyle \frac{p}{r}}}({tanh}_B\left({\displaystyle \frac{1}{2}}\sqrt{-pr}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right)\hfill \\ \hfill +{coth}_B\left({\displaystyle \frac{1}{2}}\sqrt{-pr}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right)\left)\right)]\times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4pr{\delta }^3{log}^2\left(d\right)t^{ϵ})\right).\end{array}$$

(101)

Case 5:

$$\begin{array}{c}g(y,t)=[{\alpha }_0-{\displaystyle \frac{3\delta plog\left(d\right)}{\Omega }}\left({tan}_B\left(p{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4p^2{\delta }^3{log}^2\left(d\right)t^{ϵ})\right)\right)]\hfill \\ \hfill \times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4p^2{\delta }^3{log}^2\left(d\right)t^{ϵ})\right).\end{array}$$

(102)

$$\begin{array}{c}g(y,t)=[{\alpha }_0-{\displaystyle \frac{3\delta plog\left(d\right)}{\Omega }}(-{cot}_B\left(p{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4p^2{\delta }^3{log}^2\left(d\right)t^{ϵ})\right))]\hfill \\ \hfill \times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4p^2{\delta }^3{log}^2\left(d\right)t^{ϵ})\right).\end{array}$$

(103)

$$\begin{array}{c}g(y,t)=[{\alpha }_0-{\displaystyle \frac{3\delta plog\left(d\right)}{\Omega }}({tan}_B\left(2p{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4p^2{\delta }^3{log}^2\left(d\right)t^{ϵ})\right)\pm \hfill \\ \hfill \left(\sqrt{cf}{sec}_B\left(2p{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4p^2{\delta }^3{log}^2\left(d\right)t^{ϵ})\right)\right)]\times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4p^2{\delta }^3{log}^2\left(d\right)t^{ϵ})\right).\end{array}$$

(104)

$$\begin{array}{c}g(y,t)=[{\alpha }_0-{\displaystyle \frac{3\delta plog\left(d\right)}{\Omega }}(-{cot}_B\left(2p{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4p^2{\delta }^3{log}^2\left(d\right)t^{ϵ})\right)\pm \hfill \\ \hfill \left(\sqrt{cf}{csc}_B\left(2p{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4p^2{\delta }^3{log}^2\left(d\right)t^{ϵ})\right)\right)]\times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4p^2{\delta }^3{log}^2\left(d\right)t^{ϵ})\right).\end{array}$$

(105)

$$\begin{array}{c}g(y,t)=[{\alpha }_0-{\displaystyle \frac{3\delta plog\left(d\right)}{\Omega }}({\displaystyle \frac{1}{2}}{tan}_B\left({\displaystyle \frac{1}{2}}p{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4p^2{\delta }^3{log}^2\left(d\right)t^{ϵ})\right)-\hfill \\ \hfill {\displaystyle \frac{1}{2}}{cot}_B\left({\displaystyle \frac{1}{2}}p{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4p^2{\delta }^3{log}^2\left(d\right)t^{ϵ})\right)\left)\right]\times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}+4p^2{\delta }^3{log}^2\left(d\right)t^{ϵ})\right).\end{array}$$

(106)

Case 11:

$$\begin{array}{c}g(y,t)=({\alpha }_0-{\displaystyle \frac{3\delta rlog\left(d\right)}{\Omega }}(-\left(cq\right)/(r({cosh}_B\left(q{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}-{\delta }^3{log}^2\left(d\right)q^2{t}^{ϵ})\right)-\hfill \\ \hfill {sinh}_B\left(q{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}-{\delta }^3{log}^2\left(d\right)q^2{t}^{ϵ})\right)+c\left)\right)\left)\right)\times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}-{\delta }^3{log}^2\left(d\right)q^2{t}^{ϵ})\right).\end{array}$$

(107)

$$\begin{array}{c}g(y,t)=[{\alpha }_0-{\displaystyle \frac{3\delta rlog\left(d\right)}{\Omega }}(-(q({cosh}_B\left(q{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}-{\delta }^3{log}^2\left(d\right)q^2{t}^{ϵ})\right)+\hfill \\ {sinh}_B\left(q{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}-{\delta }^3{log}^2\left(d\right)q^2{t}^{ϵ})\right)\left)\right)/\left(r\right({cosh}_B\left(q{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}\right(\delta y^{ϵ}-\\ {\delta }^3{log}^2\left(d\right)q^2{t}^{ϵ}\left)\right)+{sinh}_B\left(q{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}-{\delta }^3{log}^2\left(d\right)q^2{t}^{ϵ})\right)+f\left)\right)\left)\right]\\ \hfill \times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}-{\delta }^3{log}^2\left(d\right)q^2{t}^{ϵ})\right).\end{array}$$

(108)

Case 12:

$$\begin{array}{c}g(y,t)=[{\alpha }_0-{\displaystyle \frac{3\delta m\rho log\left(d\right)}{\Omega }}\left({\displaystyle \frac{c{d}^{\rho \frac{\Gamma (1+\varrho )}{ϵ}(\delta y^{ϵ}-{\delta }^3{log}^2\left(d\right){\rho }^2{t}^{ϵ})}}{c-mf{d}^{\rho \frac{\Gamma (1+\varrho )}{ϵ}(\delta y^{ϵ}-{\delta }^3{log}^2\left(d\right){\rho }^2{t}^{ϵ})}}}\right)]\hfill \\ \hfill \times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\delta y^{ϵ}-{\delta }^3{log}^2\left(d\right){\rho }^2{t}^{ϵ})\right).\end{array}$$

(109)

Note: Cases 6–10 result in zero solutions.

4. Improved $({\mathit{G}}^{\prime }/\mathit{G})$-Expansion Technique and Its Application

Here, we give the main steps of this technique [31].

Step 1: Assume a nonlinear fractional PDE

$$G(q,D_{M,t}^{\alpha ,\beta }q,q^2{q}_{\gamma },q_{\theta },q_{\theta \theta },q_{\gamma \gamma },q_{\gamma \theta },\dots )=0.$$

(110)

Step 2: Consider the following transformation:

$$q(\gamma ,\theta ,t)=Q\left(\eta \right),\eta =\gamma -\upsilon \theta +{\displaystyle \frac{\Gamma (\beta +1)}{\alpha }}\left(\kappa t^{\alpha }\right),$$

(111)

where $\nu$ and $\kappa$ represent the parameters. Inserting Equation (111) into Equation (110) yields the nonlinear ODE

$$H(Q,Q^2{Q}^{\prime },Q^{\prime },\dots )=0.$$

(112)

Step 3: Consider the solutions of Equation (112) given by

$$Q\left(\eta \right)=\sum _{j=0}^m{\alpha }_j{\left({\displaystyle \frac{G^{\prime }\left(\eta \right)}{G\left(\eta \right)}}\right)}^j,$$

(113)

In Equation (113), ${\alpha }_0$ and ${\alpha }_j,(j=1,2,3,\dots ,m)$ are undetermined. By using a homogenous balance scheme in Equation (112), we obtain m. A function $G=G\left(\eta \right)$ satisfies the equation

$$G{G}^{″}-{\kappa }_1{G}^2-{\kappa }_{2}G{G}^{\prime }-{\kappa }_3{\left(G^{\prime }\right)}^2=0,$$

(114)

where ${\kappa }_1$, ${\kappa }_2$, and ${\kappa }_3$ are constants.

Step 4:

Assume Equation (114) has solutions given by the following cases:

Case 1: If ${\kappa }_2\ne 0$ and $\pi ={\kappa }_2^2+4{\kappa }_1-4{\kappa }_1{\kappa }_3>0$, we have

$${\displaystyle \frac{G^{\prime }\left(\eta \right)}{G\left(\eta \right)}}={\displaystyle \frac{{\kappa }_2\sqrt{\pi }\left(C_{1}exp\left(\frac{1}{2}\eta \sqrt{\pi }\right)+C_{2}exp\left(\frac{1}{2}\eta \left(-\sqrt{\pi }\right)\right)\right)}{\left(2(1-{\kappa }_3)\right)\left(C_{1}exp\left(\frac{1}{2}\eta \sqrt{\pi }\right)-C_{2}exp\left(\frac{1}{2}\eta \left(-\sqrt{\pi }\right)\right)\right)}}+{\displaystyle \frac{{\kappa }_2}{2(1-{\kappa }_3)}}.$$

(115)

Case 2: If ${\kappa }_2\ne 0$ and $\pi ={\kappa }_2^2+4{\kappa }_1-4{\kappa }_1{\kappa }_3<0$, we have

$${\displaystyle \frac{G^{\prime }\left(\eta \right)}{G\left(\eta \right)}}={\displaystyle \frac{{\kappa }_2\sqrt{-\left(\pi \right)}(C_1\iota cos\left(\frac{1}{2}\eta \sqrt{-\left(\pi \right)}\right)-C_{2}sin\left(\frac{1}{2}\eta \sqrt{-\left(\pi \right)}\right))}{\left(2(1-{\kappa }_3)\right)(C_1\iota sin\left(\frac{1}{2}\eta \sqrt{-\left(\pi \right)}\right)+C_{2}cos\left(\frac{1}{2}\eta \sqrt{-\left(\pi \right)}\right))}}+{\displaystyle \frac{{\kappa }_2}{2(1-{\kappa }_3)}}.$$

(116)

Case 3: If ${\kappa }_2=0$ and ${\kappa }_1-{\kappa }_1{\kappa }_3\ge 0$, we have

$${\displaystyle \frac{G^{\prime }\left(\eta \right)}{G\left(\eta \right)}}={\displaystyle \frac{\sqrt{{\kappa }_1-{\kappa }_1{\kappa }_3}\left(C_{2}sin\left(\eta \sqrt{{\kappa }_1-{\kappa }_1{\kappa }_3}\right)+C_{1}cos\left(\eta \sqrt{{\kappa }_1-{\kappa }_1{\kappa }_3}\right)\right)}{(1-{\kappa }_3)\left(C_{1}sin\left(\eta \sqrt{{\kappa }_1-{\kappa }_1{\kappa }_3}\right)-C_{2}cos\left(\eta \sqrt{{\kappa }_1-{\kappa }_1{\kappa }_3}\right)\right)}}.$$

(117)

Case 4: If ${\kappa }_2=0$ and ${\kappa }_1-{\kappa }_1{\kappa }_3<0$, we have

$${\displaystyle \frac{G^{\prime }\left(\eta \right)}{G\left(\eta \right)}}={\displaystyle \frac{\sqrt{{\kappa }_1{\kappa }_3-{\kappa }_1}\left(C_1\iota cosh\left(\eta \sqrt{{\kappa }_1{\kappa }_3-{\kappa }_1}\right)-C_{2}sinh\left(\eta \sqrt{{\kappa }_1{\kappa }_3-{\kappa }_1}\right)\right)}{(1-{\kappa }_3)\left(C_1\iota sinh\left(\eta \sqrt{{\kappa }_1{\kappa }_3-{\kappa }_1}\right)-C_{2}cosh\left(\eta \sqrt{{\kappa }_1{\kappa }_3-{\kappa }_1}\right)\right)}},$$

(118)

where ${\kappa }_1$, ${\kappa }_2$, ${\kappa }_3$, $C_1$, and $C_2$ are constants.

Step 5:

Substitute Equation (113) and Equation (114) into Equation (112) and collect the coefficients of every order of $\left({\displaystyle \frac{G^{\prime }\left(\eta \right)}{G\left(\eta \right)}}\right)$. By setting each coefficient to zero, one can obtain a system of algebraic equations involving $\nu$, $\kappa$, ${\alpha }_j,(j=0,1,2,\dots ,m)$, and other parameters.

Step 6:

Use Mathematica software 13 to solve the obtained system.

Step 7:

Substitute the obtained results into Equation (113) to yield the soliton-type solutions of Equation (110).

Application

For $m=1$, Equation (113) reduces to

$$G\left(\zeta \right)={\alpha }_0+{\alpha }_1\left({\displaystyle \frac{G^{\prime }\left(\zeta \right)}{G\left(\zeta \right)}}\right),$$

(119)

where ${\alpha }_0$ and ${\alpha }_1$ are unknowns.

By substituting Equations (119) and (114) into Equation (5), we obtain the following solution set with the use of Mathematica software:

Set:

$$\left\{{\alpha }_0={\alpha }_0,{\alpha }_1=-{\displaystyle \frac{3\delta \left({\kappa }_3-1\right)}{\Omega }},\lambda ={\delta }^3{\kappa }_2^2+4{\delta }^3{\kappa }_1-4{\delta }^3{\kappa }_1{\kappa }_3\right\}.$$

(120)

Case 1:

$$\begin{array}{c}g(x,y,t)=[{\alpha }_0-{\displaystyle \frac{3\delta ({\kappa }_3-1)}{\Omega }}\left(\right({\kappa }_2\sqrt{\pi }(C_{1}exp({\displaystyle \frac{1}{2}}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\mu x^{ϵ}+\delta y^{ϵ}-({\delta }^3{\kappa }_2^2+4{\delta }^3{\kappa }_1-\hfill \\ 4{\delta }^3{\kappa }_1{\kappa }_3)t^{ϵ})\sqrt{\pi })+C_{2}exp\left({\displaystyle \frac{1}{2}}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\mu x^{ϵ}+\delta y^{ϵ}-({\delta }^3{\kappa }_2^2+4{\delta }^3{\kappa }_1-4{\delta }^3{\kappa }_1{\kappa }_3)t^{ϵ})(-\sqrt{\pi })\right)\left)\right)/\left(\right(2(1-\\ {\kappa }_3\left)\right)(C_{1}exp\left({\displaystyle \frac{1}{2}}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\mu x^{ϵ}+\delta y^{ϵ}-({\delta }^3{\kappa }_2^2+4{\delta }^3{\kappa }_1-4{\delta }^3{\kappa }_1{\kappa }_3)t^{ϵ})\sqrt{\pi }\right)-C_{2}exp\left({\displaystyle \frac{1}{2}}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}\right(\mu x^{ϵ}\\ +\delta y^{ϵ}-({\delta }^3{\kappa }_2^2+4{\delta }^3{\kappa }_1-4{\delta }^3{\kappa }_1{\kappa }_3)t^{ϵ})(-\sqrt{\pi })\left)\right))+{\displaystyle \frac{{\kappa }_2}{2(1-{\kappa }_3)}}\left)\right]\times exp(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\mu x^{ϵ}\\ \hfill +\delta y^{ϵ}-({\delta }^3{\kappa }_2^2+4{\delta }^3{\kappa }_1-4{\delta }^3{\kappa }_1{\kappa }_3)t^{ϵ}\left)\right).\end{array}$$

(121)

Case 2:

$$\begin{array}{c}g(x,y,t)=[{\alpha }_0-{\displaystyle \frac{3\delta ({\kappa }_3-1)}{\Omega }}\left(\right({\kappa }_2\sqrt{-\pi }(C_1\iota cos({\displaystyle \frac{1}{2}}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\mu x^{ϵ}+\delta y^{ϵ}-({\delta }^3{\kappa }_2^2+4{\delta }^3{\kappa }_1-\hfill \\ 4{\delta }^3{\kappa }_1{\kappa }_3)t^{ϵ})\sqrt{-\pi })-C_{2}sin\left({\displaystyle \frac{1}{2}}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\mu x^{ϵ}+\delta y^{ϵ}-({\delta }^3{\kappa }_2^2+4{\delta }^3{\kappa }_1-4{\delta }^3{\kappa }_1{\kappa }_3)t^{ϵ})\sqrt{-\pi }\right)\left)\right)/\left(\right(2(1-\\ {\kappa }_3\left)\right)(C_1\iota sin\left({\displaystyle \frac{1}{2}}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\mu x^{ϵ}+\delta y^{ϵ}-({\delta }^3{\kappa }_2^2+4{\delta }^3{\kappa }_1-4{\delta }^3{\kappa }_1{\kappa }_3)t^{ϵ})\sqrt{-\pi }\right)+C_{2}cos\left({\displaystyle \frac{1}{2}}{\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}\right(\mu x^{ϵ}\\ +\delta y^{ϵ}-({\delta }^3{\kappa }_2^2+4{\delta }^3{\kappa }_1-4{\delta }^3{\kappa }_1{\kappa }_3)t^{ϵ})\sqrt{-\pi }\left)\right))+{\displaystyle \frac{{\kappa }_2}{2(1-{\kappa }_3)}}\left)\right]\times exp(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\mu x^{ϵ}\\ \hfill +\delta y^{ϵ}-({\delta }^3{\kappa }_2^2+4{\delta }^3{\kappa }_1-4{\delta }^3{\kappa }_1{\kappa }_3)t^{ϵ}\left)\right).\end{array}$$

(122)

Case 3:

$$\begin{array}{c}g(x,y,t)=[{\alpha }_0-{\displaystyle \frac{3\delta ({\kappa }_3-1)}{\Omega }}\left(\right(\sqrt{{\kappa }_1-{\kappa }_1{\kappa }_3}(C_{2}sin({\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\mu x^{ϵ}+\delta y^{ϵ}-(4{\delta }^3{\kappa }_1-\hfill \\ 4{\delta }^3{\kappa }_1{\kappa }_3)t^{ϵ})\sqrt{{\kappa }_1(1-{\kappa }_3)})+C_{1}cos({\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\mu x^{ϵ}+\delta y^{ϵ}-(4{\delta }^3{\kappa }_1-4{\delta }^3{\kappa }_1{\kappa }_3)t^{ϵ})\\ \sqrt{{\kappa }_1(1-{\kappa }_3)}\left)\right))/((1-{\kappa }_3)(C_{1}sin({\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\mu x^{ϵ}+\delta y^{ϵ}-(4{\delta }^3{\kappa }_1-4{\delta }^3{\kappa }_1{\kappa }_3)t^{ϵ})\\ \sqrt{{\kappa }_1(1-{\kappa }_3)})-C_{2}cos\left({\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\mu x^{ϵ}+\delta y^{ϵ}-(4{\delta }^3{\kappa }_1-4{\delta }^3{\kappa }_1{\kappa }_3)t^{ϵ})\sqrt{{\kappa }_1(1-{\kappa }_3)}\right)\left)\right)\left)\right]\\ \hfill \times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\mu x^{ϵ}+\delta y^{ϵ}-(4{\delta }^3{\kappa }_1-4{\delta }^3{\kappa }_1{\kappa }_3)t^{ϵ})\right).\end{array}$$

(123)

Case 4:

$$\begin{array}{c}g(x,y,t)=[{\alpha }_0-{\displaystyle \frac{3\delta ({\kappa }_3-1)}{\Omega }}\left(\right(\sqrt{{\kappa }_1{\kappa }_3-{\kappa }_1}(C_1\iota cosh({\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\mu x^{ϵ}+\delta y^{ϵ}-(4{\delta }^3{\kappa }_1-\hfill \\ 4{\delta }^3{\kappa }_1{\kappa }_3)t^{ϵ})\sqrt{{\kappa }_1{\kappa }_3-{\kappa }_1})-C_{2}sinh({\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\mu x^{ϵ}+\delta y^{ϵ}-(4{\delta }^3{\kappa }_1-4{\delta }^3{\kappa }_1{\kappa }_3)t^{ϵ})\\ \sqrt{{\kappa }_1{\kappa }_3-{\kappa }_1}\left)\right))/((1-{\kappa }_3)(C_1\iota sinh({\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\mu x^{ϵ}+\delta y^{ϵ}-(4{\delta }^3{\kappa }_1-4{\delta }^3{\kappa }_1{\kappa }_3)t^{ϵ})\\ \sqrt{{\kappa }_1{\kappa }_3-{\kappa }_1})-C_{2}cosh\left({\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\mu x^{ϵ}+\delta y^{ϵ}-(4{\delta }^3{\kappa }_1-4{\delta }^3{\kappa }_1{\kappa }_3)t^{ϵ})\sqrt{{\kappa }_1{\kappa }_3-{\kappa }_1}\right)\left)\right)\left)\right]\\ \hfill \times exp\left(\iota {\displaystyle \frac{\Gamma (1+\varrho )}{ϵ}}(\mu x^{ϵ}+\delta y^{ϵ}-(4{\delta }^3{\kappa }_1-4{\delta }^3{\kappa }_1{\kappa }_3)t^{ϵ})\right).\end{array}$$

(124)

5. Physical Behavior of Solutions

This section presents the physical behavior of different solution types, like kink, bright, and singular types, generated using Mathematica software. Figure 1, Figure 2, Figure 3, and Figure 4 illustrate kink-type solutions. Figure 5 illustrates bright-type solutions. Figure 6 and Figure 7 illustrate singular-type solutions. The physical behavior of g(y,t) shown in Equation (50) is illustrated in three-dimensional (a), contour (b), and two-dimensional (c) graphs with $d=2; q=0.5; p=0.2; \varrho =1; r=0.5; \Omega =3; \delta =0.5; ϵ=0.9;$ and ${\alpha }_0=1.2$ in Figure 1. The physical behavior of g(y,t) shown in Equation (51) is illustrated in three-dimensional (a), contour (b), and two-dimensional (c) graphs with $d=3; q=0.5; p=0.2; \varrho =2; r=0.5; \Omega =0.2; \delta =0.5; ϵ=0.1;$ and ${\alpha }_0=1.2$ in Figure 5. The physical behavior of g(y,t) shown in Equation (52) is illustrated in three-dimensional (a), contour (b), and two-dimensional (c) graphs with $d=3; q=0.5; p=0.2; \varrho =1; r=1.5; \Omega =3; \delta =0.5; ϵ=0.7;$${\alpha }_0=-1.2; c=0.2;$ and $f=1.5$ in Figure 2. The physical behavior of g(y,t) shown in Equation (53) is illustrated in three-dimensional (a), contour (b), and two-dimensional (c) graphs with $d=5; q=0.5; p=0.2;\varrho =1;$$r=0.5;$$\Omega =1.3; \delta =0.5; ϵ=0.9; {\alpha }_0=-1.2; c=0.3;$ and $f=1.3$ in Figure 3. The physical behavior of g(y,t) shown in Equation (54) is illustrated in three-dimensional (a), contour (b), and two-dimensional (c) graphs with $d=3;$$q=0.6;p=0.3; \varrho =1; r=1.2; \Omega =0.3; \delta =-0.4; ϵ=0.1; {\alpha }_0=-0.2; c=0.3;$ and $f=1.3$ in Figure 4.

The physical behavior of g(y,t) shown in Equation (55) is illustrated in three-dimensional (a), contour (b), and two-dimensional (c) graphs with $d=5; q=0.5; p=0.1;\varrho =1; r=-0.7;\Omega =0.5;\delta =-0.3;ϵ=0.1;$ and ${\alpha }_0=-1.2$ in Figure 6. The physical behavior of g(y,t) shown in Equation (56) is illustrated in three-dimensional (a), contour (b), and two-dimensional (c) graphs with $d=3; q=0.7; p=0.2; \varrho =2; r=0.2; \Omega =0.3; \delta =0.5; ϵ=0.1;$ and ${\alpha }_0=1.2$ in Figure 7.

6. Stability Analysis

In this section, we explain the stability of the governing system. The analysis, performed using the Hamiltonian system, is tested on a few of the obtained solutions to show the stability of the corresponding system in different applications. Stability analysis has been performed for various models, such as those in [32,33]. Here, we study the stability of Equation (1). For this, we define the Hamiltonian transformation as

$$\begin{array}{c}\hfill \mathcal{M}={\displaystyle \frac{1}{2}}{\int }_{-\infty }^{\infty }{\mathit{g}}^{\mathit{2}}dy,\end{array}$$

(125)

where $\mathcal{M}$ represents the momentum factor and the possibility for power is represented by $\mathit{g}(\mathit{y},\mathit{t})$. Now, we give a necessary criterion for stable soliton solutions:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial \mathcal{M}}{\partial \lambda }}>0,\end{array}$$

(126)

where $\lambda$ represents a soliton velocity. Substituting Equation (50) into Equation (125) yields

$$\begin{array}{c}\mathcal{M}={\displaystyle \frac{1}{2}}{\int }_{-5}^5({\alpha }_0+{\displaystyle \frac{3\delta plog\left(d\right)}{\Omega }}(-{\displaystyle \frac{q}{2r}}+(\sqrt{-(q^2-4pr)}tan({\displaystyle \frac{1}{2}}\sqrt{-(q^2-4pr)}\hfill \\ \hfill (\delta y-{\delta }^3{log}^2\left(d\right)(q^2-4pr)t)\left)\right)/\left(2r\right){{)}^{-1})}^{2}dy,\end{array}$$

(127)

and using the criterion given in Equation (126), we obtain

$$\begin{array}{c}-(12\delta prlog\left(d\right)({\displaystyle \frac{qt\sqrt{4pr-q^2}{sec}^2\left(\frac{1}{2}\sqrt{4pr-q^2}({\delta }^3(-t){log}^2\left(d\right)(q^2-4pr)+5\delta )\right)}{2({tan}^2\left(\frac{1}{2}\sqrt{4pr-q^2}({\delta }^3(-t){log}^2\left(d\right)(q^2-4pr)+5\delta )\right)+1)}}\hfill \\ -{\displaystyle \frac{qt\sqrt{4pr-q^2}{sec}^2\left(\frac{1}{2}\sqrt{4pr-q^2}({\delta }^3(-t){log}^2\left(d\right)(q^2-4pr)-5\delta )\right)}{2({tan}^2\left(\frac{1}{2}\sqrt{4pr-q^2}({\delta }^3(-t){log}^2\left(d\right)(q^2-4pr)-5\delta )\right)+1)}}+\\ r\sqrt{4pr-q^2}(t\sqrt{4pr-q^2}tan\left({\displaystyle \frac{1}{2}}\sqrt{4pr-q^2}({\delta }^3(-t){log}^2\left(d\right)(q^2-4pr)+5\delta )\right)+\\ t(-\sqrt{4pr-q^2})tan\left({\displaystyle \frac{1}{2}}\sqrt{4pr-q^2}({\delta }^3(-t){log}^2\left(d\right)(q^2-4pr)-5\delta )\right)-\\ {\displaystyle \frac{2rt(4pr-q^2){sec}^2\left(\frac{1}{2}\sqrt{4pr-q^2}(-{\delta }^{3}t{log}^2\left(d\right)(q^2-4pr)+5\delta )\right)}{2r\sqrt{4pr-q^2}tan\left(\frac{1}{2}\sqrt{4pr-q^2}({\delta }^3(-t){log}^2\left(d\right)(q^2-4pr)+5\delta )\right)-q}}+\\ {\displaystyle \frac{2rt(4pr-q^2){sec}^2\left(\frac{1}{2}\sqrt{4pr-q^2}({\delta }^3(-t){log}^2\left(d\right)(q^2-4pr)-5\delta )\right)}{2r\sqrt{4pr-q^2}tan\left(\frac{1}{2}\sqrt{4pr-q^2}(-{\delta }^{3}t{log}^2\left(d\right)(q^2-4pr)-5\delta )\right)-q}}\left)\right))/(\mu \Omega \sqrt{4pr-q^2}\\ \hfill (16p{r}^3+q^2(1-4r^2)))>0.\end{array}$$

(128)

Hence, Equation (1) represents a stable nonlinear fractional equation, provided that the above condition is fulfilled.

7. Modulation Instability Analysis

Consider a steady-state solution of the (1+1)-dimensional nonlinear Estevez–Mansfield–Clarkson equation of the form [34]

$$g(y,t)=\left(G(y,t)+\sqrt{\tau }\right)e^{\iota \tau t},$$

(129)

where $\tau$ denotes the normalized optical power.

Substitute Equation (129) into Equation (1). After linearizing, we obtain

$$\iota (2\tau G^{(0,1)}+\tau G^{(3,0)})+G^{(0,2)}+G^{(3,1)}-{\tau }^{2}G-{\tau }^{5/2}=0.$$

(130)

Assume the solutions of Equation (130) are of the form

$$G(y,t)=a_1{e}^{\iota ({\theta }_{2}y-\omega t)}+a_2{e}^{-\iota ({\theta }_{2}y-\omega t)},$$

(131)

where $\omega$ and ${\theta }_2$ denote the frequency and normalized wave number of the perturbation, respectively. Substitute Equation (131) into Equation (130). By collecting the coefficients of $e^{\iota ({\theta }_{2}y-\omega t)}$ and $e^{-\iota ({\theta }_{2}y-\omega t)}$, we obtain the determinant of the coefficient matrix of $a_1$ and $a_2$.

$$\left|\begin{array}{cc}{\theta }_2^3\tau -{\theta }_2^3\omega -{\tau }^2+2\tau \omega -{\omega }^2& 0\\ 0& {\theta }_2^3(-\tau )-{\theta }_2^3\omega -{\tau }^2-2\tau \omega -{\omega }^2\end{array}\right|=0.$$

(132)

We obtain the dispersion relation after solving the above determinant of the coefficient matrix

$$\left({\theta }_2^3\tau -{\theta }_2^3\omega -{\tau }^2+2\tau \omega -{\omega }^2\right)\left({\theta }_2^3(-\tau )-{\theta }_2^3\omega -{\tau }^2-2\tau \omega -{\omega }^2\right)=0.$$

(133)

Determining the dispersion relation from Equation (133) for $\omega$ yields

$$\omega =-{\theta }_2^3\pm \sqrt{2{\theta }_2^6\tau -2{\theta }_2^6\tau -2{\theta }_2^{12}+{\theta }_2^{12}+{\theta }_2^{12}+{\tau }^2}.$$

(134)

The dispersion relation indicates the steady-state stability. When the wave number $\omega$ is imaginary, the steady-state solution will be unstable since the perturbation grows exponentially. But when the wave number $\omega$ has a real part, the steady state becomes stable against small perturbations. The steady-state solution is unstable if

$$2{\theta }_2^6\tau -2{\theta }_2^6\tau -2{\theta }_2^{12}+{\theta }_2^{12}+{\theta }_2^{12}+{\tau }^2<0.$$

(135)

The modulation instability (MI) gain spectrum (Figure 8) $G\left(\tau \right)$ is obtained as

$$G\left(\tau \right)=2Im\left(\omega \right)=\pm \sqrt{2{\theta }_2^6\tau -2{\theta }_2^{12}+{\theta }_2^{12}+{\theta }_2^{12}-2{\theta }_2^6\tau +{\tau }^2}.$$

(136)

8. Conclusions

We successfully obtained distinct kinds of exact wave solutions to the (1+1)-dimensional nonlinear Estevez–Mansfield–Clarkson (EMC) equation. We obtained different kinds of soliton solutions, including kink, bright, and singular, using the modified extended direct algebraic method and the improved $(G^{\prime }/G)$-expansion method. A novel definition of the fractional derivative, “the truncated M-fractional derivative”, is utilized for the first time for the equation under study. The obtained solutions are verified using Mathematica software by substituting the solutions back into the equation. Some obtained solutions do not exist. The results are helpful for describing waves in shallow water, fluid dynamics, and other naturally occurring phenomena. These solutions may also be useful in different fields of science and engineering. The results are represented as three-dimensional, contour, and two-dimensional plots. Stability analysis is also performed to ensure that the solutions are exact and accurate. Furthermore, modulation instability analysis is performed to study the stationary solutions of the considered model. The results will be helpful for future studies of the corresponding system. The methods utilized can be easily used to solve other nonlinear fractional partial differential equations.