Constructions of the optical solitons and other solitons to the conformable fractional Zakharov–Kuznetsov equation with power law nonlinearity

Abstract

The current research manifests kink wave answers, mixed singular optical solitons, the mixed dark-bright lump answer, the mixed dark-bright periodic wave answer, and periodic wave answers to the conformable fractional ZK model, including power law nonlinearity by plugging the revised $\left(G^{\prime }/G\right)$-expansion process. The constraint requirements for the occurrence of substantial solitons are provided. Under the selection of proper values of a, b, n, t, λ, μ and α, the 2D and 3D pictures to a few of the recorded answers are sketched. From our obtained solutions, we might decide that the investigated procedure is hugely muscular, sincere, and essential in rendering various new soliton solutions of distinct nonlinear conformable fractional evolution equations and accordingly, we shall bring it up in our future investigations.

Keywords:

• Modified $\left(G^{\prime }/G\right)$-expansion method
• $(3+1)$-dimensional conformable fractional Zakharov–Kuznetsov equation with power law nonlinearity
• conformable fractional derivative
• travelling wave solutions
• soliton solutions

2010 AMS (MOS) Subject Classifications:

• 35C07
• 35C08
• 35Q53

1. Introduction

Nowadays, the study of closed-form wave answers the nonlinear wave equation improvement ahead of its measurement. Besides, a conformable derivative nonlinear ordinary differential equation has been converted mostly to make their application in the communication scheme. Accordingly, the problem of securing the closed-form wave answers of the conformable nonlinear model attracts a lot of consideration. Nonetheless, numerous applications have been executed in the economy, quantum field theory, optical fibres, plasma physics, fluid mechanics, mathematical physics, biology, geochemistry, to mention a few. Thus, various mathematical methods have been improved to answer them, such as Lie symmetry analysis [1], the auxiliary equation method [2], the FRDTM [3], the tan$\phi \left(\xi \right)/2)$-expansion method [4], Tanh method [5], the Riccati–Bernoulli sub-ODE method [6], the exp$-\phi \left(\xi \right)$-expansion method [7–9], extended trial equation method [10], Fractional Fan sub-equation method [11], new generalized $\left(G^{\prime }/G\right)$-expansion method [12–15], exponential rational function method [16], $\left(G^{\prime }/G\right)$-expansion method [17–19], modified extended tanh method [20], improved $\left(G^{\prime }/G\right)$-expansion method [21], differential transform method [22], the Painleve analysis [23], fractional homotopy method [24], Truncation method [25], Semi-Inverse variational principle [26], the Feng's first integral method [27], the unified method [28], $\left(\frac{G^{\prime }}{G^2}\right)$-expansion method [29], singular manifold method [30], singular manifold method [31], homotopy perturbation transform method [32], Collocation method [33], separation of variables method [34], Lagrange multiplier method [35], fractional Adams– Bashforth–Moulton method [36], Chebyshev wavelet method [37], Jacobi elliptic function method [38], Kudryashov method [39], exp-function method [40], sub-equation method [41], space spectral time-fractional Adam–Bashforth– Moulton method [42], exp-function method [43], the phase field method [44], fractional homotopy analysis transform method [45] and many more [57].

In this paper, the modified $\left(G^{\prime }/G\right)$-expansion process will obtain soliton answers of the following conformable fractional ZK model, including power law nonlinearity [28,46]. We are considering that using above model [28,46], (1) $\begin{array}{r}\frac{{\mathrm{\partial }}^{\alpha }u}{\mathrm{\partial }{t}^{\alpha }}+a{u}^n\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+b\frac{\mathrm{\partial }}{\mathrm{\partial }x}\left(\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{z}^2}\right)=0,\end{array}$(1) where a and b are constants, n is the power law nonlinearity parameter, $\frac{{\mathrm{\partial }}^{\alpha }u}{\mathrm{\partial }{t}^{\alpha }}$ is the evolution term, $u^n\frac{\mathrm{\partial }u}{\mathrm{\partial }x}$ is the nonlinearity and $\frac{\mathrm{\partial }}{\mathrm{\partial }x}\left(\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{z}^2}\right)$ is dispersion. Solitons are the outcome of a rule between dispersion and nonlinearity. The above model typically appears in the analysis of plasma physics. Matebese et al. [46] noted the above model through the three analytical methods, such as the $\left(G^{\prime }/G\right)$-expansion method, the extended tanh function method, and the ansatz method. Besides, Aminikhah et al. [47] attempted to discover the same model when $\alpha =1$ plugging the functional variable method. The particular case where n = 1 and $\alpha =1$ provides the $(3+1)$-dimensional Zakharov–Kuznetsov equation. Section 2 provides a few fundamental aspects and the knowledge of the conformable fractional calculus theory. The novel closed-form wave answers of the recommended model are discussed in Section 4. The last section conveys the conclusions and future tasks.

2. Conformable fractional derivative

This section gives a few essential characteristics and the knowledge of the conformable fractional calculus theory which can be seen in  [48–55].

Definition 2.1

we consider that $f:(0,\mathrm{\infty })\to \mathbb{R}$, therefore, the conformable fractional derivative of f of order α is expressed as (2) $\begin{array}{r}{}_t{D}^{\alpha }f\left(t\right)=\underset{ϵ\to 0}{lim}\frac{f(t+ϵt^{1-\alpha })-f\left(t\right)}{ϵ},\end{array}$(2) for all t>0, $\alpha \in (0,1).$

If f is α-differentiable in some $(0,a),a>0$ and $\underset{t\to 0^{+}}{lim}{(}_t{D}^{\alpha }f\left(t\right))$ exists, then in accordance with the definition, we obtain (3) $\begin{array}{r}{}_t{D}^{\alpha }\left(f\right)\left(0\right)=\underset{t\to 0^{+}}{lim}{(}_t{D}^{\alpha }f\left(t\right)).\end{array}$(3)

The novel definition convinces the characteristics manifested in the following theorem.

Theorem 2.2

We consider that $\alpha \in (0,1]$, and f, g be α-differentiable at a point t, such that

• ${}_t{D}^{\alpha }(af+bg)=a{(}_t{D}^{\alpha }f)+b{(}_t{D}^{\alpha }g),\mathrm{\forall }a,b\in \mathbb{R}$.

• ${}_t{D}^{\alpha }\left(t^{\mu }\right)=\mu t^{\mu -\alpha },forall\mu \in \mathbb{R}.$

• ${}_t{D}^{\alpha }(f+g)=f{(}_t{D}^{\alpha }g)+g{(}_t{D}^{\alpha }f).$

• ${}_t{D}^{\alpha }\left(\frac{f}{g}\right)=\frac{g{(}_t{D}^{\alpha }f)-f{(}_t{D}^{\alpha }g)}{g^2}.$

In addition, if f is differentiable, then ${}_t{D}^{\alpha }f\left(t\right)=t^{1-\alpha }\frac{df}{dt}.$

Abdeljawad [50] established the chain rule under the conformable fractional derivatives.

Theorem 2.3

We consider that $f:(0,\mathrm{\infty })\to \mathbb{R}$ is a function, for example, f is differentiable and also α-differentiable. Let us consider that g is a function defined in the range of f and also differentiable. Then one has the following rule: (4) $\begin{array}{r}{}_t{D}^{\alpha }f\left(fog\right)\left(t\right)=t^{1-\alpha }{g}^{\prime }\left(t\right)f^{\prime }\left(g\right(t\left)\right).\end{array}$(4)

2.1. The fractional complex transformation

This section implements the complex fractional transformation for the fractional-order PDE: (5) $\begin{array}{r}L\left(u,\frac{{\mathrm{\partial }}^{\alpha }u}{\mathrm{\partial }{t}^{\alpha }},\frac{\mathrm{\partial }u}{\mathrm{\partial }x},\frac{\mathrm{\partial }u}{\mathrm{\partial }y},\frac{{\mathrm{\partial }}^{2\alpha }u}{\mathrm{\partial }{t}^{2\alpha }},\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{t}^2}\right)=0,t\ge 0,0<\alpha \le 1,\end{array}$(5) where $L=u(x,y,t)$. For Equation (5(5) $\begin{array}{r}L\left(u,\frac{{\mathrm{\partial }}^{\alpha }u}{\mathrm{\partial }{t}^{\alpha }},\frac{\mathrm{\partial }u}{\mathrm{\partial }x},\frac{\mathrm{\partial }u}{\mathrm{\partial }y},\frac{{\mathrm{\partial }}^{2\alpha }u}{\mathrm{\partial }{t}^{2\alpha }},\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{t}^2}\right)=0,t\ge 0,0<\alpha \le 1,\end{array}$(5) ), let $u=u\left(\xi \right)=u(x,y,z,t),\xi =x+y+z-V\frac{t^{\alpha }}{\alpha }$ and obtain $\frac{{\mathrm{\partial }}^{\alpha }}{\mathrm{\partial }{t}^{\alpha }}=-V\frac{\mathrm{\partial }}{\mathrm{\partial }\xi }$, $\frac{\mathrm{\partial }}{\mathrm{\partial }x}=\frac{\mathrm{\partial }}{\mathrm{\partial }\xi }$, $\frac{\mathrm{\partial }}{\mathrm{\partial }y}=\frac{\mathrm{\partial }}{\mathrm{\partial }\xi }$, $\frac{{\mathrm{\partial }}^{2\alpha }}{\mathrm{\partial }{t}^{2\alpha }}=V^2\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\xi }^2}$ $\dots$, where V stands for the travelling wave speed. Then, Equation (5(5) $\begin{array}{r}L\left(u,\frac{{\mathrm{\partial }}^{\alpha }u}{\mathrm{\partial }{t}^{\alpha }},\frac{\mathrm{\partial }u}{\mathrm{\partial }x},\frac{\mathrm{\partial }u}{\mathrm{\partial }y},\frac{{\mathrm{\partial }}^{2\alpha }u}{\mathrm{\partial }{t}^{2\alpha }},\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{t}^2}\right)=0,t\ge 0,0<\alpha \le 1,\end{array}$(5) ) becomes (6) $\begin{array}{r}Q\left(u,\frac{\mathrm{\partial }u}{\mathrm{\partial }\xi },\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{\xi }^2},\dots \right)=0.\end{array}$(6)

3. New soliton solutions for the $(3+1)$-dimensional conformable fractional Zakharov–Kuznetsov equation with power law nonlinearity

To represent the idea of the modified $\left(G^{\prime }/G\right)$-expansion method [56], we apply the way on the (3 + 1)-dimensional conformable fractional Zakharov–Kuznetsov equation with power law nonlinearity [28,46]. Let us consider that (7) $\begin{array}{r}\frac{{\mathrm{\partial }}^{\alpha }u}{\mathrm{\partial }{t}^{\alpha }}+a{u}^n\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+b\frac{\mathrm{\partial }}{\mathrm{\partial }x}\left(\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{z}^2}\right)=0.\end{array}$(7) Let (8) $\begin{array}{r}u=u\left(\xi \right)=u(x,y,z,t),\xi =x+y+z-V\frac{t^{\alpha }}{\alpha }.\end{array}$(8) From Equations (7(7) $\begin{array}{r}\frac{{\mathrm{\partial }}^{\alpha }u}{\mathrm{\partial }{t}^{\alpha }}+a{u}^n\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+b\frac{\mathrm{\partial }}{\mathrm{\partial }x}\left(\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{z}^2}\right)=0.\end{array}$(7) ) and (8(8) $\begin{array}{r}u=u\left(\xi \right)=u(x,y,z,t),\xi =x+y+z-V\frac{t^{\alpha }}{\alpha }.\end{array}$(8) ), we achieve: (9) $\begin{array}{r}-V\frac{\mathrm{\partial }u}{\mathrm{\partial }\xi }+a{u}^n\frac{\mathrm{\partial }u}{\mathrm{\partial }\xi }+b\frac{\mathrm{\partial }}{\mathrm{\partial }\xi }\left(\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{\xi }^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{\xi }^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{\xi }^2}\right)=0.\end{array}$(9) Integrating the above equation, we have (10) $\begin{array}{r}-Vu+\frac{a}{n+1}{u}^{n+1}+3b\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{\xi }^2}=0.\end{array}$(10) Plugging $u=v^{\frac{1}{n}}$, Equation (10(10) $\begin{array}{r}-Vu+\frac{a}{n+1}{u}^{n+1}+3b\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{\xi }^2}=0.\end{array}$(10) ) becomes (11) $\begin{array}{rl}& 3bn(n+1)v\frac{{\mathrm{\partial }}^{2}v}{\mathrm{\partial }{\xi }^2}+3b(n^2-1){\left(\frac{\mathrm{\partial }v}{\mathrm{\partial }\xi }\right)}^2\\ & -n^2(1+n)V{v}^2+a{n}^2{v}^3=0,n\ne 0,\pm 1.\end{array}$(11) In accordance with the modified $\left(G^{\prime }/G\right)$-expansion method [56], by plugging homogeneous balance rule between $v\frac{{\mathrm{\partial }}^{2}v}{\mathrm{\partial }{\xi }^2}$ and $v^3$ from the equation (11(11) $\begin{array}{rl}& 3bn(n+1)v\frac{{\mathrm{\partial }}^{2}v}{\mathrm{\partial }{\xi }^2}+3b(n^2-1){\left(\frac{\mathrm{\partial }v}{\mathrm{\partial }\xi }\right)}^2\\ & -n^2(1+n)V{v}^2+a{n}^2{v}^3=0,n\ne 0,\pm 1.\end{array}$(11) ), it yields M = 2. Therefore, Equation (11(11) $\begin{array}{rl}& 3bn(n+1)v\frac{{\mathrm{\partial }}^{2}v}{\mathrm{\partial }{\xi }^2}+3b(n^2-1){\left(\frac{\mathrm{\partial }v}{\mathrm{\partial }\xi }\right)}^2\\ & -n^2(1+n)V{v}^2+a{n}^2{v}^3=0,n\ne 0,\pm 1.\end{array}$(11) ) reduces to (12) $\begin{array}{r}v\left(\xi \right)=A_2{F}^2+A_{1}F+A_0+A_{-1}{F}^{-1}+A_{-2}{F}^{-2},\end{array}$(12) where $A_0$, $A_1$, $A_2$, $A_{-1}$ and $A_{-2}$ are constants.

Plugging Equation (12(12) $\begin{array}{r}v\left(\xi \right)=A_2{F}^2+A_{1}F+A_0+A_{-1}{F}^{-1}+A_{-2}{F}^{-2},\end{array}$(12) ) into Equation (11(11) $\begin{array}{rl}& 3bn(n+1)v\frac{{\mathrm{\partial }}^{2}v}{\mathrm{\partial }{\xi }^2}+3b(n^2-1){\left(\frac{\mathrm{\partial }v}{\mathrm{\partial }\xi }\right)}^2\\ & -n^2(1+n)V{v}^2+a{n}^2{v}^3=0,n\ne 0,\pm 1.\end{array}$(11) ) and then calculating each coefficients of $F^i(\pm 1,\pm 2,\dots \dots .,\pm m)$ to zeros, we obtain the following relations:

• Phase 1: Let $V=-\frac{3b({\lambda }^{2}n-4\mu n-n+1)}{n^2},A_0=\frac{3b({\lambda }^2-4\mu )(n+1)}{2an},A_1=0,A_2=0,A_{-1}=0$, and $A_{-2}=-\frac{9b({\lambda }^4-8{\lambda }^2\mu +16{\mu }^2)(n+1)}{8an}$ and use the values of the Phase 1 into Equation (12(12) $\begin{array}{r}v\left(\xi \right)=A_2{F}^2+A_{1}F+A_0+A_{-1}{F}^{-1}+A_{-2}{F}^{-2},\end{array}$(12) ), we obtain: $\begin{array}{rl}{u}_{11}(x,t)& =\left[\frac{-3b({\lambda }^2-4\mu )(n+1)}{2an}\right.\\ & \times \left\{2+3csc{h}^2\left(\frac{\sqrt{{\lambda }^2-4\mu }}{2}(x+y+z\right.\right.\\ & {\left.\left.\left.-\frac{3b({\lambda }^{2}n-4\mu n-n+1))}{n^2}\frac{t^{\alpha }}{\alpha }\right)\right\}\right]}^{\frac{1}{n}}.\\ u_{12}(x,t)& =\left[\frac{-3b({\lambda }^2-4\mu )(n+1)}{2an}\right.\\ & \times \left\{2+3sec{h}^2\left(\frac{\sqrt{{\lambda }^2-4\mu }}{2}(x+y+z\right.\right.\\ & {\left.\left.\left.-\frac{3b({\lambda }^{2}n-4\mu n-n+1))}{n^2}\frac{t^{\alpha }}{\alpha }\right)\right\}\right]}^{\frac{1}{n}}.\\ u_{13}(x,t)& =\left[\frac{-3b({\lambda }^2-4\mu )(n+1)}{2an}\right.\\ & \times \left\{2-3cs{c}^2\left(\frac{\sqrt{4\mu -{\lambda }^2}}{2}(x+y+z\right.\right.\\ & {\left.\left.\left.-\frac{3b({\lambda }^{2}n-4\mu n-n+1))}{n^2}\frac{t^{\alpha }}{\alpha }\right)\right\}\right]}^{\frac{1}{n}}.\\ u_{14}(x,t)& =\left[\frac{-3b({\lambda }^2-4\mu )(n+1)}{2an}\right.\\ & \times \left\{2-3se{c}^2\left(\frac{\sqrt{4\mu -{\lambda }^2}}{2}(x+y+z\right.\right.\\ & {\left.\left.\left.-\frac{3b({\lambda }^{2}n-4\mu n-n+1))}{n^2}\frac{t^{\alpha }}{\alpha }\right)\right\}\right]}^{\frac{1}{n}}.\\ u_{15}(x,t)& =\left[\frac{3b({\lambda }^2-4\mu )(n+1)}{2an}\right.\\ & -\frac{9b({\lambda }^4-8{\lambda }^2\mu +16{\mu }^2)(n+1)}{8an}\\ & {\left.\frac{1}{(x+y+z-\frac{3b({\lambda }^{2}n-4\mu n-n+1)}{n^2}\frac{t^{\alpha }}{\alpha }{)}^2}\right]}^{\frac{1}{n}}.\end{array}$

• Phase 2: Let $V=\frac{3b({\lambda }^{2}n-4\mu n+n-1)}{n^2},A_0=\frac{9b({\lambda }^2-4\mu )(n+1)}{2an},A_1=0,A_2=0,A_{-1}=0$, and $A_{-2}=-\frac{9b({\lambda }^4-8{\lambda }^2\mu +16{\mu }^2)(n+1)}{8an}$ and plug the values of the Phase 2 into Equation (12(12) $\begin{array}{r}v\left(\xi \right)=A_2{F}^2+A_{1}F+A_0+A_{-1}{F}^{-1}+A_{-2}{F}^{-2},\end{array}$(12) ), we get: $\begin{array}{rl}{u}_{21}(x,t)& =\left[\frac{9b({\lambda }^2-4\mu )(n+1)}{2an}\right.\\ & \times csc{h}^2\left(\frac{\sqrt{{\lambda }^2-4\mu }}{2}(x+y+z\right.\\ & {\left.\left.\left.-\frac{3b({\lambda }^{2}n-4\mu n-n+1)}{n^2}\frac{t^{\alpha }}{\alpha }\right)\right)\right]}^{\frac{1}{n}}.\\ u_{22}(x,t)& =\left[\frac{9b({\lambda }^2-4\mu )(n+1)}{2an}sec{h}^2\right.\\ & \times \left(\frac{\sqrt{{\lambda }^2-4\mu }}{2}(x+y+z\right.\\ & {\left.\left.\left.-\frac{3b({\lambda }^{2}n-4\mu n-n+1)}{n^2}\frac{t^{\alpha }}{\alpha }\right)\right)\right]}^{\frac{1}{n}}.\\ u_{23}(x,t)& =\left[\frac{9b({\lambda }^2-4\mu )(n+1)}{2an}\right.\\ & \times cs{c}^2\left(\frac{\sqrt{4\mu -{\lambda }^2}}{2}(x+y+z\right.\\ & {\left.\left.\left.-\frac{3b({\lambda }^{2}n-4\mu n-n+1)}{n^2}\frac{t^{\alpha }}{\alpha }\right)\right)\right]}^{\frac{1}{n}}.\\ u_{24}(x,t)& =\left[\frac{9b({\lambda }^2-4\mu )(n+1)}{2an}\right.\\ & \times se{c}^2\left(\frac{\sqrt{4\mu -{\lambda }^2}}{2}(x+y+z\right.\\ & {\left.\left.\left.-\frac{3b({\lambda }^{2}n-4\mu n-n+1)}{n^2}\frac{t^{\alpha }}{\alpha }\right)\right)\right]}^{\frac{1}{n}}.\\ u_{25}(x,t)& =\left[\frac{3b({\lambda }^2-4\mu )(n+1)}{2an}\right.\\ & \times -\frac{9b({\lambda }^4-8{\lambda }^2\mu +16{\mu }^2)(n+1)}{8an}\\ & {\left.\frac{1}{{\left(x+y+z-\frac{3b({\lambda }^{2}n-4\mu n-n+1)}{n^2}\frac{t^{\alpha }}{\alpha }\right)}^2}\right]}^{\frac{1}{n}}.\end{array}$

• Phase 3: Let $V=\frac{3b({\lambda }^{2}n-4\mu n+n-1)}{n^2},A_0=\frac{9b({\lambda }^2-4\mu )(n+1)}{2an}{A}_1=0A_2=-\frac{18b(n+1)}{an}{A}_{-1}=0$, and $A_{-2}=0$ and plug the values of the Phase 3 into Equation (12(12) $\begin{array}{r}v\left(\xi \right)=A_2{F}^2+A_{1}F+A_0+A_{-1}{F}^{-1}+A_{-2}{F}^{-2},\end{array}$(12) ), we obtain: $\begin{array}{rl}& u_{31}(x,t)\\ & =\left[\frac{9b({\lambda }^2-4\mu )(n+1)}{2an}-\frac{18b(n+1)({\lambda }^2-4\mu )}{4an}\right.\\ & {\tanh }^2\left(\frac{\sqrt{{\lambda }^2-4\mu }}{2}\left(x+y+z\phantom{\frac{3b({\lambda }^{2}n-4\mu n+n-1)}{n^2}}\right.\right.\\ & -{\left.\left.\left.\frac{3b({\lambda }^{2}n-4\mu n+n-1)}{n^2}\frac{t^{\alpha }}{\alpha }\right)\right)\right]}^{\frac{1}{n}}.\\ & u_{32}(x,t)\\ & =\left[\frac{9b({\lambda }^2-4\mu )(n+1)}{2an}-\frac{18b(n+1)({\lambda }^2-4\mu )}{4an}\right.\\ & {\coth }^2\left(\frac{\sqrt{{\lambda }^2-4\mu }}{2}\left(x+y+z\phantom{\frac{3b({\lambda }^{2}n-4\mu n+n-1)}{n^2}}\right.\right.\\ & -{\left.\left.\left.\frac{3b({\lambda }^{2}n-4\mu n+n-1)}{n^2}\frac{t^{\alpha }}{\alpha }\right)\right)\right]}^{\frac{1}{n}}.\\ & u_{33}(x,t)\\ & =\left[\frac{9b({\lambda }^2-4\mu )(n+1)}{2an}-\frac{18b(n+1)({\lambda }^2-4\mu )}{4an}\right.\\ & {\tan }^2\left(\frac{\sqrt{{\lambda }^2-4\mu }}{2}\left(x+y+z\phantom{\frac{3b({\lambda }^{2}n-4\mu n+n-1)}{n^2}}\right.\right.\\ & -{\left.\left.\left.\frac{3b({\lambda }^{2}n-4\mu n+n-1)}{n^2}\frac{t^{\alpha }}{\alpha }\right)\right)\right]}^{\frac{1}{n}}.\\ & u_{34}(x,t)\\ & =\left[\frac{9b({\lambda }^2-4\mu )(n+1)}{2an}-\frac{18b(n+1)({\lambda }^2-4\mu )}{4an}\right.\\ & {\cot }^2\left(\frac{\sqrt{{\lambda }^2-4\mu }}{2}\left(x+y+z\phantom{\frac{3b({\lambda }^{2}n-4\mu n+n-1)}{n^2}}\right.\right.\\ & -{\left.\left.\left.\frac{3b({\lambda }^{2}n-4\mu n+n-1)}{n^2}\frac{t^{\alpha }}{\alpha }\right)\right)\right]}^{\frac{1}{n}}.\\ & u_{35}(x,t)\\ & =\left[\frac{9b({\lambda }^2-4\mu )(n+1)}{2an}-\frac{18b(n+1)({\lambda }^2-4\mu )}{4an}\right.\\ & {\left.\frac{1}{(x+y+z-\frac{3b({\lambda }^{2}n-4\mu n+n-1)}{n^2}\frac{t^{\alpha }}{\alpha }{)}^2}\right]}^{\frac{1}{n}}.\end{array}$

• Phase 4: Let $V=\frac{3b({\lambda }^{2}n-16\mu n+n-1)}{n^2},A_0=\frac{9b({\lambda }^2-4\mu )(n+1)}{an},A_1=0,A_2=-\frac{18b(n+1)}{an},A_{-1}=0$, and $A_{-2}=-\frac{9b({\lambda }^4-8{\lambda }^2\mu +16{\mu }^2)(n+1)}{8an}.$

  Similarly, we can provide new five soliton answers to the case mentioned above of the studied model, which are omitted for assistance.

• Phase 5: Let $V=-\frac{3b({\lambda }^{2}n-4\mu n-n+1)}{n^2},A_0=\frac{3b({\lambda }^2-4\mu )(n+1)}{2an},A_1=0,A_2=-\frac{18b(n+1)}{an},A_{-1}=0$, and $A_{-2}=0.$

  Similarly, we can provide new five soliton answers to the case mentioned above of the studied model, which are omitted for assistance.

• Phase 6: Let $V=-\frac{3b({\lambda }^{2}n-16\mu n-n+1)}{n^2},A_0=-\frac{3b({\lambda }^2-4\mu )(n+1)}{an}{A}_1=0A_2=-\frac{18b(n+1)}{an}{A}_{-1}=0$, and $A_{-2}=-\frac{9b({\lambda }^4-8{\lambda }^2\mu +16{\mu }^2)(n+1)}{8an}.$

  Similarly, we can provide new five soliton answers to the case mentioned above of the studied model, which are omitted for assistance.

A graphical description is an essential tool for analysis and to communicate the answers to the problems lucidly. When working the calculation in daily life, we require the fundamental knowledge of securing the use of graphs. Hence, the graphical displays of some of the obtained solutions are demonstrated in Figures 1–6.

Figure 1. The $3D$ shape for $u_{12}$ under the constant values of $\lambda =3$, $\mu =1$, a = 0.1, b = 0.2, n = 2, $\alpha =0.35$, as well as t = 0.1 for the 2D shape. (a) Real $2D$ shape. (b) Imaginary $2D$ (c) shape Real $3D$ surface and Imaginary $3D$ surface.

Figure 2. The $3D$ shape for $u_{14}$ under the constant values of $\lambda =2$, $\mu =2$, a = 0.1, b = 0.2, n = 2, $\alpha =0.35$, as well as t = 0.1 for the 2D shape. (a) Real $2D$ shape. (b) Imaginary $2D$ shape. (c) Real $3D$ surface and Imaginary $3D$ surface.

Figure 3. The $3D$ shape for $u_{22}$ under the constant values of $\lambda =3$, $\mu =1$, a = 0.1, b = 0.2, n = 2, $\alpha =0.35$ as well as t = 0.1 for the 2D shape. (a) Real $2D$ shape. (b) Imaginary $2D$ shape. (c) Real $3D$ surface and (d) Imaginary $3D$ surface.

Figure 4. The $3D$ shape for $u_{23}$ under the constant values of $\lambda =2$, $\mu =2$, a = 0.1, b = 0.2, n = 2, $\alpha =0.35$ as well as t = 0.1 for the 2D shape. (a) Real $2D$ shape. (b) Imaginary $2D$ shape. (c) Real $3D$ surface and (d) Imaginary $3D$ surface.

Figure 5. The $3D$ shape for $u_{32}$ under the constant values of $\lambda =3$, $\mu =1$, a = 0.1, b = 0.2, n = 2, $\alpha =0.35$ as well as t = 0.1 for the 2D shape. (a) Real $2D$ shape. (b) Imaginary $2D$ shape. (c) Real $3D$ surface and (d) Imaginary $3D$ surface.

Figure 6. The $3D$ shape for $u_{35}$ under the constant values of $\lambda =2$, $\mu =1$, a = 0.1, b = 0.2, n = 2, $\alpha =0.35$ as well as t = 0.1 for the 2D shape. (a) Real $2D$ shape. (b) Imaginary $2D$ shape. (c) Real $3D$ surface and (d) Imaginary $3D$ surface.

4. Conclusions and future work

We have examined the modified $\left(G^{\prime }/G\right)$-expansion process for generating closed-form wave answers of the conformable fractional ZK equation, including power law nonlinearity. This scheme permits us to establish more many nonlinear conformable fractional evolution models in mathematical sciences by the examined models. As a consequence, various new types of closed-form wave answers are achieved. From our expert answers, we strongly conclude that the studied idea is great brawny, reliable, and crucial in executing numerous new closed-form wave answers of different nonlinear conformable FDEs. Finally, we mention that the process can be made to various distinct nonlinear conformable FDEs that occur in the area of mathematical physics.

Acknowledgments

The authors would like to acknowledge CAS-TWAS President's fellowship programme.

Disclosure statement

No potential conflict of interest was reported by the authors.