Discovery of New Exact Wave Solutions to the M-Fractional Complex Three Coupled Maccari’s System by Sardar Sub-Equation Scheme

Abstract

In this paper, we succeed at discovering the new exact wave solutions to the truncated M-fractional complex three coupled Maccari’s system by utilizing the Sardar sub-equation scheme. The obtained solutions are in the form of trigonometric and hyperbolic forms. These solutions have many applications in nonlinear optics, fiber optics, deep water-waves, plasma physics, mathematical physics, fluid mechanics, hydrodynamics and engineering, where the propagation of nonlinear waves is important. Achieved solutions are verified with the use of Mathematica software. Some of the achieved solutions are also described graphically by 2-dimensional, 3-dimensional and contour plots with the help of Maple software. The gained solutions are helpful for the further development of a concerned model. Finally, this technique is simple, fruitful and reliable to handle nonlinear fractional partial differential equations (NLFPDEs).

1. Introduction

Fractional Calculus (FC) is a generalization of classical calculus concerned with operations of integration and differentiation of non-integer (fractional) order. Since the 19th century, the theory of fractional calculus developed rapidly, mostly as a foundation for a number of applied disciplines, including fractional geometry, fractional differential equations (FDE) and fractional dynamics. The applications of FC are very wide nowadays. It is safe to say that almost no discipline of modern engineering, and science in general, remains untouched by the tools and techniques of fractional calculus. For example, wide and fruitful applications can be found in rheology, viscoelasticity, acoustics, optics, chemical and statistical physics, robotics, control theory, electrical and mechanical engineering, bio-engineering etc. In fact, one could argue that real world processes are fractional order systems in general. The main reason for the success of FC applications is that these new fractional-order models are often more accurate than integer-order ones, i.e., there are more degrees of freedom in the fractional order model than in the corresponding classical one. Fractional calculus is a field of mathematics study that grew out of the traditional definitions of calculus integral and derivative operators in much the same way fractional exponents are an outgrowth of exponents with integer values. A fractional equation (FE) is a differential equation that contains fractional derivatives or integrals. The awareness of the importance of this kind of equation has grown continually in the last decade. Many applications have become apparent: wave propagation in a complex or porous media [1], the fractional complex Ginzburg–Landau model [2], fractional order modified Duffing systems [3], the fractional order Boussinesq–Like equations occurring in physical sciences and engineering [4], symmetric regularized long-wave (SRLW) equations arising in long water flow models [5] and extended forced Korteweg–de Vries equations with variable coefficients in fluid or plasma [6]. There are many types of fractional order derivatives, i.e., confirmable fractional derivatives [7], Beta fractional derivatives [8], Caputo–Fabrizio fractional derivatives [9], Atangana–Baleanu–Riemann derivatives [10] and truncated M-fractional derivatives [11,12].

Different types of phenomenon occurring chemically, biologically and economically, among others, are represented as non-linear partial differential equations (NLPDEs). Many different methods have been developed to gain the analytical wave solutions of these NLPDEs, i.e., the optical soliton solutions of coupled nonlinear Schrödinger equations have been gained with the use of the Kudryashov R function technique [13], some new kinds of optical soliton solutions of time-fractional perturbed nonlinear Schrödinger equations have been achieved by using the generalized Kudryashov scheme [14], by applying the modified auxiliary equation technique, optical wave solutions of time-fractional resonant non-linear Schrödinger equations have been obtained [15] and new optical wave solutions for time-fractional perturbed non-linear Schrödinger equations have been achieved by utilizing the improved $tan\left(\varphi \right(\zeta /2\left)\right)$-expansion scheme [16].

It is well known that there is a strict connection between symmetry and exact wave solutions. Most of the existing techniques to solve partial/fractional differential equations for finding the soliton or solitary solutions are, in substance, a case of symmetry reduction, including nonclassical symmetry and Lie symmetries, etc. The study of partial differential equations—specifically, those derived from applied mathematics—displays the elegance of symmetry analysis best. Therefore, if the classical symmetries of the governing equation can be found, it will be shown that the equations can have integrability, that is, their exact solutions can be found. Schrödinger-type equations have symmetries in many ways; for this reason, symmetry is the key to nature and soliton theory.

We utilize the Sardar sub-equation method for our research work. This method is one of the strong methods for solving nonlinear evolution equations. In this method, if we select the special values of parameters, then we can acquire the traveling wave solutions. The solutions obtained by this method are the solutions gained by the first integral method, the trial equation method and the functional variable method. Several new traveling wave solutions are attained, including generalized hyperbolic and trigonometric functions. This method has so many applications in the literature. For example, some solitons solutions of the perturbed Fokas–Lenells model have been obtained with the use of the Sardar sub-equation method [17], new kinds of soliton wave solutions of the (2+1)-dimensional Sawada–Kotera model have been gained by this method [18], the dark, bright and singular optical wave solutions of the higher order non-linear Schrödinger have been gained by using the Sardar sub-equation scheme [19] and the singular, bright, dark, periodic singular, combined dark-bright solitons and other wave solutions of the strain wave equation have been achieved with the use of this model [20].

Our study model is a complex three coupled Maccari’s system along with truncated M-fractional derivative. Different types of exact wave solutions for this model have been obtained with the help of different methods in the literature. Instantly, the dark, bright and complex optical wave solutions have been achieved by using extended rational sine–cosine and rational sinh–cosh techniques [21], trigonometric, rational and hyperbolic solutions have been attained with the use of the unified solver technique [22], optical soliton solutions have been gained by utilizing the planar dynamic system scheme [23], some new traveling wave solutions have been obtained by applying the modified trial equation technique [24], the periodic, double periodic, dark, bright, combined soliton solutions etc. have been obtained with the help of the generalized auxiliary equation mapping technique [25], the trigonometric and hyperbolic trigonometric function solutions have been attained by using the modified $exp(-\varphi (\xi \left)\right)$-function method [26], some exact wave solutions in the form of trigonometric, hyperbolic trigonometric and rational functions have been gained by utilizing the He’s semi-inverse and $(G^{\prime }/G)$-expansion methods [27] and solitary wave solutions have been gained by applying the $exp(-\varphi (\xi \left)\right)$-function method [28].

The main aim of this research is to discover new exact wave solutions to the truncated M-fractional complex three coupled Maccari’s system with the help of the Sardar sub-equation method.

The motivation of this paper is to explain the effect of the M-fractional derivative on the solutions for the space-time fractional complex three coupled Maccari’s system that are gained with the use of the Sardar sub-equation method. The significance of the M-fractional derivative is that it fulfills both properties of the integer and fractional order derivatives. This derivative generalizes the many fractional derivatives and satisfies important properties of the integer-order derivatives. Via our method, we can observe some elementary relationships between nonlinear fractional partial differential equations (NLFPDEs) and other simple nonlinear ordinary differential equations (NLODEs). It has been found that with the use of simple schemes and solvable ODEs, different types of exact-wave solutions for some complicated NLFPDEs can be easily obtained. The solutions attained are newer than the existing solutions in the literature.

This paper consists of different sections. In Section 2, we describe the main steps of our concerned method Sardar sub-equation. In Section 3, we explain our concerned model and its mathematical analysis. In Section 4, we apply the method to gain the new exact wave solutions for our concerned model. In Section 5, we explain the obtained solutions through 2-D, 3-D and contour graphs. In Section 6, we give the conclusion of our research work.

2. Description of Sardar Sub-Equation Method

Here, we explain the fundamental points of the Sardar sub-equation method [29]. Let us consider the non-linear fractional partial differential equation:

$$F(g,D_{M,t}^{\alpha ,\gamma }g,D_{M,2x}^{2\alpha ,\gamma }g,D_{M,xt}^{2\alpha ,\gamma }g,g{D}_{M,2t}^{2\alpha ,\gamma }g,D_{M,xy}^{2\alpha ,\gamma }g,...)=0.$$

(1)

Here, $g=g(x,y,t)$ represents a wave function.

Applying the wave transformations gives as follow:

$$g(x,y,t)=G\left(\zeta \right),\zeta =\frac{\Gamma (1+\gamma )}{\alpha }(\lambda x^{\alpha }+\kappa y^{\alpha }+\mu t^{\alpha }),$$

(2)

where parameter $\lambda$ represents the wave number, $\kappa$ shows the frequency and $\mu$ denotes the speed of soliton.

We obtain a non-linear ordinary differential equation (ODE), shown as:

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

(3)

Consider that Equation (3) possesses the results in the given shape:

$$G\left(\zeta \right)=\sum _{i=0}^m{b}_i{\psi }^i\left(\zeta \right),$$

(4)

where m is a natural number.

Here, $\psi \left(\zeta \right)$ fulfills the ODE, shown as:

$${\psi }^{{}^{\prime }}\left(\zeta \right)=\sqrt{\sigma +\kappa {\psi }^2\left(\zeta \right)+{\psi }^4\left(\zeta \right)},$$

(5)

where $\sigma$ and $\kappa$ are constants.

Using Equation (4) into Equation (3) with Equation (5) and collecting the coefficients of each power of ${\psi }^i$. Putting the co-efficient of each power equal to 0, we gain a set of algebraic equations in the term $b_i$, $\lambda$, $\mu$. By solving the obtained system of equations, we obtain the values of the parameters.

Case 1: If $\kappa >0$ and $\sigma =0$, then

$${\psi }_1^{\pm }=\pm \sqrt{-\kappa ab}sec{h}_{ab}\left(\sqrt{\kappa }\zeta \right),$$

(6)

$${\psi }_2^{\pm }=\pm \sqrt{\kappa ab}csc{h}_{ab}\left(\sqrt{\kappa }\zeta \right),$$

(7)

where $sec{h}_{ab}\left(\zeta \right)=\frac{2}{a{e}^{\zeta }+b{e}^{-\zeta }}$, $csc{h}_{ab}\left(\zeta \right)=\frac{2}{a{e}^{\zeta }-b{e}^{-\zeta }}.$

Case 2: If $\kappa <0$ and $\sigma =0$, then

$${\psi }_3^{\pm }=\pm \sqrt{-\kappa ab}{sec}_{ab}\left(\sqrt{-\kappa }\zeta \right),$$

(8)

$${\psi }_4^{\pm }=\pm \sqrt{-\kappa ab}{csc}_{ab}\left(\sqrt{-\kappa }\zeta \right),$$

(9)

where ${sec}_{ab}\left(\zeta \right)=\frac{2}{a{e}^{i\zeta }+b{e}^{-i\zeta }}$, ${csc}_{ab}\left(\zeta \right)=\frac{2i}{a{e}^{i\zeta }-b{e}^{-i\zeta }}.$

Case 3: If $\kappa <0$ and $\sigma =\frac{{\kappa }^2}{4}$, then

$${\psi }_5^{\pm }=\pm \sqrt{-\frac{\kappa }{2}}{tanh}_{ab}\left(\sqrt{-\frac{\kappa }{2}}\zeta \right),$$

(10)

$${\psi }_6^{\pm }=\pm \sqrt{-\frac{\kappa }{2}}{coth}_{ab}\left(\sqrt{-\frac{\kappa }{2}}\zeta \right),$$

(11)

$${\psi }_7^{\pm }=\pm \sqrt{-\frac{\kappa }{2}}({tanh}_{ab}\left(\sqrt{-2\kappa }\zeta \right)\pm \iota \sqrt{ab}sec{h}_{ab}\left(\sqrt{-2\kappa }\zeta \right)),$$

(12)

$${\psi }_8^{\pm }=\pm \sqrt{-\frac{\kappa }{2}}({coth}_{ab}\left(\sqrt{-2\kappa }\zeta \right)\pm \sqrt{ab}csc{h}_{ab}\left(\sqrt{-2\kappa }\zeta \right)),$$

(13)

$${\psi }_9^{\pm }=\pm \sqrt{-\frac{\kappa }{8}}({tanh}_{ab}\left(\sqrt{-\frac{\kappa }{8}}\zeta \right)+{coth}_{ab}\left(\sqrt{-\frac{\kappa }{8}}\zeta \right)),$$

(14)

where ${tanh}_{ab}\left(\zeta \right)=\frac{a{e}^{\zeta }-b{e}^{-\zeta }}{a{e}^{\zeta }+b{e}^{-\zeta }}$, ${coth}_{ab}\left(\zeta \right)=\frac{a{e}^{\zeta }+b{e}^{-\zeta }}{a{e}^{\zeta }-b{e}^{-\zeta }}.$

Case 4: If $\kappa >0$ and $\sigma =\frac{{\kappa }^2}{4}$, then

$${\psi }_{10}^{\pm }=\pm \sqrt{\frac{\kappa }{2}}{tan}_{ab}\left(\sqrt{\frac{\kappa }{2}}\zeta \right),$$

(15)

$${\psi }_{11}^{\pm }=\pm \sqrt{\frac{\kappa }{2}}{cot}_{ab}\left(\sqrt{\frac{\kappa }{2}}\zeta \right),$$

(16)

$${\psi }_{12}^{\pm }=\pm \sqrt{\frac{\kappa }{2}}({tan}_{ab}\left(\sqrt{2\kappa }\zeta \right)\pm \sqrt{ab}{sec}_{ab}\left(\sqrt{2\kappa }\zeta \right)),$$

(17)

$${\psi }_{13}^{\pm }=\pm \sqrt{\frac{\kappa }{2}}({cot}_{ab}\left(\sqrt{2\kappa }\zeta \right)\pm \sqrt{ab}{csc}_{ab}\left(\sqrt{2\kappa }\zeta \right)),$$

(18)

$${\psi }_{14}^{\pm }=\pm \sqrt{\frac{\kappa }{8}}({tan}_{ab}\left(\sqrt{\frac{\kappa }{8}}\zeta \right)+{cot}_{ab}\left(\sqrt{\frac{\kappa }{8}}\zeta \right)),$$

(19)

where ${tan}_{ab}\left(\zeta \right)=\frac{a{e}^{\iota \zeta }-b{e}^{-\iota \zeta }}{i(a{e}^{\iota \zeta }+b{e}^{-\iota \zeta })}$, ${cot}_{ab}\left(\zeta \right)=\frac{i(a{e}^{\iota \zeta }+b{e}^{-\iota \zeta })}{a{e}^{\iota \zeta }-b{e}^{-\iota \zeta }}.$

3. The Governing Model and Its Mathematical Treatment

The three coupled non-linear Maccari’s system explains how isolated waves are propagated in a finite region of space, in optical communications, hydrodynamics and plasma physics. Assume the M-fractional three coupled non-linear Maccari’s system shown in [30]:

$$\begin{array}{c}i{D}_{M,t}^{\alpha ,\gamma }U+U_{xx}+ZU=0,\\ \hspace{1em} i{D}_{M,t}^{\alpha ,\gamma }V+V_{xx}+ZV=0,\\ \hspace{1em} i{D}_{M,t}^{\alpha ,\gamma }W+W_{xx}+ZW=0,\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} D_{M,t}^{\alpha ,\gamma }Z+Z_y{+\left(\right|U+V+W|}^2{)}_x=0,\end{array}$$

(20)

where

$$D_{M,t}^{\alpha ,\gamma }U\left(t\right)=\underset{\tau \to 0}{lim}\frac{U\left(t{E}_{\gamma }\left(\tau t^{1-\alpha }\right)\right)-U\left(t\right)}{\tau },0<\alpha \le 1,\gamma >0.$$

(21)

Here, $E_{\gamma }(.)$ represents the truncated Mittag–Leffler function of one parameter given in [31,32].

Let us consider the following transformations:

$$\begin{array}{c}U(x,y,t)=U\left(\zeta \right)\times exp\left(\iota ({\theta }_{1}x+\mu y+\tau \frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right),\hfill \\ V(x,y,t)=V\left(\zeta \right)\times exp\left(\iota ({\theta }_{1}x+\mu y+\tau \frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right),\\ W(x,y,t)=W\left(\zeta \right)\times exp\left(\iota ({\theta }_{1}x+\mu y+\tau \frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right),\\ \hfill \hspace{1em}\hspace{1em}\hspace{1em} Z(x,y,t)=Z\left(\zeta \right)where\zeta =\lambda (x+y-2\beta \frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha }).\end{array}$$

(22)

Here, ${\theta }_1$, $\sigma$, $\tau$, $\mu$ and $\beta$ are the unknowns and $\vartheta$ is the arbitrary constant.

Substituting Equation (22) into Equation (20) yields the real and imaginary parts:

Real parts:

$$\begin{array}{c} {\lambda }^2{U}^{″}-(\tau +{\theta }_1^2)U+UZ=0,\hfill \\ {\lambda }^2{V}^{″}-(\tau +{\theta }_1^2)V+VZ=0,\\ {\lambda }^2{W}^{″}-(\tau +{\theta }_1^2)W+WZ=0,\\ \hfill \hspace{1em}\hspace{1em}\hspace{1em} \lambda (1-2\beta )Z^{\prime }+\lambda {\left({(U+V+W)}^2\right)}^{\prime }=0.\end{array}$$

(23)

Imaginary parts:

$$(-2\beta +2{\theta }_1)U^{\prime }=0,(-2\beta +2{\theta }_1)V^{\prime }=0,(-2\beta +2{\theta }_1)W^{\prime }=0.$$

(24)

By Equation (24), imply $\beta ={\theta }_1$.

Integrating the fourth equation of system (23) yields

$$Z=-\frac{{(U+V+W)}^2}{1-2{\theta }_1}.$$

(25)

Putting Equation (25) into the first three equations of the system (23), we obtain

$$\begin{array}{c}{\lambda }^2{U}^{″}-(\tau +{\theta }_1^2)U-\frac{{(U+V+W)}^2}{1-2{\theta }_1}U=0,\\ {\lambda }^2{V}^{″}-(\tau +{\theta }_1^2)V-\frac{{(U+V+W)}^2}{1-2{\theta }_1}V=0,\\ {\lambda }^2{W}^{″}-(\tau +{\theta }_1^2)W-\frac{{(U+V+W)}^2}{1-2{\theta }_1}W=0.\hfill \end{array}$$

(26)

Taking $V={\theta }_2$ U and taking $W=c$ U, we obtain

$${\lambda }^2{U}^{″}-(\tau +{\theta }_1^2)U-\frac{{(1+{\theta }_2+c)}^2}{1-2{\theta }_1}{U}^3=0.$$

(27)

4. Solutions through Sardar Sub-Equation Method

By applying the homogeneous balance technique and balancing $U^{″}$ and $U^3$ in Equation (27), we gain

$$3m=m+2,$$

(28)

$$m=1.$$

(29)

So, Equation (4) changes into the given form for m = 1.

$$G\left(\zeta \right)=b_0+b_1\psi \left(\zeta \right).$$

(30)

Putting Equation (30) into Equation (27) with Equation (5). By collecting co-efficients of every order of $\psi \left(\zeta \right)$ and taking them equal to 0, we obtain a set of algebraic equations. By solving the obtained set of equations using Mathematica software, we gain the following sets.

Set 1:

$$\left\{b_0=0,b_1=\mp \frac{\sqrt{2}\sqrt{\left(1-2{\theta }_1\right){\lambda }^2}}{\sqrt{\left(c+{\theta }_2+1\right){}^2}},\tau =\kappa {\lambda }^2-{\theta }_1^2\right\}.$$

(31)

Case 1

$$U(x,y,t)=\mp \frac{\sqrt{2\left(1-2{\theta }_1\right)}\lambda }{\left(c+{\theta }_2+1\right)}(\pm \sqrt{-\kappa ab}sec{h}_{ab}\left(\sqrt{\kappa }\zeta \right))\times exp\left(\iota ({\theta }_{1}x+\mu y+(\kappa {\lambda }^2-{\theta }_1^2)\frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right).$$

(32)

$$V(x,y,t)=\mp \frac{{\theta }_2\sqrt{2\left(1-2{\theta }_1\right)}\lambda }{\left(c+{\theta }_2+1\right)}(\pm \sqrt{-\kappa ab}sec{h}_{ab}\left(\sqrt{\kappa }\zeta \right))\times exp\left(\iota ({\theta }_{1}x+\mu y+(\kappa {\lambda }^2-{\theta }_1^2)\frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right).$$

(33)

$$W(x,y,t)=\mp \frac{c\sqrt{2\left(1-2{\theta }_1\right)}\lambda }{\left(c+{\theta }_2+1\right)}(\pm \sqrt{-\kappa ab}sec{h}_{ab}\left(\sqrt{\kappa }\zeta \right))\times exp\left(\iota ({\theta }_{1}x+\mu y+(\kappa {\lambda }^2-{\theta }_1^2)\frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right).$$

(34)

$$Z(x,y,t)=-2{\lambda }^2{(\pm \sqrt{-\kappa ab}sec{h}_{ab}\left(\sqrt{\kappa }\zeta \right))}^2.$$

(35)

$$U(x,y,t)=\mp \frac{\sqrt{2\left(1-2{\theta }_1\right)}\lambda }{\left(c+{\theta }_2+1\right)}(\pm \sqrt{\kappa ab}csc{h}_{ab}\left(\sqrt{\kappa }\zeta \right))\times exp\left(\iota ({\theta }_{1}x+\mu y+(\kappa {\lambda }^2-{\theta }_1^2)\frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right).$$

(36)

$$V(x,y,t)=\mp \frac{{\theta }_2\sqrt{2\left(1-2{\theta }_1\right)}\lambda }{\left(c+{\theta }_2+1\right)}(\pm \sqrt{\kappa ab}csc{h}_{ab}\left(\sqrt{\kappa }\zeta \right))\times exp\left(\iota ({\theta }_{1}x+\mu y+(\kappa {\lambda }^2-{\theta }_1^2)\frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right).$$

(37)

$$W(x,y,t)=\mp \frac{c\sqrt{2\left(1-2{\theta }_1\right)}\lambda }{\left(c+{\theta }_2+1\right)}(\pm \sqrt{\kappa ab}csc{h}_{ab}\left(\sqrt{\kappa }\zeta \right))\times exp\left(\iota ({\theta }_{1}x+\mu y+(\kappa {\lambda }^2-{\theta }_1^2)\frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right).$$

(38)

$$Z(x,y,t)=-2{\lambda }^2{(\pm \sqrt{\kappa ab}csc{h}_{ab}\left(\sqrt{\kappa }\zeta \right))}^2.$$

(39)

Case 2:

$$U(x,y,t)=\mp \frac{\sqrt{2\left(1-2{\theta }_1\right)}\lambda }{\left(c+{\theta }_2+1\right)}(\pm \sqrt{-\kappa ab}{sec}_{ab}\left(\sqrt{-\kappa }\zeta \right))\times exp\left(\iota ({\theta }_{1}x+\mu y+(\kappa {\lambda }^2-{\theta }_1^2)\frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right).$$

(40)

$$V(x,y,t)=\mp \frac{{\theta }_2\sqrt{2\left(1-2{\theta }_1\right)}\lambda }{\left(c+{\theta }_2+1\right)}(\pm \sqrt{-\kappa ab}{sec}_{ab}\left(\sqrt{-\kappa }\zeta \right))\times exp\left(\iota ({\theta }_{1}x+\mu y+(\kappa {\lambda }^2-{\theta }_1^2)\frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right).$$

(41)

$$W(x,y,t)=\mp \frac{c\sqrt{2\left(1-2{\theta }_1\right)}\lambda }{\left(c+{\theta }_2+1\right)}(\pm \sqrt{-\kappa ab}{sec}_{ab}\left(\sqrt{-\kappa }\zeta \right))\times exp\left(\iota ({\theta }_{1}x+\mu y+(\kappa {\lambda }^2-{\theta }_1^2)\frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right).$$

(42)

$$Z(x,y,t)=-2{\lambda }^2{(\pm \sqrt{-\kappa ab}{sec}_{ab}\left(\sqrt{-\kappa }\zeta \right))}^2.$$

(43)

$$U(x,y,t)=\mp \frac{\sqrt{2\left(1-2{\theta }_1\right)}\lambda }{\left(c+{\theta }_2+1\right)}(\pm \sqrt{-\kappa ab}{csc}_{ab}\left(\sqrt{-\kappa }\zeta \right))\times exp\left(\iota ({\theta }_{1}x+\mu y+(\kappa {\lambda }^2-{\theta }_1^2)\frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right).$$

(44)

$$V(x,y,t)=\mp \frac{{\theta }_2\sqrt{2\left(1-2{\theta }_1\right)}\lambda }{\left(c+{\theta }_2+1\right)}(\pm \sqrt{-\kappa ab}{csc}_{ab}\left(\sqrt{-\kappa }\zeta \right))\times exp\left(\iota ({\theta }_{1}x+\mu y+(\kappa {\lambda }^2-{\theta }_1^2)\frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right).$$

(45)

$$W(x,y,t)=\mp \frac{c\sqrt{2\left(1-2{\theta }_1\right)}\lambda }{\left(c+{\theta }_2+1\right)}(\pm \sqrt{-\kappa ab}{csc}_{ab}\left(\sqrt{-\kappa }\zeta \right))\times exp\left(\iota ({\theta }_{1}x+\mu y+(\kappa {\lambda }^2-{\theta }_1^2)\frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right).$$

(46)

$$Z(x,y,t)=-2{\lambda }^2{(\pm \sqrt{-\kappa ab}{csc}_{ab}\left(\sqrt{-\kappa }\zeta \right))}^2.$$

(47)

Case 3:

$$U(x,y,t)=\mp \frac{\sqrt{2\left(1-2{\theta }_1\right)}\lambda }{\left(c+{\theta }_2+1\right)}(\pm \sqrt{-\frac{\kappa }{2}}{tanh}_{ab}\left(\sqrt{-\frac{\kappa }{2}}\zeta \right))\times exp\left(\iota ({\theta }_{1}x+\mu y+(\kappa {\lambda }^2-{\theta }_1^2)\frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right).$$

(48)

$$V(x,y,t)=\mp \frac{{\theta }_2\sqrt{2\left(1-2{\theta }_1\right)}\lambda }{\left(c+{\theta }_2+1\right)}(\pm \sqrt{-\frac{\kappa }{2}}{tanh}_{ab}\left(\sqrt{-\frac{\kappa }{2}}\zeta \right))\times exp\left(\iota ({\theta }_{1}x+\mu y+(\kappa {\lambda }^2-{\theta }_1^2)\frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right).$$

(49)

$$W(x,y,t)=\mp \frac{c\sqrt{2\left(1-2{\theta }_1\right)}\lambda }{\left(c+{\theta }_2+1\right)}(\pm \sqrt{-\frac{\kappa }{2}}{tanh}_{ab}\left(\sqrt{-\frac{\kappa }{2}}\zeta \right))\times exp\left(\iota ({\theta }_{1}x+\mu y+(\kappa {\lambda }^2-{\theta }_1^2)\frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right).$$

(50)

$$Z(x,y,t)={\lambda }^2\kappa {\left({tanh}_{ab}\left(\sqrt{-\frac{\kappa }{2}}\zeta \right)\right)}^2.$$

(51)

$$U(x,y,t)=\mp \frac{\sqrt{2\left(1-2{\theta }_1\right)}\lambda }{\left(c+{\theta }_2+1\right)}(\pm \sqrt{-\frac{\kappa }{2}}{coth}_{ab}\left(\sqrt{-\frac{\kappa }{2}}\zeta \right))\times exp\left(\iota ({\theta }_{1}x+\mu y+(\kappa {\lambda }^2-{\theta }_1^2)\frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right).$$

(52)

$$V(x,y,t)=\mp \frac{{\theta }_2\sqrt{2\left(1-2{\theta }_1\right)}\lambda }{\left(c+{\theta }_2+1\right)}(\pm \sqrt{-\frac{\kappa }{2}}{coth}_{ab}\left(\sqrt{-\frac{\kappa }{2}}\zeta \right))\times exp\left(\iota ({\theta }_{1}x+\mu y+(\kappa {\lambda }^2-{\theta }_1^2)\frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right).$$

(53)

$$W(x,y,t)=\mp \frac{c\sqrt{2\left(1-2{\theta }_1\right)}\lambda }{\left(c+{\theta }_2+1\right)}(\pm \sqrt{-\frac{\kappa }{2}}{coth}_{ab}\left(\sqrt{-\frac{\kappa }{2}}\zeta \right))\times exp\left(\iota ({\theta }_{1}x+\mu y+(\kappa {\lambda }^2-{\theta }_1^2)\frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right).$$

(54)

$$Z(x,y,t)={\lambda }^2\kappa {\left({coth}_{ab}\left(\sqrt{-\frac{\kappa }{2}}\zeta \right)\right)}^2.$$

(55)

$$\begin{array}{c}U(x,y,t)=\mp \frac{\sqrt{2\left(1-2{\theta }_1\right)}\lambda }{\left(c+{\theta }_2+1\right)}(\pm \sqrt{-\frac{\kappa }{2}}({tanh}_{ab}\left(\sqrt{-2\kappa }\zeta \right)\pm \iota \sqrt{ab}sec{h}_{ab}\left(\sqrt{-2\kappa }\zeta \right)))\hfill \\ \hfill \times exp\left(\iota ({\theta }_{1}x+\mu y+(\kappa {\lambda }^2-{\theta }_1^2)\frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right).\end{array}$$

(56)

$$\begin{array}{c}V(x,y,t)=\mp \frac{{\theta }_2\sqrt{2\left(1-2{\theta }_1\right)}\lambda }{\left(c+{\theta }_2+1\right)}(\pm \sqrt{-\frac{\kappa }{2}}({tanh}_{ab}\left(\sqrt{-2\kappa }\zeta \right)\pm \iota \sqrt{ab}sec{h}_{ab}\left(\sqrt{-2\kappa }\zeta \right)))\hfill \\ \hfill \times exp\left(\iota ({\theta }_{1}x+\mu y+(\kappa {\lambda }^2-{\theta }_1^2)\frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right).\end{array}$$

(57)

$$\begin{array}{c}W(x,y,t)=\mp \frac{c\sqrt{2\left(1-2{\theta }_1\right)}\lambda }{\left(c+{\theta }_2+1\right)}(\pm \sqrt{-\frac{\kappa }{2}}({tanh}_{ab}\left(\sqrt{-2\kappa }\zeta \right)\pm \iota \sqrt{ab}sec{h}_{ab}\left(\sqrt{-2\kappa }\zeta \right)))\hfill \\ \hfill \times exp\left(\iota ({\theta }_{1}x+\mu y+(\kappa {\lambda }^2-{\theta }_1^2)\frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right).\end{array}$$

(58)

$$Z(x,y,t)={\lambda }^2\kappa {({tanh}_{ab}\left(\sqrt{-2\kappa }\zeta \right)\pm \iota \sqrt{ab}sec{h}_{ab}\left(\sqrt{-2\kappa }\zeta \right))}^2.$$

(59)

$$\begin{array}{c}U(x,y,t)=\mp \frac{\sqrt{2\left(1-2{\theta }_1\right)}\lambda }{\left(c+{\theta }_2+1\right)}(\pm \sqrt{-\frac{\kappa }{2}}({coth}_{ab}\left(\sqrt{-2\kappa }\zeta \right)\pm \sqrt{ab}csc{h}_{ab}\left(\sqrt{-2\kappa }\zeta \right)))\hfill \\ \hfill \times exp\left(\iota ({\theta }_{1}x+\mu y+(\kappa {\lambda }^2-{\theta }_1^2)\frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right).\end{array}$$

(60)

$$\begin{array}{c}V(x,y,t)=\mp \frac{{\theta }_2\sqrt{2\left(1-2{\theta }_1\right)}\lambda }{\left(c+{\theta }_2+1\right)}(\pm \sqrt{-\frac{\kappa }{2}}({coth}_{ab}\left(\sqrt{-2\kappa }\zeta \right)\pm \sqrt{ab}csc{h}_{ab}\left(\sqrt{-2\kappa }\zeta \right)))\hfill \\ \hfill \times exp\left(\iota ({\theta }_{1}x+\mu y+(\kappa {\lambda }^2-{\theta }_1^2)\frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right).\end{array}$$

(61)

$$\begin{array}{c}W(x,y,t)=\mp \frac{c\sqrt{2\left(1-2{\theta }_1\right)}\lambda }{\left(c+{\theta }_2+1\right)}(\pm \sqrt{-\frac{\kappa }{2}}({coth}_{ab}\left(\sqrt{-2\kappa }\zeta \right)\pm \sqrt{ab}csc{h}_{ab}\left(\sqrt{-2\kappa }\zeta \right)))\hfill \\ \hfill \times exp\left(\iota ({\theta }_{1}x+\mu y+(\kappa {\lambda }^2-{\theta }_1^2)\frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right).\end{array}$$

(62)

$$Z(x,y,t)={\lambda }^2\kappa {({coth}_{ab}\left(\sqrt{-2\kappa }\zeta \right)\pm \sqrt{ab}csc{h}_{ab}\left(\sqrt{-2\kappa }\zeta \right))}^2.$$

(63)

$$\begin{array}{c}U(x,y,t)=\mp \frac{\sqrt{2\left(1-2{\theta }_1\right)}\lambda }{\left(c+{\theta }_2+1\right)}(\pm \sqrt{-\frac{\kappa }{8}}({tanh}_{ab}\left(\sqrt{-\frac{\kappa }{8}}\zeta \right)+{coth}_{ab}\left(\sqrt{-\frac{\kappa }{8}}\zeta \right)))\hfill \\ \hfill \times exp\left(\iota ({\theta }_{1}x+\mu y+(\kappa {\lambda }^2-{\theta }_1^2)\frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right).\end{array}$$

(64)

$$\begin{array}{c}V(x,y,t)=\mp \frac{{\theta }_2\sqrt{2\left(1-2{\theta }_1\right)}\lambda }{\left(c+{\theta }_2+1\right)}(\pm \sqrt{-\frac{\kappa }{8}}({tanh}_{ab}\left(\sqrt{-\frac{\kappa }{8}}\zeta \right)+{coth}_{ab}\left(\sqrt{-\frac{\kappa }{8}}\zeta \right)))\hfill \\ \hfill \times exp\left(\iota ({\theta }_{1}x+\mu y+(\kappa {\lambda }^2-{\theta }_1^2)\frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right).\end{array}$$

(65)

$$\begin{array}{c}W(x,y,t)=\mp \frac{c\sqrt{2\left(1-2{\theta }_1\right)}\lambda }{\left(c+{\theta }_2+1\right)}(\pm \sqrt{-\frac{\kappa }{8}}({tanh}_{ab}\left(\sqrt{-\frac{\kappa }{8}}\zeta \right)+{coth}_{ab}\left(\sqrt{-\frac{\kappa }{8}}\zeta \right)))\hfill \\ \hfill \times exp\left(\iota ({\theta }_{1}x+\mu y+(\kappa {\lambda }^2-{\theta }_1^2)\frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right).\end{array}$$

(66)

$$Z(x,y,t)=\frac{\kappa {\lambda }^2}{4}{({tanh}_{ab}\left(\sqrt{-\frac{\kappa }{8}}\zeta \right)+{coth}_{ab}\left(\sqrt{-\frac{\kappa }{8}}\zeta \right))}^2.$$

(67)

Case 4:

$$U(x,y,t)=\mp \frac{\sqrt{2\left(1-2{\theta }_1\right)}\lambda }{\left(c+{\theta }_2+1\right)}(\pm \sqrt{\frac{\kappa }{2}}{tan}_{ab}\left(\sqrt{\frac{\kappa }{2}}\zeta \right))\times exp\left(\iota ({\theta }_{1}x+\mu y+(\kappa {\lambda }^2-{\theta }_1^2)\frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right).$$

(68)

$$V(x,y,t)=\mp \frac{{\theta }_2\sqrt{2\left(1-2{\theta }_1\right)}\lambda }{\left(c+{\theta }_2+1\right)}(\pm \sqrt{\frac{\kappa }{2}}{tan}_{ab}\left(\sqrt{\frac{\kappa }{2}}\zeta \right))\times exp\left(\iota ({\theta }_{1}x+\mu y+(\kappa {\lambda }^2-{\theta }_1^2)\frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right).$$

(69)

$$W(x,y,t)=\mp \frac{c\sqrt{2\left(1-2{\theta }_1\right)}\lambda }{\left(c+{\theta }_2+1\right)}(\pm \sqrt{\frac{\kappa }{2}}{tan}_{ab}\left(\sqrt{\frac{\kappa }{2}}\zeta \right))\times exp\left(\iota ({\theta }_{1}x+\mu y+(\kappa {\lambda }^2-{\theta }_1^2)\frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right).$$

(70)

$$Z(x,y,t)=-{\lambda }^2\kappa {\left({tan}_{ab}\left(\sqrt{\frac{\kappa }{2}}\zeta \right)\right)}^2.$$

(71)

$$U(x,y,t)=\mp \frac{\sqrt{2\left(1-2{\theta }_1\right)}\lambda }{\left(c+{\theta }_2+1\right)}(\pm \sqrt{\frac{\kappa }{2}}{cot}_{ab}\left(\sqrt{\frac{\kappa }{2}}\zeta \right))\times exp\left(\iota ({\theta }_{1}x+\mu y+(\kappa {\lambda }^2-{\theta }_1^2)\frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right).$$

(72)

$$V(x,y,t)=\mp \frac{{\theta }_2\sqrt{2\left(1-2{\theta }_1\right)}\lambda }{\left(c+{\theta }_2+1\right)}(\pm \sqrt{\frac{\kappa }{2}}{cot}_{ab}\left(\sqrt{\frac{\kappa }{2}}\zeta \right))\times exp\left(\iota ({\theta }_{1}x+\mu y+(\kappa {\lambda }^2-{\theta }_1^2)\frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right).$$

(73)

$$W(x,y,t)=\mp \frac{c\sqrt{2\left(1-2{\theta }_1\right)}\lambda }{\left(c+{\theta }_2+1\right)}(\pm \sqrt{\frac{\kappa }{2}}{cot}_{ab}\left(\sqrt{\frac{\kappa }{2}}\zeta \right))\times exp\left(\iota ({\theta }_{1}x+\mu y+(\kappa {\lambda }^2-{\theta }_1^2)\frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right).$$

(74)

$$Z(x,y,t)=-{\lambda }^2\kappa {\left({cot}_{ab}\left(\sqrt{\frac{\kappa }{2}}\zeta \right)\right)}^2.$$

(75)

$$\begin{array}{c}U(x,y,t)=\mp \frac{\sqrt{2\left(1-2{\theta }_1\right)}\lambda }{\left(c+{\theta }_2+1\right)}(\pm \sqrt{\frac{\kappa }{2}}({tan}_{ab}\left(\sqrt{2\kappa }\zeta \right)\pm \sqrt{ab}{sec}_{ab}\left(\sqrt{2\kappa }\zeta \right)))\hfill \\ \hfill \times exp\left(\iota ({\theta }_{1}x+\mu y+(\kappa {\lambda }^2-{\theta }_1^2)\frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right).\end{array}$$

(76)

$$\begin{array}{c}V(x,y,t)=\mp \frac{{\theta }_2\sqrt{2\left(1-2{\theta }_1\right)}\lambda }{\left(c+{\theta }_2+1\right)}(\pm \sqrt{\frac{\kappa }{2}}({tan}_{ab}\left(\sqrt{2\kappa }\zeta \right)\pm \sqrt{ab}{sec}_{ab}\left(\sqrt{2\kappa }\zeta \right)))\hfill \\ \hfill \times exp\left(\iota ({\theta }_{1}x+\mu y+(\kappa {\lambda }^2-{\theta }_1^2)\frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right).\end{array}$$

(77)

$$\begin{array}{c}W(x,y,t)=\mp \frac{c\sqrt{2\left(1-2{\theta }_1\right)}\lambda }{\left(c+{\theta }_2+1\right)}(\pm \sqrt{\frac{\kappa }{2}}({tan}_{ab}\left(\sqrt{2\kappa }\zeta \right)\pm \sqrt{ab}{sec}_{ab}\left(\sqrt{2\kappa }\zeta \right)))\hfill \\ \hfill \times exp\left(\iota ({\theta }_{1}x+\mu y+(\kappa {\lambda }^2-{\theta }_1^2)\frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right).\end{array}$$

(78)

$$Z(x,y,t)=-{\lambda }^2\kappa {({tan}_{ab}\left(\sqrt{2\kappa }\zeta \right)\pm \sqrt{ab}{sec}_{ab}\left(\sqrt{2\kappa }\zeta \right))}^2.$$

(79)

$$\begin{array}{c}U(x,y,t)=\mp \frac{\sqrt{2\left(1-2{\theta }_1\right)}\lambda }{\left(c+{\theta }_2+1\right)}(\pm \sqrt{\frac{\kappa }{2}}({cot}_{ab}\left(\sqrt{2\kappa }\zeta \right)\pm \sqrt{ab}{csc}_{ab}\left(\sqrt{2\kappa }\zeta \right)))\hfill \\ \hfill \times exp\left(\iota ({\theta }_{1}x+\mu y+(\kappa {\lambda }^2-{\theta }_1^2)\frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right).\end{array}$$

(80)

$$\begin{array}{c}V(x,y,t)=\mp \frac{{\theta }_2\sqrt{2\left(1-2{\theta }_1\right)}\lambda }{\left(c+{\theta }_2+1\right)}(\pm \sqrt{\frac{\kappa }{2}}({cot}_{ab}\left(\sqrt{2\kappa }\zeta \right)\pm \sqrt{ab}{csc}_{ab}\left(\sqrt{2\kappa }\zeta \right)))\hfill \\ \hfill \times exp\left(\iota ({\theta }_{1}x+\mu y+(\kappa {\lambda }^2-{\theta }_1^2)\frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right).\end{array}$$

(81)

$$\begin{array}{c}W(x,y,t)=\mp \frac{c\sqrt{2\left(1-2{\theta }_1\right)}\lambda }{\left(c+{\theta }_2+1\right)}(\pm \sqrt{\frac{\kappa }{2}}({cot}_{ab}\left(\sqrt{2\kappa }\zeta \right)\pm \sqrt{ab}{csc}_{ab}\left(\sqrt{2\kappa }\zeta \right)))\hfill \\ \hfill \times exp\left(\iota ({\theta }_{1}x+\mu y+(\kappa {\lambda }^2-{\theta }_1^2)\frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right).\end{array}$$

(82)

$$Z(x,y,t)=-{\lambda }^2\kappa {({cot}_{ab}\left(\sqrt{2\kappa }\zeta \right)\pm \sqrt{ab}{csc}_{ab}\left(\sqrt{2\kappa }\zeta \right))}^2.$$

(83)

$$\begin{array}{c}U(x,y,t)=\mp \frac{\sqrt{2\left(1-2{\theta }_1\right)}\lambda }{\left(c+{\theta }_2+1\right)}(\pm \sqrt{\frac{\kappa }{8}}({tan}_{ab}\left(\sqrt{\frac{\kappa }{8}}\zeta \right)+{cot}_{ab}\left(\sqrt{\frac{\kappa }{8}}\zeta \right)))\hfill \\ \hfill \times exp\left(\iota ({\theta }_{1}x+\mu y+(\kappa {\lambda }^2-{\theta }_1^2)\frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right).\end{array}$$

(84)

$$\begin{array}{c}V(x,y,t)=\mp \frac{{\theta }_2\sqrt{2\left(1-2{\theta }_1\right)}\lambda }{\left(c+{\theta }_2+1\right)}(\pm \sqrt{\frac{\kappa }{8}}({tan}_{ab}\left(\sqrt{\frac{\kappa }{8}}\zeta \right)+{cot}_{ab}\left(\sqrt{\frac{\kappa }{8}}\zeta \right)))\hfill \\ \hfill \times exp\left(\iota ({\theta }_{1}x+\mu y+(\kappa {\lambda }^2-{\theta }_1^2)\frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right).\end{array}$$

(85)

$$\begin{array}{c}W(x,y,t)=\mp \frac{c\sqrt{2(1-2{\theta }_1)}\lambda }{(c+{\theta }_2+1)}(\pm \sqrt{\frac{\kappa }{8}}({tan}_{ab}\left(\sqrt{\frac{\kappa }{8}}\zeta \right)+{cot}_{ab}\left(\sqrt{\frac{\kappa }{8}}\zeta \right)))\hfill \\ \hfill \times exp\left(\iota ({\theta }_{1}x+\mu y+(\kappa {\lambda }^2-{\theta }_1^2)\frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha })\right)+\vartheta \left)\right).\end{array}$$

(86)

$$Z(x,y,t)=-\frac{{\lambda }^2\kappa }{4}{({tan}_{ab}\left(\sqrt{\frac{\kappa }{8}}\zeta \right)+{cot}_{ab}\left(\sqrt{\frac{\kappa }{8}}\zeta \right))}^2.$$

(87)

where $\zeta =\lambda (x+y-2{\theta }_1\frac{\Gamma (\gamma +1)}{\alpha }{t}^{\alpha }).$

5. Graphical Representation of Solutions

In this section we present the graphical behavior of the solutions in the 3D, contour and 2D surfaces. The effect of the fractional order derivatives is also shown by the following graphs that are not given in [30]. The fractional order derivative gives more appropriate results than the integer order derivatives.

Bright soliton solution: we gave graphs for the solution shown in Equation (32) in Figure 1. Figure 1a 3D, Figure 1b in contour and Figure 1c 2D with different values of $t=-1$ is represented by the red line, $t=0$ is represented by the black line and $t=1$ is represented by the yellow with ${\theta }_1=0.003,{\theta }_2=0.01,\lambda =0.5,\beta =0.5,c=1,\gamma =1,$ $\mu =-1,\vartheta =0,$ $\kappa =0.1,y=0,\alpha =0.9,$ $-10<t<10$ and $-10<x<10$.

Periodic wave solution: we gave graphs for solution (40) in Figure 2. Figure 2a 3D, Figure 2b in contour and Figure 2c 2D with different values of $t=-1$ is represented by the red line, $t=0$ is represented by the black line and $t=1$ is represented by the yellow with ${\theta }_1=0.003,{\theta }_2=0.01,\lambda =0.5,\beta =0.5,c=1,\gamma =1,\mu =-1,\vartheta =0,\kappa =-0.1,y=0,\alpha =0.9,$ $-10<t<10$ and $-10<x<10$.

Dark soliton solution: we gave graphs for solution (48) in Figure 3. Figure 3a 3D, Figure 3b in contour and Figure 3c 2D with different values of $t=-1$ is represented by the red line, $t=0$ is represented by the black line and $t=1$ is represented by the green with ${\theta }_1=0.3,{\theta }_2=0.5,\lambda =0.5,\beta =0.05,c=1,\gamma =1,\mu =0.1,\vartheta =0,\kappa =-0.1,y=0,\alpha =0.9,$ $-10<t<10$ and $-10<x<10$.

Periodic wave solution: we gave graphs for solution (68) in Figure 4. Figure 4a 3D, Figure 4b in contour and Figure 4c 2D with different values of $t=-1$ is represented by the red line, $t=0$ is represented by the black line and $t=1$ is represented by the green with ${\theta }_1=0.03,{\theta }_2=0.001,\lambda =0.5,\beta =0.05,c=1,\gamma =1,\mu =-0.1,\vartheta =0,\kappa =0.1,y=0,\alpha =0.9,$ $-10<t<10$ and $-10<x<10$.

6. Conclusions

Overall, this paper contributes to our understanding of the truncated M-fractional complex three coupled Maccari’s system and provides a useful technique for handling nonlinear partial differential equations. This paper describes the successful application of the Sardar sub-equation method to obtain new optical wave solutions for the truncated M-fractional complex three coupled Maccari’s system. The obtained solutions are useful for future studies of the concerned model and provide insights into the behavior of optical waves in complex media. As far as we know, for the first time, we used this method for this model so that all the solutions are new and cannot be found in the literature to our best knowledge. The solutions were verified using Mathematica software and were also described graphically using 2-dimensional, 3-dimensional, and contour plots through Maple software. The obtained results in this work have some interesting applications in some plasma environments, such as new earth’s ionoshere zone. The Sardar sub-equation method is shown to be a simple, fruitful and reliable technique for handling nonlinear partial differential equations. We will extend the proposed method for fractional models in a future work.