Stability Analysis, Modulation Instability, and Beta-Time Fractional Exact Soliton Solutions to the Van der Waals Equation

Abstract

The study consists of the distinct types of the exact soliton solutions to an important model called the beta-time fractional (1 + 1)-dimensional non-linear Van der Waals equation. This model is used to explain the motion of molecules and materials. The Van der Waals equation explains the phase separation phenomenon. Noncovalent Van der Waals or dispersion forces usually have an effect on the structure, dynamics, stability, and function of molecules and materials in different branches of science, including biology, chemistry, materials science, and physics. Solutions are obtained, including dark, dark-singular, periodic wave, singular wave, and many more exact wave solutions by using the modified extended tanh function method. Using the fractional derivatives makes different solutions different from the existing solutions. The gained results will be of high importance in the interaction of quantum-mechanical fluctuations, granular matters, and other applications of the Van der Waals equation. The solutions may be useful in distinct fields of science and civil engineering, as well as some basic physical ones like those studied in geophysics. The results are verified and represented by two-dimensional, three-dimensional, and contour graphs by using Mathematica software. The obtained results are newer than the existing results. Stability analysis is also performed to check the stability of the concerned model. Furthermore, modulation instability is studied to study the stationary solutions of the concerned model. The results will be helpful in future studies of the concerned system. In the end, we can say that the method used is straightforward and dynamic, and it will be a useful tool for debating tough issues in a wide range of fields.

1. Introduction

Mathematical modeling has much importance in applied sciences and engineering. Many models in many branches of science and engineering have been developed, including the new generalized Bogoyavlensky–Konopelchenko model [1], the Oskolkov model [2], nonlinear fractional partial differential equations [3], fourth-order parabolic partial differential equations [4], the longitudinal wave equation [5], the Biswas–Arshed model [6], the Kundu–Mukherjee–Naskar model [7], the Single-Joint Robot Arm model [8], and the Landau–Ginzburg–Higgs equation [9]. Only a few years ago, the use of fractional derivatives in mathematical modeling played a significant role [10]. With the help of these nonlinear evolution equations (NLEEs), nonlinear science is utilized to explain various nonlinear phenomena and complex interaction behaviors in fluid mechanics, bioinformatics, traffic flow, and financial systems [11,12,13,14,15]. Nonlinear fractional partial differential equations are a more prominent way to represent any naturally occurring phenomenon, including the multimodal learning paradigm [16], multimodal hybrid parallel networks [17], the neural architecture method [18], the accurate automated extraction of coseismic deformation [19], and improved BPNN method [20]. Various mathematical models are represented in the form of involving different fractional derivatives, such as the conformable nonlinear coupled Schrödinger model [21], the beta fractional modified Benjamin Bona Mahony model [22], the Truncated M-fractional Westervelt model [23], and many more.

There are distinct techniques to obtain the exact soliton solutions, for instance, the generalized double auxiliary equation technique [24], the new Extended Direct Algebraic scheme [25], the Ricatti equation mapping technique [26], Paul Painleve’s technique [27], the improved Fan sub-equation approach [28], the enhanced algebraic method [29], the exponential rational function technique [30], the multiple exp-function scheme [31], the Hirota bilinear approach [32], the Jacobi elliptic function technique [33], the generalized exponential rational function approach [34], the modified Sardar sub-equation scheme [35], the improved modified extended tanh function technique [36], and the new extended hyperbolic function method [37].

A neural network identifier was built to estimate the unknown system dynamics, and a critic neural network was constructed to solve the Hamiltonian-Jacobi-Bellman equation associated with the optimal control problem [38]. The closed-form solitary wave solutions for the Ginzburg-Landau equation were obtained [39]. Two efficient techniques were employed to explore these solutions for the Chaffee-Infante equation [40]. The method integrates logarithmic transformations and the symbolic structures were used to obtain several trigonometric, hyperbolic, and rational expressions to nonlinear Zakharov system [41]. A generalization of the regularized long-wave equation was considered, and the existences of smooth soliton, peakon, and periodic solutions were established via the complete discrimination system for polynomial method and the bifurcation method [42]. The linear structure of Sharma-Tasso-Olver-Burgers equation was presented and a general form of the multi-wave solution was obtained [43]. A new closed-form design of wideband infinite impulse response digital integrators using numerical integration rules that were constructed via trigonometric interpolation [44].

Some of the nonlinear models examined using the mentioned methods in literature are as follows: a fast nonsingleton fuzzy predictive [45], the fuzzy neural network [46], the fractional-order fuzzy control [47], optimal deep learning control [48] and optical deposition method [49]. A notable models in this context is the nonlinear equations which offers a more precise interpretation of complex nonlinear models such as multimodal engineering optimization problems [50], deep convolutional neural networks [51], finger photoplethys-mography [52], multi-objects detection [53], and robust niching chimp optimization [54]. The nonlinear models belong to a class of nonlinear dynamical systems that have specific applications and implications in various fields and can exhibit rich dynamics, including chimp optimization method [55], improved chimp optimization method [56], an innovative algorithm for multi-objective problems [57], vision-language learning technique [58], virtual multiple quasi method [59].

Many approaches have been suggested by researchers in this context. Some of the methods that have attracted attention recently are as follows: network traffic detection model [60], generalized matrix completion [61], a robust observer method [62] quasi-Z-source inverter method [63], and fuzzy logic system [64].

In this paper, we used a simple and useful scheme: the modified extended tanh function scheme. The concerned scheme has been used for different models. For instance, the modified extended tanh function method is used for the Lakshmanan–Porsezian–Daniel equation [65], the coupled Klein–Gordon–Zakharov model [66], the Bogoyavlenskii model [67], the generalized Bugures–Fisher model [68], the Sharma–Tasso–Olveer equation [69], the complex Ginzburg-Landau equation [70], the phi-four equation [71], and the Sasa–Satsuma equation [72].

The basic aim of this paper is to explore new kinds of exact soliton solutions to the (1 + 1)-dimensional nonlinear Van der Waals equation along the beta-time fractional derivative. Additionally, stability analysis and modulation instability analysis are used to verify the stability and validity of the obtained solutions.

This paper has different sections: the governing equation and its mathematical analysis are given in Section 2, the modified extended tanh function scheme and its application are explained in Section 3, the graphic representation is shown in Section 4, the physical interpretation is given in Section 5, the stability analysis is given in Section 6, the modulation instability is explained in Section 7, and the conclusion is given in Section 8.

1.1. $\beta$-Time Derivative and Its Characteristics

Definition 1.

Consider $u(t)$ is defined for all non-negative t. Therefore, the beta-time fractional derivative of the function u of order θ is shown as follows [73]

$$\begin{array}{c}\hfill D^{\theta }(u(t))={\displaystyle \frac{d^{\theta }u(t)}{d{t}^{\theta }}}=\underset{ϵ\to 0}{lim}{\displaystyle \frac{u(t+ϵ{(t+\frac{1}{\Gamma (\theta )})}^{1-\theta })-u(t)}{ϵ}},0<\theta \le 1.\end{array}$$

Some of the main characteristics are given in [73,74,75].

1.2. Characteristics

Suppose $v(t)$ and $u(t)$ are the $\theta$—time differentiable functions ∀ t$>0$ and $\theta \in (0,1]$. Then

$$\begin{array}{c}\hfill i.D^{\theta }(av(t)+bu(t))=a{D}^{\theta }(v(t))+b{D}^{\theta }(u(t)),\forall a,b\in R.\end{array}$$

$$\begin{array}{c}\hfill ii.D^{\theta }(v(t)u(t))=u(t)D^{\theta }(v(t))+v(t)D^{\theta }(u(t)).\end{array}$$

$$\begin{array}{c}\hfill iii.D^{\theta }({\displaystyle \frac{v(t)}{u(t)}})={\displaystyle \frac{u(t)D^{\theta }(v(t))-v(t)D^{\theta }(u(t))}{{(u(t))}^2}}.\end{array}$$

$$\begin{array}{c}\hfill iv.D^{\theta }(v(t))={(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{1-\theta }{\displaystyle \frac{dv(t)}{d\theta }}.\end{array}$$

2. Model Representation and Mathematical Analysis

Consider a nonlinear (1 + 1)-dimensional Van der Waals equation given as [76]

$$u_{tt}+u_{xxxx}-\alpha u_{xxt}-{(u^3)}_{xx}-\beta u_{xx}=0,$$

(1)

where $u=u(x,t)$ is a wave profile depending on the spatial variable x and temporal variable t, while $\alpha$ and $\beta$ are the non-zero constants. $\alpha$ represents an effective viscosity, while $\beta$ denotes a bifurcation parameter. Equation (1) is linked with natural phenomena and industrial applications. This equation also plays an important role in geophysics, civil engineering, pharmaceuticals, etc. Different techniques are used for Equation (1) in the literature, including the extended $(G^{\prime }/G)$-expansion scheme [76], the generalized Kudryashov technique [77], the simplest equation scheme [78], the extended sinh-Gordon equation expansion technique [79], and many more.

Suppose the nonlinear (1 + 1)-dimensional Van der Waals equation in the beta-time fractional derivative is given as

$${\displaystyle \frac{{\partial }^{2\theta }u}{\partial t^{2\theta }}}+{\displaystyle \frac{{\partial }^{4\theta }u}{\partial x^{4\theta }}}-\alpha {\displaystyle \frac{{\partial }^{2\theta }}{\partial x^{2\theta }}}({\displaystyle \frac{{\partial }^{\theta }u}{\partial t^{\theta }}})-{\displaystyle \frac{{\partial }^{2\theta }}{\partial x^{2\theta }}}{u}^3-\beta {\displaystyle \frac{{\partial }^{2\theta }u}{\partial x^{2\theta }}}=0.$$

(2)

Assume a wave transformation is given as

$$u(x,t)=U(\zeta ),\zeta =(\mu {\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }-\omega {\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }),$$

(3)

where $U^{″}$ represents the profile function, which describes the shape of the wave, while $\mu$ and $\omega$ are the non-zero constants.

By using Equation (3) into Equation (2) and integrating twice with respect to $\zeta$, one gains

$$\left({\omega }^2-\beta {\mu }^2\right)U-{\mu }^2{U}^3+\alpha {\mu }^2\omega U^{\prime }+{\mu }^4{U}^{″}=0.$$

(4)

We obtain a value of m by applying the homogenous balance scheme to Equation (4) as follows:

By balancing the terms $U^{″}$ and $U^3$ we obtain

$$3m=m+2,\mathrm{so}m=1.$$

Next, we will explain the methods and find the exact wave solitons of Equation (4).

3. Description of METhF Technique

Here, some of the main steps of the method are given. Assuming a conformable fractional NLPDE:

$$\chi (v,v^2,\dots ,{\displaystyle \frac{{\partial }^{\alpha }v}{\partial t^{\alpha }}},{\displaystyle \frac{{\partial }^{2\alpha }v}{\partial t^{2\alpha }}},\dots ,{\displaystyle \frac{\partial v}{\partial x}},{\displaystyle \frac{{\partial }^{2}v}{\partial x^2}},\dots )=0,$$

(5)

where $v=v(x,t)$ shows a wave profile. Consider the following transformation

$$v(x,t)=V(\zeta ),\zeta =\mu (x-\nu {\displaystyle \frac{t^{\alpha }}{\alpha }}),$$

(6)

where $\nu$ denotes a soliton speed. Using Equation (6) in Equation (5), yields

$$\chi (V,V^2,\mu V^{\prime },\mu {\nu }^2{V}^{″},\dots )=0.$$

(7)

Consider the solution of Equation (7) is shown as

$$V(\zeta )=a_0+\sum _{j=1}^m{a}_j{\phi }^j(\zeta )+\sum _{j=1}^m{b}_j{\phi }^{-j}(\zeta ),$$

(8)

where $a_0,a_j,b_j,(j=1,2,3,\dots ,m)$ are undetermined.

$\phi (\zeta )$ fulfills the following:

$${\phi }^{\prime }(\zeta )=\delta +{\phi }^2(\zeta ),$$

(9)

where $\delta$ is a constant, and the solution of Equation (9) is shown in [80]:

If $\delta$ is negative, then we have

$$\phi (\zeta )=-\sqrt{-\delta }tanh(\sqrt{-\delta }\zeta ),$$

(10)

or

$$\phi (\zeta )=-\sqrt{-\delta }coth(\sqrt{-\delta }\zeta ).$$

(11)

If $\delta$ is zero, then we have

$$\phi (\zeta )={\displaystyle \frac{-1}{\zeta }}.$$

(12)

If $\delta$ is positive, then we get

$$\phi (\zeta )=\sqrt{\delta }tan(\sqrt{\delta }\zeta ),$$

(13)

or

$$\phi (\zeta )=-\sqrt{\delta }cot(\sqrt{\delta }\zeta ).$$

(14)

Put Equations (8) and (9) into Equation (7). Collecting the co-efficient of $\phi (\zeta )$ of every power and putting it to zero, one obtains sets of equations including $a_0,a_j,b_j(j=1,2,3,\dots ,m)$, and $\delta$. By manumitting the obtained equations, we obtain the results for Equation (5).

Application to the METhF Technique

For $m=1$, the Equation (8) reduces to

$$U(\zeta )=a_0+a_1\phi (\zeta )+b_1\phi {(\zeta )}^{-1},$$

(15)

Applying Equation (15) into Equation (4) along with Equation (9), one obtains a system containing $a_0$, $a_1$, $b_1$, $\omega$ and $\delta$. Solving the system yields the following sets.

Set 1:

$$\{a_0=\mp {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}},a_1=0,b_1=\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta \delta }}{\sqrt{2}(2{\alpha }^2-9)}},\omega ={\displaystyle \frac{3\alpha \beta }{2(2{\alpha }^2-9)\sqrt{-\delta }}},\mu ={\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(8{\alpha }^2-36)\delta }}}\}.$$

(16)

Case 1: For $\delta <0$:

$$\begin{array}{c}u(x,t)=\mp {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta \delta }}{\sqrt{2}(2{\alpha }^2-9)}}(-\sqrt{-\delta }tanh(\sqrt{-\delta }({\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(8{\alpha }^2-36)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ \hfill -{\displaystyle \frac{3\alpha \beta }{2(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }{)))}^{-1},\end{array}$$

(17)

or

$$\begin{array}{c}u(x,t)=\mp {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta \delta }}{\sqrt{2}(2{\alpha }^2-9)}}(-\sqrt{-\delta }coth(\sqrt{-\delta }({\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(8{\alpha }^2-36)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ \hfill -{\displaystyle \frac{3\alpha \beta }{2(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }{)))}^{-1}.\end{array}$$

(18)

Case 3: For $\delta >0$:

$$\begin{array}{c}u(x,t)=\mp {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta \delta }}{\sqrt{2}(2{\alpha }^2-9)}}(\sqrt{\delta }tan(\sqrt{\delta }({\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(8{\alpha }^2-36)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ \hfill -{\displaystyle \frac{3\alpha \beta }{2(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }{)))}^{-1},\end{array}$$

(19)

or

$$\begin{array}{c}u(x,t)=\mp {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta \delta }}{\sqrt{2}(2{\alpha }^2-9)}}(-\sqrt{\delta }cot(\sqrt{\delta }({\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(8{\alpha }^2-36)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ \hfill -{\displaystyle \frac{3\alpha \beta }{2(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }{)))}^{-1}.\end{array}$$

(20)

Set 2:

$$\{a_0=\pm {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}},a_1=\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta }}{(2{\alpha }^2-9)\sqrt{2\delta }}},b_1=0,\omega ={\displaystyle \frac{3\alpha \beta }{2(2{\alpha }^2-9)\sqrt{-\delta }}},\mu ={\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(8{\alpha }^2-36)\delta }}}\}.$$

(21)

Case 1: For $\delta <0$:

$$\begin{array}{c}u(x,t)=\pm {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta }}{(2{\alpha }^2-9)\sqrt{2\delta }}}(-\sqrt{-\delta }tanh(\sqrt{-\delta }({\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(8{\alpha }^2-36)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ \hfill -{\displaystyle \frac{3\alpha \beta }{2(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }))),\end{array}$$

(22)

or

$$\begin{array}{c}u(x,t)=\pm {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta }}{(2{\alpha }^2-9)\sqrt{2\delta }}}(-\sqrt{-\delta }coth(\sqrt{-\delta }({\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(8{\alpha }^2-36)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ \hfill -{\displaystyle \frac{3\alpha \beta }{2(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }))).\end{array}$$

(23)

Case 3: For $\delta >0$:

$$\begin{array}{c}u(x,t)=\pm {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta }}{(2{\alpha }^2-9)\sqrt{2\delta }}}(\sqrt{\delta }tan(\sqrt{\delta }({\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(8{\alpha }^2-36)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ \hfill -{\displaystyle \frac{3\alpha \beta }{2(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }))),\end{array}$$

(24)

or

$$\begin{array}{c}u(x,t)=\pm {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta }}{(2{\alpha }^2-9)\sqrt{2\delta }}}(-\sqrt{\delta }cot(\sqrt{\delta }({\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(8{\alpha }^2-36)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ \hfill -{\displaystyle \frac{3\alpha \beta }{2(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }))).\end{array}$$

(25)

Set 3:

$$\begin{array}{c}\{a_0=\pm {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}},a_1=0,b_1=\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta \delta }}{\sqrt{2}(2{\alpha }^2-9)}},\omega =-{\displaystyle \frac{3\alpha \beta }{2(2{\alpha }^2-9)\sqrt{-\delta }}},\hfill \\ \hfill \mu ={\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(8{\alpha }^2-36)\delta }}}\}.\end{array}$$

(26)

Case 1: For $\delta <0$:

$$\begin{array}{c}u(x,t)=\pm {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta \delta }}{\sqrt{2}(2{\alpha }^2-9)}}(-\sqrt{-\delta }tanh(\sqrt{-\delta }({\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(8{\alpha }^2-36)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ \hfill +{\displaystyle \frac{3\alpha \beta }{2(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }{)))}^{-1},\end{array}$$

(27)

or

$$\begin{array}{c}u(x,t)=\pm {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta \delta }}{\sqrt{2}(2{\alpha }^2-9)}}(-\sqrt{-\delta }coth(\sqrt{-\delta }({\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(8{\alpha }^2-36)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ \hfill +{\displaystyle \frac{3\alpha \beta }{2(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }{)))}^{-1}.\end{array}$$

(28)

Case 3: For $\delta >0$:

$$\begin{array}{c}u(x,t)=\pm {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta \delta }}{\sqrt{2}(2{\alpha }^2-9)}}(\sqrt{\delta }tan(\sqrt{\delta }({\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(8{\alpha }^2-36)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ \hfill +{\displaystyle \frac{3\alpha \beta }{2(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }{)))}^{-1},\end{array}$$

(29)

or

$$\begin{array}{c}u(x,t)=\pm {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta \delta }}{\sqrt{2}(2{\alpha }^2-9)}}(-\sqrt{\delta }cot(\sqrt{\delta }({\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(8{\alpha }^2-36)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ \hfill +{\displaystyle \frac{3\alpha \beta }{2(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }{)))}^{-1}.\end{array}$$

(30)

Set 4:

$$\begin{array}{c}\{a_0=\mp {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}},a_1=\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta }}{(2{\alpha }^2-9)\sqrt{2\delta }}},b_1=0,\omega =-{\displaystyle \frac{3\alpha \beta }{2(2{\alpha }^2-9)\sqrt{-\delta }}},\hfill \\ \hfill \mu ={\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(8{\alpha }^2-36)\delta }}}\}.\end{array}$$

(31)

Case 1: For $\delta <0$:

$$\begin{array}{c}u(x,t)=\mp {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta }}{(2{\alpha }^2-9)\sqrt{2\delta }}}(-\sqrt{-\delta }tanh(\sqrt{-\delta }({\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(8{\alpha }^2-36)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ \hfill +{\displaystyle \frac{3\alpha \beta }{2(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }))),\end{array}$$

(32)

or

$$\begin{array}{c}u(x,t)=\mp {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta }}{(2{\alpha }^2-9)\sqrt{2\delta }}}(-\sqrt{-\delta }coth(\sqrt{-\delta }({\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(8{\alpha }^2-36)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ \hfill +{\displaystyle \frac{3\alpha \beta }{2(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }))).\end{array}$$

(33)

Case 3: For $\delta >$:

$$\begin{array}{c}u(x,t)=\mp {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta }}{(2{\alpha }^2-9)\sqrt{2\delta }}}(\sqrt{\delta }tan(\sqrt{\delta }({\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(8{\alpha }^2-36)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ \hfill +{\displaystyle \frac{3\alpha \beta }{2(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }))),\end{array}$$

(34)

or

$$\begin{array}{c}u(x,t)=\mp {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta }}{(2{\alpha }^2-9)\sqrt{2\delta }}}(-\sqrt{\delta }cot(\sqrt{\delta }({\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(8{\alpha }^2-36)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ \hfill +{\displaystyle \frac{3\alpha \beta }{2(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }))).\end{array}$$

(35)

Set 5:

$$\begin{array}{c}\{a_0=\mp {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}},a_1=0,b_1=\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta \delta }}{\sqrt{2}(2{\alpha }^2-9)}},\omega ={\displaystyle \frac{3\alpha \beta }{2(2{\alpha }^2-9)\sqrt{-\delta }}},\hfill \\ \hfill \mu =-{\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(8{\alpha }^2-36)\delta }}}\}.\end{array}$$

(36)

Case 1: For $\delta <0$:

$$\begin{array}{c}u(x,t)=\mp {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta \delta }}{\sqrt{2}(2{\alpha }^2-9)}}(-\sqrt{-\delta }tanh(\sqrt{-\delta }(-{\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(8{\alpha }^2-36)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ \hfill -{\displaystyle \frac{3\alpha \beta }{2(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }{)))}^{-1},\end{array}$$

(37)

or

$$\begin{array}{c}u(x,t)=\mp {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta \delta }}{\sqrt{2}(2{\alpha }^2-9)}}(-\sqrt{-\delta }coth(\sqrt{-\delta }(-{\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(8{\alpha }^2-36)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ \hfill -{\displaystyle \frac{3\alpha \beta }{2(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }{)))}^{-1}.\end{array}$$

(38)

Case 3: For $\delta >0$:

$$\begin{array}{c}u(x,t)=\mp {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta \delta }}{\sqrt{2}(2{\alpha }^2-9)}}(\sqrt{\delta }tan(\sqrt{\delta }(-{\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(8{\alpha }^2-36)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ \hfill -{\displaystyle \frac{3\alpha \beta }{2(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }{)))}^{-1},\end{array}$$

(39)

or

$$\begin{array}{c}u(x,t)=\mp {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta \delta }}{\sqrt{2}(2{\alpha }^2-9)}}(-\sqrt{\delta }cot(\sqrt{\delta }(-{\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(8{\alpha }^2-36)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ \hfill -{\displaystyle \frac{3\alpha \beta }{2(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }{)))}^{-1}.\end{array}$$

(40)

Set 6:

$$\begin{array}{c}\{a_0=\pm {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}},a_1=\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta }}{(2{\alpha }^2-9)\sqrt{2\delta }}},b_1=0,\omega ={\displaystyle \frac{3\alpha \beta }{2(2{\alpha }^2-9)\sqrt{-\delta }}},\hfill \\ \hfill \mu =-{\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(8{\alpha }^2-36)\delta }}}\}.\end{array}$$

(41)

Case 1: For $\delta <0$:

$$\begin{array}{c}u(x,t)=\pm {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta }}{(2{\alpha }^2-9)\sqrt{2\delta }}}(-\sqrt{-\delta }tanh(\sqrt{-\delta }(-{\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(8{\alpha }^2-36)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ \hfill -{\displaystyle \frac{3\alpha \beta }{2(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }))),\end{array}$$

(42)

or

$$\begin{array}{c}u(x,t)=\pm {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta }}{(2{\alpha }^2-9)\sqrt{2\delta }}}(-\sqrt{-\delta }coth(\sqrt{-\delta }(-{\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(8{\alpha }^2-36)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ \hfill -{\displaystyle \frac{3\alpha \beta }{2(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }))).\end{array}$$

(43)

Case 3: For $\delta >0$:

$$\begin{array}{c}u(x,t)=\pm {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta }}{(2{\alpha }^2-9)\sqrt{2\delta }}}(\sqrt{\delta }tan(\sqrt{\delta }(-{\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(8{\alpha }^2-36)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ \hfill -{\displaystyle \frac{3\alpha \beta }{2(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }))),\end{array}$$

(44)

or

$$\begin{array}{c}u(x,t)=\pm {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta }}{(2{\alpha }^2-9)\sqrt{2\delta }}}(-\sqrt{\delta }cot(\sqrt{\delta }(-{\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(8{\alpha }^2-36)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ \hfill -{\displaystyle \frac{3\alpha \beta }{2(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }))).\end{array}$$

(45)

Set 7:

$$\begin{array}{c}\{a_0=\pm {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}},a_1=0,b_1=\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta \delta }}{\sqrt{2}(2{\alpha }^2-9)}},\omega =-{\displaystyle \frac{3\alpha \beta }{2(2{\alpha }^2-9)\sqrt{-\delta }}},\hfill \\ \hfill \mu =-{\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(8{\alpha }^2-36)\delta }}}\}.\end{array}$$

(46)

Case 1: For $\delta <0$:

$$\begin{array}{c}u(x,t)=\pm {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta \delta }}{\sqrt{2}(2{\alpha }^2-9)}}(-\sqrt{-\delta }tanh(\sqrt{-\delta }(-{\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(8{\alpha }^2-36)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ \hfill +{\displaystyle \frac{3\alpha \beta }{2(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }{)))}^{-1},\end{array}$$

(47)

or

$$\begin{array}{c}u(x,t)=\pm {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta \delta }}{\sqrt{2}(2{\alpha }^2-9)}}(-\sqrt{-\delta }coth(\sqrt{-\delta }(-{\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(8{\alpha }^2-36)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ \hfill +{\displaystyle \frac{3\alpha \beta }{2(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }{)))}^{-1}.\end{array}$$

(48)

Case 3: For $\delta >0$:

$$\begin{array}{c}u(x,t)=\pm {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta \delta }}{\sqrt{2}(2{\alpha }^2-9)}}(\sqrt{\delta }tan(\sqrt{\delta }(-{\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(8{\alpha }^2-36)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ \hfill +{\displaystyle \frac{3\alpha \beta }{2(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }{)))}^{-1},\end{array}$$

(49)

or

$$\begin{array}{c}u(x,t)=\pm {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta \delta }}{\sqrt{2}(2{\alpha }^2-9)}}(-\sqrt{\delta }cot(\sqrt{\delta }(-{\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(8{\alpha }^2-36)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ \hfill +{\displaystyle \frac{3\alpha \beta }{2(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }{)))}^{-1}.\end{array}$$

(50)

Set 8:

$$\begin{array}{c}\{a_0=\mp {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}},a_1=\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta }}{(2{\alpha }^2-9)\sqrt{2\delta }}},b_1=0,\omega =-{\displaystyle \frac{3\alpha \beta }{2(2{\alpha }^2-9)\sqrt{-\delta }}},\hfill \\ \hfill \mu =-{\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(8{\alpha }^2-36)\delta }}}\}.\end{array}$$

(51)

Case 1: For $\delta <0$:

$$\begin{array}{c}u(x,t)=\mp {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta }}{(2{\alpha }^2-9)\sqrt{2\delta }}}(-\sqrt{-\delta }tanh(\sqrt{-\delta }(-{\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(8{\alpha }^2-36)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ \hfill +{\displaystyle \frac{3\alpha \beta }{2(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }))),\end{array}$$

(52)

or

$$\begin{array}{c}u(x,t)=\mp {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta }}{(2{\alpha }^2-9)\sqrt{2\delta }}}(-\sqrt{-\delta }coth(\sqrt{-\delta }(-{\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(8{\alpha }^2-36)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ \hfill +{\displaystyle \frac{3\alpha \beta }{2(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }))).\end{array}$$

(53)

Case 3: For $\delta >0$:

$$\begin{array}{c}u(x,t)=\mp {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta }}{(2{\alpha }^2-9)\sqrt{2\delta }}}(\sqrt{\delta }tan(\sqrt{\delta }(-{\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(8{\alpha }^2-36)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ \hfill +{\displaystyle \frac{3\alpha \beta }{2(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }))),\end{array}$$

(54)

or

$$\begin{array}{c}u(x,t)=\mp {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta }}{(2{\alpha }^2-9)\sqrt{2\delta }}}(-\sqrt{\delta }cot(\sqrt{\delta }(-{\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(8{\alpha }^2-36)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ \hfill +{\displaystyle \frac{3\alpha \beta }{2(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }))).\end{array}$$

(55)

Set 9:

$$\begin{array}{c}\{a_0=\mp {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}},a_1=\mp {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta }}{2(2{\alpha }^2-9)\sqrt{2\delta }}},b_1=\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta \delta }}{2\sqrt{2}(2{\alpha }^2-9)}},\omega ={\displaystyle \frac{3\alpha \beta }{4(2{\alpha }^2-9)\sqrt{-\delta }}},\hfill \\ \hfill \mu ={\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(32{\alpha }^2-144)\delta }}}\}.\end{array}$$

(56)

Case 1: For $\delta <0$:

$$\begin{array}{c}u(x,t)=\mp {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\mp {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta }}{2(2{\alpha }^2-9)\sqrt{2\delta }}}(-\sqrt{-\delta }tanh(\sqrt{-\delta }({\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(32{\alpha }^2-144)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ -{\displaystyle \frac{3\alpha \beta }{4(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta })))\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta \delta }}{2\sqrt{2}(2{\alpha }^2-9)}}(-\sqrt{-\delta }tanh(\sqrt{-\delta }({\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(32{\alpha }^2-144)\delta }}}\\ \hfill {\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }-{\displaystyle \frac{3\alpha \beta }{4(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }{)))}^{-1},\end{array}$$

(57)

or

$$\begin{array}{c}u(x,t)=\mp {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\mp {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta }}{2(2{\alpha }^2-9)\sqrt{2\delta }}}(-\sqrt{-\delta }coth(\sqrt{-\delta }({\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(32{\alpha }^2-144)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ -{\displaystyle \frac{3\alpha \beta }{4(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta })))\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta \delta }}{2\sqrt{2}(2{\alpha }^2-9)}}(-\sqrt{-\delta }coth(\sqrt{-\delta }({\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(32{\alpha }^2-144)\delta }}}\\ \hfill {\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }-{\displaystyle \frac{3\alpha \beta }{4(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }{)))}^{-1}.\end{array}$$

(58)

Case 3: For $\delta >0$:

$$\begin{array}{c}u(x,t)=\mp {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\mp {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta }}{2(2{\alpha }^2-9)\sqrt{2\delta }}}(\sqrt{\delta }tan(\sqrt{\delta }({\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(32{\alpha }^2-144)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ -{\displaystyle \frac{3\alpha \beta }{4(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta })))\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta \delta }}{2\sqrt{2}(2{\alpha }^2-9)}}(\sqrt{\delta }tan(\sqrt{\delta }({\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(32{\alpha }^2-144)\delta }}}\\ \hfill {\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }-{\displaystyle \frac{3\alpha \beta }{4(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }{)))}^{-1},\end{array}$$

(59)

or

$$\begin{array}{c}u(x,t)=\mp {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\mp {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta }}{2(2{\alpha }^2-9)\sqrt{2\delta }}}(-\sqrt{\delta }cot(\sqrt{\delta }({\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(32{\alpha }^2-144)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ -{\displaystyle \frac{3\alpha \beta }{4(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta })))\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta \delta }}{2\sqrt{2}(2{\alpha }^2-9)}}(-\sqrt{\delta }cot(\sqrt{\delta }({\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(32{\alpha }^2-144)\delta }}}\\ \hfill {\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }-{\displaystyle \frac{3\alpha \beta }{4(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }{)))}^{-1}.\end{array}$$

(60)

Set 10:

$$\begin{array}{c}\{a_0=\mp {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}},a_1=\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta }}{2(2{\alpha }^2-9)\sqrt{2\delta }}},b_1=\mp {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta \delta }}{2\sqrt{2}(2{\alpha }^2-9)}},\omega =-{\displaystyle \frac{3\alpha \beta }{4(2{\alpha }^2-9)\sqrt{-\delta }}},\hfill \\ \hfill \mu ={\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(32{\alpha }^2-144)\delta }}}\}.\end{array}$$

(61)

Case 1: For $\delta <0$:

$$\begin{array}{c}u(x,t)=\mp {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta }}{2(2{\alpha }^2-9)\sqrt{2\delta }}}(-\sqrt{-\delta }tanh(\sqrt{-\delta }({\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(32{\alpha }^2-144)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ +{\displaystyle \frac{3\alpha \beta }{4(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta })))\mp {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta \delta }}{2\sqrt{2}(2{\alpha }^2-9)}}(-\sqrt{-\delta }tanh(\sqrt{-\delta }({\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(32{\alpha }^2-144)\delta }}}\\ \hfill {\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }+{\displaystyle \frac{3\alpha \beta }{4(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }{)))}^{-1},\end{array}$$

(62)

or

$$\begin{array}{c}u(x,t)=\mp {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta }}{2(2{\alpha }^2-9)\sqrt{2\delta }}}(-\sqrt{-\delta }coth(\sqrt{-\delta }({\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(32{\alpha }^2-144)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ +{\displaystyle \frac{3\alpha \beta }{4(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta })))\mp {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta \delta }}{2\sqrt{2}(2{\alpha }^2-9)}}(-\sqrt{-\delta }coth(\sqrt{-\delta }({\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(32{\alpha }^2-144)\delta }}}\\ \hfill {\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }+{\displaystyle \frac{3\alpha \beta }{4(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }{)))}^{-1}.\end{array}$$

(63)

Case 3: For $\delta >0$:

$$\begin{array}{c}u(x,t)=\mp {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta }}{2(2{\alpha }^2-9)\sqrt{2\delta }}}(\sqrt{\delta }tan(\sqrt{\delta }({\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(32{\alpha }^2-144)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ +{\displaystyle \frac{3\alpha \beta }{4(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta })))\mp {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta \delta }}{2\sqrt{2}(2{\alpha }^2-9)}}(\sqrt{\delta }tan(\sqrt{\delta }({\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(32{\alpha }^2-144)\delta }}}\\ \hfill {\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }+{\displaystyle \frac{3\alpha \beta }{4(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }{)))}^{-1},\end{array}$$

(64)

or

$$\begin{array}{c}u(x,t)=\mp {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta }}{2(2{\alpha }^2-9)\sqrt{2\delta }}}(-\sqrt{\delta }cot(\sqrt{\delta }({\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(32{\alpha }^2-144)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ +{\displaystyle \frac{3\alpha \beta }{4(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta })))\mp {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta \delta }}{2\sqrt{2}(2{\alpha }^2-9)}}(-\sqrt{\delta }cot(\sqrt{\delta }({\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(32{\alpha }^2-144)\delta }}}\\ \hfill {\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }+{\displaystyle \frac{3\alpha \beta }{4(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }{)))}^{-1}.\end{array}$$

(65)

Set 11:

$$\begin{array}{c}\{a_0=\mp {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}},a_1=\mp {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta }}{2(2{\alpha }^2-9)\sqrt{2\delta }}},b_1=\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta \delta }}{2\sqrt{2}(2{\alpha }^2-9)}},\omega ={\displaystyle \frac{3\alpha \beta }{4(2{\alpha }^2-9)\sqrt{-\delta }}},\hfill \\ \hfill \mu =-{\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(32{\alpha }^2-144)\delta }}}\}.\end{array}$$

(66)

Case 1: For $\delta <0$:

$$\begin{array}{c}u(x,t)=\mp {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\mp {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta }}{2(2{\alpha }^2-9)\sqrt{2\delta }}}(-\sqrt{-\delta }tanh(\sqrt{-\delta }(-{\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(32{\alpha }^2-144)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ -{\displaystyle \frac{3\alpha \beta }{4(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta })))\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta \delta }}{2\sqrt{2}(2{\alpha }^2-9)}}(-\sqrt{-\delta }tanh(\sqrt{-\delta }(-{\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(32{\alpha }^2-144)\delta }}}\\ \hfill {\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }-{\displaystyle \frac{3\alpha \beta }{4(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }{)))}^{-1},\end{array}$$

(67)

or

$$\begin{array}{c}u(x,t)=\mp {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\mp {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta }}{2(2{\alpha }^2-9)\sqrt{2\delta }}}(-\sqrt{-\delta }coth(\sqrt{-\delta }(-{\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(32{\alpha }^2-144)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ -{\displaystyle \frac{3\alpha \beta }{4(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta })))\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta \delta }}{2\sqrt{2}(2{\alpha }^2-9)}}(-\sqrt{-\delta }coth(\sqrt{-\delta }(-{\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(32{\alpha }^2-144)\delta }}}\\ \hfill {\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }-{\displaystyle \frac{3\alpha \beta }{4(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }{)))}^{-1}.\end{array}$$

(68)

Case 3: For $\delta >0$:

$$\begin{array}{c}u(x,t)=\mp {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\mp {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta }}{2(2{\alpha }^2-9)\sqrt{2\delta }}}(\sqrt{\delta }tan(\sqrt{\delta }(-{\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(32{\alpha }^2-144)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ -{\displaystyle \frac{3\alpha \beta }{4(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta })))\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta \delta }}{2\sqrt{2}(2{\alpha }^2-9)}}(\sqrt{\delta }tan(\sqrt{\delta }(-{\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(32{\alpha }^2-144)\delta }}}\\ \hfill {\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }-{\displaystyle \frac{3\alpha \beta }{4(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }{)))}^{-1},\end{array}$$

(69)

or

$$\begin{array}{c}u(x,t)=\mp {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\mp {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta }}{2(2{\alpha }^2-9)\sqrt{2\delta }}}(-\sqrt{\delta }cot(\sqrt{\delta }(-{\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(32{\alpha }^2-144)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ -{\displaystyle \frac{3\alpha \beta }{4(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta })))\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta \delta }}{2\sqrt{2}(2{\alpha }^2-9)}}(-\sqrt{\delta }cot(\sqrt{\delta }(-{\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(32{\alpha }^2-144)\delta }}}\\ \hfill {\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }-{\displaystyle \frac{3\alpha \beta }{4(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }{)))}^{-1}.\end{array}$$

(70)

Set 12:

$$\begin{array}{c}\{a_0=\mp {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}},a_1=\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta }}{2(2{\alpha }^2-9)\sqrt{2\delta }}},b_1=\mp {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta \delta }}{2\sqrt{2}(2{\alpha }^2-9)}},\omega =-{\displaystyle \frac{3\alpha \beta }{4(2{\alpha }^2-9)\sqrt{-\delta }}},\hfill \\ \hfill \mu =-{\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(32{\alpha }^2-144)\delta }}}\}.\end{array}$$

(71)

Case 1: For $\delta <0$:

$$\begin{array}{c}u(x,t)=\mp {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta }}{2(2{\alpha }^2-9)\sqrt{2\delta }}}(-\sqrt{-\delta }tanh(\sqrt{-\delta }(-{\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(32{\alpha }^2-144)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ +{\displaystyle \frac{3\alpha \beta }{4(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta })))\mp {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta \delta }}{2\sqrt{2}(2{\alpha }^2-9)}}(-\sqrt{-\delta }tanh(\sqrt{-\delta }(-{\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(32{\alpha }^2-144)\delta }}}\\ \hfill {\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }+{\displaystyle \frac{3\alpha \beta }{4(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }{)))}^{-1},\end{array}$$

(72)

or

$$\begin{array}{c}u(x,t)=\mp {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta }}{2(2{\alpha }^2-9)\sqrt{2\delta }}}(-\sqrt{-\delta }coth(\sqrt{-\delta }(-{\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(32{\alpha }^2-144)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ +{\displaystyle \frac{3\alpha \beta }{4(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta })))\mp {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta \delta }}{2\sqrt{2}(2{\alpha }^2-9)}}(-\sqrt{-\delta }coth(\sqrt{-\delta }(-{\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(32{\alpha }^2-144)\delta }}}\\ \hfill {\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }+{\displaystyle \frac{3\alpha \beta }{4(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }{)))}^{-1}.\end{array}$$

(73)

Case 3: For $\delta >0$:

$$\begin{array}{c}u(x,t)=\mp {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta }}{2(2{\alpha }^2-9)\sqrt{2\delta }}}(\sqrt{\delta }tan(\sqrt{\delta }(-{\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(32{\alpha }^2-144)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ +{\displaystyle \frac{3\alpha \beta }{4(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta })))\mp {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta \delta }}{2\sqrt{2}(2{\alpha }^2-9)}}(\sqrt{\delta }tan(\sqrt{\delta }(-{\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(32{\alpha }^2-144)\delta }}}\\ \hfill {\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }+{\displaystyle \frac{3\alpha \beta }{4(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }{)))}^{-1},\end{array}$$

(74)

or

$$\begin{array}{c}u(x,t)=\mp {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta }}{2(2{\alpha }^2-9)\sqrt{2\delta }}}(-\sqrt{\delta }cot(\sqrt{\delta }(-{\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(32{\alpha }^2-144)\delta }}}{\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }\hfill \\ +{\displaystyle \frac{3\alpha \beta }{4(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta })))\mp {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta \delta }}{2\sqrt{2}(2{\alpha }^2-9)}}(-\sqrt{\delta }cot(\sqrt{\delta }(-{\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(32{\alpha }^2-144)\delta }}}\\ \hfill {\displaystyle \frac{1}{\theta }}{(x+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }+{\displaystyle \frac{3\alpha \beta }{4(2{\alpha }^2-9)\sqrt{-\delta }}}{\displaystyle \frac{1}{\theta }}{(t+{\displaystyle \frac{1}{\Gamma (\theta )}})}^{\theta }{)))}^{-1}.\end{array}$$

(75)

4. Graphical Description

In this section, the authors explain the attained results using two-dimensional, three-dimensional, and contour plots. The effect of the fractional derivative is also shown by representing graphs for $\theta =0.5$, $\theta =0.7$, and $\theta =1.0$.

5. Physical Interpretations

In this section, we will explain the physical behavior of some of the gained results to the beta-time fractional (1+1)-dimensional non-linear Van der Waals equation.

Figure 1 shows the dark wave solution for the values $\alpha =1.3$, $\beta =-0.03$, and $\delta =-0.8$. Figure 1a shows a 2-D plot for $-15<x<15$ when $\theta =1$, with a blue curve if t is 5, an orange curve if the value of t is 10, and a green curve if the value of t is 15. Figure 1b denotes a 2-dimensional graph for $-15<x<15$ and $t\in (5,15)$, while the red curve if $\theta =0.5$, the black curve if $\theta =0.7$, and the blue curve if $\theta =1$. Figure 1c shows a 3-D plot at $\theta =0.6$ with $t\in (5,15)$. Figure 1d denotes a contour graph at $\theta =0.6$ with $t\in (5,15)$. In all figures, we plotted the analytical solution in different times according to made procedures. These results enable us to can discover some features of dark wave solutions.

Figure 2 shows the singular wave solution for the values $\alpha =2.7$, $\beta =0.35$, and $\delta =-0.5$. Figure 2a shows a 2-D plot for $-25<x<25$ when $\theta =1$, with a blue curve if t is 5, an orange curve if the value of t is 10, and a green curve if the value of t is 15. Figure 2b denotes a 2-dimensional graph for $-25<x<25$ and $t\in (5,15)$, with a red curve if $\theta =0.5$, a black curve if $\theta =0.7$, and a blue curve if $\theta =1$. Figure 2c shows a 3-D plot at $\theta =0.6$ with $t\in (5,15)$. Figure 2d denotes a contour graph at $\theta =0.6$ with $t\in (5,15)$. In all figures, we plotted the analytical solution in different times according to made procedures. These results enable us to can discover some features of singular wave solutions.

Figure 3 shows the periodic wave solution for the values $\alpha =2.7$, $\beta =-0.45$, and $\delta =0.5$. Figure 3a shows a 2-D plot for $-15<x<15$ when $\theta =1$, with a blue curve if t is 5, an orange curve if the value of t is 10, and a green curve if the value of t is 15. Figure 3b denotes a 2-dimensional graph for $-15<x<15$ and $t\in (5,15)$, with a red curve if $\theta =0.5$, a black curve if $\theta =0.7$, and a blue curve if $\theta =1$. Figure 3c shows a 3-D plot at $\theta =0.6$ with $t\in (5,15)$. Figure 3d denotes a contour graph at $\theta =0.6$ with $t\in (5,15)$. In all figures, we plotted the analytical solution in different times according to made procedures. These results enable us to can discover some features of periodic wave solutions.

Figure 4 shows the singular-dark wave solution for the values $\alpha =2.7$, $\beta =0.35$, and $\delta =-0.2$. Figure 4a shows a 2-D plot for $-40<x<40$ when $\theta =1$, with a blue curve if t is 5, an orange curve if the value of t is 10, and a green curve if the value of t is 15. Figure 4b denotes a 2-dimensional graph for $-40<x<40$ and $t\in (5,15)$, with a red curve if $\theta =0.5$, a black curve if $\theta =0.7$, and a blue curve if $\theta =1$. Figure 4c shows a 3-D plot at $\theta =0.6$ with $t\in (5,15)$. Figure 4d denotes a contour graph at $\theta =0.6$ with $t\in (5,15)$. The parameter $\alpha$ denotes the viscosity effect; a high value of $\alpha$ makes the flow of fluids sluggishly, and a low value makes the flow of fluids more easily. This same behavior is also observed in our obtained results when we increase the value of $\alpha$, where we obtain the different curves of our results. In all figures, we plotted the analytical solution in different times according to made procedures. These results enable us to can discover some features of singular-dark wave solutions.

6. Stability Analysis

In this section, we will explain the stability of the governing system. This analysis is performed by using the Hamiltonian system, which is tested on a few of the obtained solutions to show the stability of the concerning system in different applications. Stability analyses have been performed for various models such as [81,82]. Here, we study the Equation (1) stability, for which we define the Hamiltonian transformation as

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

(76)

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

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

(77)

where $\omega$ represents a soliton velocity; putting Equation (22) into Equation (76) yields

$$\begin{array}{c}\mathcal{M}={\displaystyle \frac{1}{2}}{\int }_{-7}^7(\pm {\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{18-4{\alpha }^2}}}\pm {\displaystyle \frac{\alpha \sqrt{2{\alpha }^2-9}\sqrt{\beta }}{(2{\alpha }^2-9)\sqrt{2\delta }}}(-\sqrt{-\delta }tanh(\sqrt{-\delta }({\displaystyle \frac{\alpha \sqrt{\beta }}{\sqrt{(8{\alpha }^2-36)\delta }}}x\hfill \\ \hfill -{\displaystyle \frac{3\alpha \beta }{2(2{\alpha }^2-9)\sqrt{-\delta }}}{t)))))}^{2}dx,\end{array}$$

(78)

By using the criterion is given in Equation (77), we obtain the following

$$\begin{array}{c}{\displaystyle \frac{1}{2}}((2\sqrt{2}\alpha \sqrt{\beta }\sqrt{(2{\alpha }^2-9)\delta }(\sqrt{-\delta }ttanh({\displaystyle \frac{1}{2}}\sqrt{-\delta }({\displaystyle \frac{2t(3\alpha \beta )}{2(2{\alpha }^2-9)\sqrt{-\delta }}}-{\displaystyle \frac{7\alpha \sqrt{\beta }}{\sqrt{(2{\alpha }^2-9)\delta }}}))\hfill \\ -\sqrt{-\delta }ttanh({\displaystyle \frac{1}{2}}\sqrt{-\delta }({\displaystyle \frac{7\alpha \sqrt{\beta }}{\sqrt{(2{\alpha }^2-9)\delta }}}+{\displaystyle \frac{2t(3\alpha \beta )}{2(2{\alpha }^2-9)\sqrt{-\delta }}}))))/(\sqrt{18-4{\alpha }^2}\sqrt{2{\alpha }^2-9}\sqrt{\delta })\\ +(\alpha \sqrt{\beta }\sqrt{(2{\alpha }^2-9)\delta }(\sqrt{-\delta }t{\sec \mathrm{h}}^2(\sqrt{-\delta }({\displaystyle \frac{t(-(3\alpha \beta ))}{2(2{\alpha }^2-9)\sqrt{-\delta }}}-{\displaystyle \frac{7\alpha \sqrt{\beta }}{2\sqrt{(2{\alpha }^2-9)\delta }}}))\\ \hfill -\sqrt{-\delta }t{\sec \mathrm{h}}^2(\sqrt{-\delta }({\displaystyle \frac{7\alpha \sqrt{\beta }}{2\sqrt{(2{\alpha }^2-9)\delta }}}-{\displaystyle \frac{t(3\alpha \beta )}{2(2{\alpha }^2-9)\sqrt{-\delta }}}))))/((2{\alpha }^2-9)\sqrt{-\delta }))>0.\end{array}$$

(79)

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

7. Modulation Instability Analysis

Considering a steady-state solution of the (1 + 1)-dimensional non-linear Van der Waals equation is of the form [83],

$$u(x,t)=\left(U(x,t)+\sqrt{\tau }\right)e^{\iota \tau t},$$

(80)

where $\tau$ denotes the normalized optical power.

Putting Equation (80) into Equation (1). After linearizing, we obtain

$$-{\tau }^{5/2}+2\iota \tau U_t+U_{\mathrm{tt}}-{\tau }^{2}U-\alpha \iota \tau U_{\mathrm{xx}}-\beta U_{\mathrm{xx}}-\alpha U_{\mathrm{xxt}}+U_{\mathrm{xxxx}}=0.$$

(81)

Assuming the result of the Equation (81) given as

$$U(x,t)=A_1{e}^{\iota (px-qt)}+A_2{e}^{-\iota (px-qt)},$$

(82)

where p and q denote the frequency and normalized wave number of perturbation, respectively. We substitute Equation (82) into Equation (81). By collecting the coefficients of $e^{\iota (px-qt)}$ and $e^{\iota (px-qt)}$, we obtain the dispersion relation after solving the determinant of the coefficient matrix.

$$p^8+2\beta p^6-{\alpha }^2{p}^4{\tau }^2+{\beta }^2{p}^4+{\alpha }^2{p}^4{q}^2-2p^4{q}^2-2p^4{\tau }^2-2\beta p^2{\tau }^2-2\beta p^2{q}^2+q^4-2q^2{\tau }^2+{\tau }^4=0=0.$$

(83)

Determine the dispersion relation from Equation (83) for q, which yields

$$q=\pm \sqrt{-{\displaystyle \frac{1}{2}}{\alpha }^2{p}^4+p^4+\beta p^2+{\displaystyle \frac{1}{2}}\sqrt{{\alpha }^4{p}^8-4{\alpha }^2{p}^8-4{\alpha }^2\beta p^6+16p^4{\tau }^2+16\beta p^2{\tau }^2}+{\tau }^2}.$$

(84)

The gained dispersion relation shows steady-state stability. When a wave number q is imaginary, then steady-state the result will be unstable since the perturbation increases exponentially. But when a wave number q is real, then the steady state is converted into stability against small perturbations. The steady-state result is unstable if

$$p^4-{\displaystyle \frac{1}{2}}{\alpha }^2{p}^4+\beta p^2+{\displaystyle \frac{1}{2}}\sqrt{{\alpha }^4{p}^8-4{\alpha }^2{p}^8-4{\alpha }^2\beta p^6+16p^4{\tau }^2+16\beta p^2{\tau }^2}+{\tau }^2<0.$$

(85)

The modulation instability (MI) gain spectrum $G(p)$ is obtained as

$$G(p)=2Im(q)=\pm \sqrt{p^4-{\displaystyle \frac{1}{2}}{\alpha }^2{p}^4+\beta p^2+{\displaystyle \frac{1}{2}}\sqrt{{\alpha }^4{p}^8-4{\alpha }^2{p}^8-4{\alpha }^2\beta p^6+16p^4{\tau }^2+16\beta p^2{\tau }^2}+{\tau }^2}.$$

(86)

The gained dispersion is provided and presented in Figure 5 to investigate the steady-state stability.

8. Conclusions

This paper contains the distinct types of exact soliton solutions of the (1 + 1)-dimensional non-linear Van der Waals equation along the beta-time fractional derivative. We obtain bright, dark, dark-solitary, dark-singular, periodic wave, and many more exact wave solutions by using the modified extended tanh function method. The use of fractional derivatives makes the solutions different from the existing solutions. The obtained results are represented by two-dimensional, three-dimensional, and contour graphs. Stability analysis is carried out to check the stability of the model. Additionally, modulation instability is performed to study the stationary solutions of the governing model. Results are helpful in the progress of the concerned system. The gained results will be of high importance in the interaction of quantum-mechanical fluctuations, granular matters, and other fields of Van der Waals equation applications. The achieved results are also useful in various naturally occurring phenomena, industries, geophysics, civil engineering, pharmaceutical, and many others. It is suggested that the method used is also useful for the other nonlinear models of different fields of science and engineering.