Exact Solitary Wave Solutions and Sensitivity Analysis of the Fractional (3+1)D KdV–ZK Equation

Abstract

The present paper examines a novel exact solution to nonlinear fractional partial differential equations (FDEs) through the Sardar sub-equation method (SSEM) coupled with Jumarie’s Modified Riemann–Liouville derivative (JMRLD). We take the (3+1)-dimensional space–time fractional modified Korteweg-de Vries (KdV) -Zakharov-Kuznetsov (ZK) equation as a case study, which describes some intricate phenomena of wave behavior in plasma physics and fluid dynamics. With the implementation of SSEM, we yield new solitary wave solutions and explicitly examine the role of the fractional-order parameter in the dynamics of the solutions. In addition, the sensitivity analysis of the results is conducted in the Galilean transformation in order to ensure that the obtained results are valid and have physical significance. Besides expanding the toolbox of analytical methods to address high-dimensional nonlinear FDEs, the proposed method helps to better understand how fractional-order dynamics affect the nonlinear wave phenomenon. The results are compared to known methods and a discussion about their possible applications and limitations is given. The results show the effectiveness and flexibility of SSEM along with JMRLD in forming new categories of exact solutions to nonlinear fractional models.

1. Introduction

Partial differential equations play both an effective and active role in modeling natural phenomena and complex phenomena in theoretical and applied physics. Partial differential equations (PDEs) that deal simultaneously with multiple functions and their partial derivatives with respect to different variables are basic tools for modeling most physical phenomena, such as fluid dynamics, heat transfer, and structural mechanics [1,2,3,4,5]. Depending on the properties of their solutions and the type of governing equations, PDEs are usually divided into three types: elliptic, parabolic, and hyperbolic. A classical example of a phenomenon described by an elliptic PDE is the temperature distribution of a steady-state object. Parabolic PDEs including the heat equation are termed as time-dependent phenomena with the effort of heat in a solid body. Hyperbolic partial differential equations similar to the wave equation describe the propagation of waves including electromagnetic and sound waves. Numerical methods are often employed to approximate solutions to PDEs because of their well-known difficulty in solving them. Solving PDEs and understanding characteristics such as their existence, uniqueness, and regularity are important aspects of the subject. PDEs are important tools in engineering for simulating, modeling, and optimizing complex schemes [6]. They are applied to problems in image processing, electro magnetics, fluid dynamics, heat transport, structural mechanics, and optimization.

Fractional differential equations (FDEs) are differential equations that have derivatives of any order. Examples of natural phenomena modeled by partial differential equations appear in chemical physics, viscoelasticity, control theory, fluid flow, rheology, diffusive transport, electrical networks, probability and statistics, and dynamical processes in self-similar and porous structures. Fractional derivatives decrease into two main categories: Riemann–Liouville and caputo fractional derivatives (CFD), depending on the derivative’s formulation and order. The Riemann–Liouville integral is a generalization of the integral operator for integer-order derivatives, which is used to form Riemann–Liouville fractional derivatives (RLFD). The Caputo derivative, defined by the fractional integral, is a generalization of the classical integer-order derivative and is commonly used in modeling fractional differential equations because of its ability to incorporate conventional initial conditions in physical systems. The decision between the two forms of fractional derivative depends on the specific problem being solved. Each has advantages and applications of its own. Because fractional differential equations have so many applications across varied domains, they have been carefully investigated. For example, the Wang and Zhang [7] studied a family of Hadamard fractional derivatives containing nonlinear fractional-order differential impulsive systems [8]. Naz et al. [9] explored the modified exponential approach using a fractional-order longitudinal wave equation. Muhammad et al. [10] investigated the fractional Tzitzeica equation for modeling complex phenomena by using the $(G^{\prime }/G,1/G)$-expansion technique. Systems of FDEs have been solved using the differential transform method, which yields approximate analytical solutions in the form of convergent series with readily computed components.

The progress of weakly dispersive and weakly nonlinear wave packets in definite physical schemes is described by the modified Korteweg–de Vries (KdV)–Zakharov–Kuznetsov equation, which is a nonlinear PDE. It is an extension of the traditional KdV equation that adds higher-order nonlinearity and distribution, between other effects. Applications for the modified KdV–Zakharov–Kuznetsov equation can found in fluid dynamics, plasma physics, and nonlinear optics, among other areas. Research indicates its potential to estimate and deal with wave dynamics in real-world applications, as well as its ability to provide insights into the behavior of wave phenomena in complex systems.

The (3+1)-dimensional space-time fractional modified Kdv–Zakharov–Kuznetsov equation [11] is written as follows:

$$\begin{array}{c}\hfill D_t^{\alpha }v+\gamma v^2{D}_x^{\alpha }v+D_x^{3\alpha }v+D_x^{\alpha }{D}_y^{2\alpha }v+D_x^{\alpha }{D}_z^{2\alpha }v=0,0<\alpha \le 1,\end{array}$$

(1)

where $\gamma$ and $\alpha$ are arbitrary constants with $0<\alpha \le 1$. The problem lies in multiple space dimensions. The MKdV–ZK equation appears as weakly 2-dimensional variations of the Mkdv equation [12]. In plasma, Mkdv is a model that can be derived for the development of ion-acoustic perturbation along two negative ion components of various temperatures.

In order to find exact solutions for the fractional MKdV–ZK equation, specific effective methods have been advanced. The extended direct algebraic method [13], Series solution [14], Generalized Arnous method [15], and Lie symmetry analysis [16] are some of these methods. These methods provide opportunities to consider the equation’s analysis and solution, contributing insightful information about its behavior and possible application in various scientific and technical domains.

This paper suggests using JMRLD [17] together with SSEM to obtain exact solutions for nonlinear FDEs. This study’s main goal to cover the capabilities of this method by using them to produce exact solutions for FDEs in space and time that relate to JMRLD. One of the important limitations of the classical Riemann–Liouville derivative, resolved by Jumarie modification of the Riemann–Liouville fractional derivative, is the guarantee that the fractional derivative of any constant is zero and that it leads to a more coherent extension of classical calculus to fractional orders. In addition, this formulation maintains some fundamental functional relationships, like the Mittag–Leffler function being analogous to the exponential one when differentiated, enabling analytical solutions to fractional differential equations. The aim of this paper is to expand the field of mathematical modeling and analysis by improving our command and application of exact solution techniques for FDEs. The JMRLD of order $\alpha$ is defined by the following expression [18] is

$$\begin{array}{c}\hfill D_x^{\alpha }\mathcal{F}\left(x\right)=\left\{\begin{array}{c}{\displaystyle \frac{1}{\mathrm{\Gamma }(1-\alpha )}}{\displaystyle \frac{d}{dx}}{\int }_0^x{(x-\xi )}^{-\alpha }(\mathcal{F}\left(\xi \right)-\mathcal{F}\left(0\right))d\xi ,0<\alpha <1,\hfill \\ {\left({\mathcal{F}}^{\left(\mathfrak{n}\right)}\left(x\right)\right)}^{(\alpha -\mathfrak{n})},\mathfrak{n}\le \alpha <\mathfrak{n}+1,\mathfrak{n}\le 1,\hfill \end{array}\right.\end{array}$$

(2)

where $\mathcal{F}:\mathcal{R}\to \mathcal{R}$ describes a function that is continuous and satisfies the following properties.

Property 1.

Let us now consider the case $\mathcal{F}\left(x\right)$, which refers to the function that is continuous on $\mathcal{R}\to \mathcal{R}$. The following equality holds for integral [18]:

$$\begin{array}{c}\hfill {\left(dx\right)}^{\alpha }:D_x^{\alpha }\mathcal{F}\left(x\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}{\int }_0^x{(x-\xi )}^{\alpha -1}f\left(\xi \right)d\xi ={\displaystyle \frac{1}{\mathrm{\Gamma }(1+\alpha )}}{\displaystyle \frac{d}{dx}}{\int }_0^{x}f\left(\xi \right){\left(d\xi \right)}^{\alpha },0<\alpha \le 1.\end{array}$$

Property 2.

$D_x^{\alpha }{x}^r={\displaystyle \frac{\mathrm{\Gamma }(1+r)}{\mathrm{\Gamma }(1+r-\alpha )}}{x}^{r-\alpha }.$

Property 3.

$D_x^{\alpha }\left(c\mathcal{F}\left(x\right)\right)=c{D}_x^{\alpha }\mathcal{F}\left(x\right)$

Property 4.

$D_x^{\alpha }a\mathcal{F}\left(x\right)+b\mathcal{G}\left(x\right)=a{D}_x^{\alpha }\mathcal{F}\left(x\right)+b{D}_x^{\alpha }\mathcal{G}\left(x\right),$ where a and b are constants.

Property 5.

$D_x^{\alpha }c=0$, where c is a constant.

Sensitivity analysis is an essential tool for understanding the behavior and reliability of complex models [19,20,21,22]. It identifies the parameters to check that have an influence on outcomes, allowing researchers and decision-makers to focus their attention where it matters [23,24,25]. By assessing how small changes in inputs affect results, it evaluates the robustness and stability of a system, highlighting potential weaknesses or areas of uncertainty [26,27]. In this way, sensitivity analysis not only guides the improvement and validation of models, but also supports more confident and informed decision-making across fields such as engineering, finance, environmental studies, and scientific research [28,29,30,31,32].

The layout of the paper is as follows: Section 2 discusses the Methodology, Section 3 covers the Execution of the Methodology, Section 4 presents the Simulation and Discussion, Section 5 provides the Sensitivity Analysis of the Model, Section 6 concludes the study, and, finally, Section 7 outlines the future scope of the study.

2. Methodology

In this section, the Sardar sub-equation technique has been explained, which can be applied to various differential equations including FDEs to derive analytic solutions. An efficient analytical method that is frequently employed to solve nonlinear partial differential equations (PDEs) that arise in mathematical physics and applied sciences is the Sardar sub-equation method (SSM). Bright, dark, singular, periodic, and combined soliton solutions are among the many types of exact solutions that SSM can systematically produce. These solutions are important for the understanding of complex nonlinear waves phenomena in various physical environments such as fluid dynamics, quantum mechanics, and plasma physics and optics [33,34,35,36,37]. Let us briefly explain this methodology.

• Step 1: Consider NLPDE as follows:

$$\begin{array}{c}\hfill P(v, v_x, v_t, v_y, v_z, v_{tt}, v_{xx}, v_{zz}, v_{yy}, \dots )=0,\end{array}$$

(3)

where P is a polynomial function along partial derivatives.

• Step 2: Utilize the following traveling wave transformation:

$$\begin{array}{c}\hfill v(x,y,z,t)=H\left(\chi \right),\chi ={\displaystyle \frac{m{x}^{\alpha }}{\mathrm{\Gamma }(1+\alpha )}}+{\displaystyle \frac{n{y}^{\alpha }}{\mathrm{\Gamma }(1+\alpha )}}+{\displaystyle \frac{p{z}^{\alpha }}{\mathrm{\Gamma }(1+\alpha )}}-{\displaystyle \frac{q{t}^{\alpha }}{\mathrm{\Gamma }(1+\alpha )}}.\end{array}$$

(4)

• Step 3: Apply (4) into (3) to obtain the following ODE:

$$\begin{array}{c}\hfill Q(H,H^{\prime },H^{″},H^{‴},\dots )=0.\end{array}$$

(5)

• Step 4: The solution of (3) is as follows:

$$\begin{array}{c}\hfill H\left(\chi \right)=\sum _{j=0}^P{\kappa }_j{\mathrm{\Pi }}^j\left(\chi \right),\end{array}$$

(6)

where ${\kappa }_j,j=0,1,2,\dots ,P$ are coefficients with constant values, and ${\mathrm{\Pi }}^{\prime }\left(\chi \right)$, defined as follows:

$$\begin{array}{c}\hfill {\mathrm{\Pi }}^{\prime }\left(\chi \right)=\sqrt{\tau +\delta \mathrm{\Pi }{\left(\chi \right)}^2+\mathrm{\Pi }{\left(\chi \right)}^4},\end{array}$$

(7)

where $\tau$ and $\delta$ are real constants.

• Step 5: The solutions of (5) are given as follows:

• Case I: If $\delta >0$ and $\tau =0$, we have:

$$\begin{array}{c}{\mathrm{\Pi }}_1^{\pm }\left(\chi \right)=\pm \sqrt{-fg\delta }sec{h}_{fg}\left(\sqrt{\delta }\chi \right),\hfill \\ {\mathrm{\Pi }}_2^{\pm }\left(\chi \right)=\pm \sqrt{fg\delta }csc{h}_{fg}\left(\sqrt{\delta }\chi \right),\hfill \end{array}$$

where $sec{h}_{fg}\left(\chi \right)={\displaystyle \frac{2}{f{e}^{\chi }+g{e}^{-\chi }}}$ and $csc{h}_{fg}\left(\chi \right)={\displaystyle \frac{2}{f{e}^{\chi }-g{e}^{-\chi }}}$

• Case II: If $\delta <0$ and $\tau =0$, we have:

$$\begin{array}{c}{\mathrm{\Pi }}_3^{\pm }\left(\chi \right)=\pm \sqrt{-fg\delta }{sec}_{fg}\left(\sqrt{-\delta }\chi \right),\hfill \\ {\mathrm{\Pi }}_4^{\pm }\left(\chi \right)=\pm \sqrt{-fg\delta }{csc}_{fg}\left(\sqrt{-\delta }\chi \right),\hfill \end{array}$$

where ${sec}_{fg}\left(\chi \right)={\displaystyle \frac{2}{f{e}^{\iota \chi }+g{e}^{-\iota \chi }}},$ and ${csc}_{fg}\left(\chi \right)={\displaystyle \frac{2\iota }{f{e}^{\iota \chi }-g{e}^{-\iota \chi }}}$

• Case III: If $\delta <0$ and $\tau ={\displaystyle \frac{{\delta }^2}{4}}$, afterward:

$$\begin{array}{c}{\mathrm{\Pi }}_5^{\pm }\left(\chi \right)=\pm \sqrt{{\displaystyle \frac{-\delta }{2}}}{tanh}_{fg}\left(\sqrt{{\displaystyle \frac{-\delta }{2}}}\chi \right),\hfill \\ {\mathrm{\Pi }}_6^{\pm }\left(\chi \right)=\pm \sqrt{{\displaystyle \frac{-\delta }{2}}}{coth}_{fg}\left(\sqrt{{\displaystyle \frac{-\delta }{2}}}\chi \right),\hfill \\ {\mathrm{\Pi }}_7^{\pm }\left(\chi \right)=\pm \sqrt{{\displaystyle \frac{-\delta }{2}}}\left({tanh}_{fg}\left(\sqrt{-2\delta }\chi \right)\pm \iota \sqrt{fg}sec{h}_{fg}\left(\sqrt{-2\delta }\chi \right)\right),\hfill \\ {\mathrm{\Pi }}_8^{\pm }\left(\chi \right)=\pm \sqrt{{\displaystyle \frac{-\delta }{2}}}\left({coth}_{fg}\left(\sqrt{-2\delta }\chi \right)\pm \sqrt{fg}csc{h}_{fg}\left(\sqrt{-2\delta }\chi \right)\right),\hfill \\ {\mathrm{\Pi }}_9^{\pm }\left(\chi \right)=\pm \sqrt{{\displaystyle \frac{-\delta }{8}}}\left({tanh}_{fg}\left(\sqrt{{\displaystyle \frac{-\delta }{8}}}\chi \right)+{coth}_{fg}\left(\sqrt{{\displaystyle \frac{-\delta }{8}}}\chi \right)\right),\hfill \end{array}$$

where,

${tanh}_{fg}\left(\chi \right)={\displaystyle \frac{f{e}^{\chi }-g{e}^{-\chi }}{f{e}^{\chi }+g{e}^{-\chi }}}$, ${coth}_{fg}\left(\chi \right)={\displaystyle \frac{f{e}^{\chi }+g{e}^{-\chi }}{f{e}^{\chi }-g{e}^{-\chi }}}$

• Case IV: If $\delta >0$ and $\tau ={\displaystyle \frac{{\delta }^2}{4}}$, afterward:

$$\begin{array}{c}{\mathrm{\Pi }}_{10}^{\pm }\left(\chi \right)=\pm \sqrt{{\displaystyle \frac{\delta }{2}}}{tan}_{fg}\left(\sqrt{{\displaystyle \frac{\delta }{2}}}\chi \right),\hfill \\ {\mathrm{\Pi }}_{11}^{\pm }\left(\chi \right)=\pm \sqrt{{\displaystyle \frac{\delta }{2}}}{cot}_{fg}\left(\sqrt{{\displaystyle \frac{\delta }{2}}}\chi \right),\hfill \\ {\mathrm{\Pi }}_{12}^{\pm }\left(\chi \right)=\pm \sqrt{{\displaystyle \frac{\delta }{2}}}\left({tan}_{fg}\left(\sqrt{2\delta }\chi \right)\pm \sqrt{fg}{sec}_{fg}\left(\sqrt{2\delta }\chi \right)\right),\hfill \\ {\mathrm{\Pi }}_{13}^{\pm }\left(\chi \right)=\pm \sqrt{{\displaystyle \frac{\delta }{2}}}\left({cot}_{fg}\left(\sqrt{2\delta }\chi \right)\pm \sqrt{fg}{csc}_{fg}\left(\sqrt{2\delta }\chi \right)\right),\hfill \\ {\mathrm{\Pi }}_{14}^{\pm }\left(\chi \right)=\pm \sqrt{{\displaystyle \frac{\delta }{8}}}\left({tan}_{fg}\left(\sqrt{{\displaystyle \frac{\delta }{8}}}\chi \right)+{cot}_{fg}\left(\sqrt{{\displaystyle \frac{\delta }{8}}}\chi \right)\right),\hfill \end{array}$$

where,

${tan}_{fg}\left(\chi \right)=-\iota {\displaystyle \frac{f{e}^{\iota \chi }-g{e}^{-\iota \chi }}{f{e}^{\iota \chi }+g{e}^{-\iota \chi }}}$, ${cot}_{fg}\left(\chi \right)=\iota {\displaystyle \frac{f{e}^{\iota \chi }+g{e}^{-\iota \chi }}{f{e}^{\iota \chi }-g{e}^{-\iota \chi }}}.$

• Step 6: To collect all powers of ${\mathrm{\Pi }}^j\left(\chi \right)$, put (6) into (5) along (7). Set the coefficients of all polynomials equal to zero and then solve algebraically to obtain the constant values. All analytic results are collected by Mathematica11.0.

3. Execution of Methodology

In this section, we apply the Sardar sub-equation to find the analytic solution of the (3+1)-dimensional fractional Mkdv–ZK equation. Fractional differential equations have regained breadth because they capture memory and hereditary aspects of many different processes in science and engineering, and have become essential in modeling these phenomena. Here, we note the fascinating (3+1)-dimensional fractional modified KdV–ZK equation, which has both many interesting physical implications and whose analytic features are perplexing. The following traveling wave transformation is used to convert PDE into ODE:

$$\begin{array}{c}\hfill v(x,y,z,t)=H\left(\chi \right),\chi ={\displaystyle \frac{m{x}^{\alpha }}{\mathrm{\Gamma }(1+\alpha )}}+{\displaystyle \frac{n{y}^{\alpha }}{\mathrm{\Gamma }(1+\alpha )}}+{\displaystyle \frac{p{z}^{\alpha }}{\mathrm{\Gamma }(1+\alpha )}}-{\displaystyle \frac{q{t}^{\alpha }}{\mathrm{\Gamma }(1+\alpha )}},\end{array}$$

(8)

where $m,n,p$, and q are non-zero constants. Substitute the values of (8) with all required derivatives into (1) to reduce into following ordinary differential equation (ODE).

$$\begin{array}{c}\hfill -q{H}^{\prime }+\gamma m{H}^2{H}^{\prime }+(m^3+m{n}^2+m{p}^2)H^{‴}=0.\end{array}$$

By integrating the above equation w.r.t $\chi$, we have

$$\begin{array}{c}\hfill -qH+{\displaystyle \frac{\gamma m}{3}}{H}^3+(m^3+m{n}^2+m{p}^2)H^{″}+C=0,\end{array}$$

(9)

where C is the integrating constant. Balance the highest derivative and nonlinear term such as $H^3$, and $H^{″}$ yields $P=1$. Thus, (5) is reduced into the following:

$$\begin{array}{c}\hfill H\left(\chi \right)={\kappa }_0+{\kappa }_1\mathrm{\Pi }\left(\chi \right),\end{array}$$

(10)

where ${\kappa }_{0}and{\kappa }_1$ are constants. Substituting (10) into (9) and then comparing the polynomial coefficients of all powers of $\mathrm{\Pi }\left(\chi \right)$ is equal to zero. We obtain

$${\kappa }_0=0,{\kappa }_1=\pm {\displaystyle \frac{\sqrt{-6}\sqrt{m}\sqrt{m^2+n^2+p^2}}{\sqrt{\gamma \alpha }}},\delta ={\displaystyle \frac{q}{m\left(m^2+n^2+p^2\right)}},C=0.$$

(11)

The analytic solutions of (1) are listed below.

• Case I: If $\delta >0$, and $\tau =0$, we have

$$\begin{array}{ccc}\hfill v_{1,1}^{\pm }& =& \pm \left({\displaystyle \frac{\sqrt{-6}\sqrt{m^2+n^2+p^2}}{\sqrt{\gamma }}}\right)\sqrt{-{\displaystyle \frac{fgq}{m\left(m^2+n^2+p^2\right)}}}{\sec \mathrm{h}}_{fg}\left(\sqrt{{\displaystyle \frac{q}{m\left(m^2+n^2+p^2\right)}}}\chi \right),\hfill \\ \hfill v_{1,2}^{\pm }& =& \pm \left({\displaystyle \frac{\sqrt{-6}\sqrt{m^2+n^2+p^2}}{\sqrt{\gamma }}}\right)\sqrt{{\displaystyle \frac{fgq}{m\left(m^2+n^2+p^2\right)}}}{\csc \mathrm{h}}_{fg}\left(\sqrt{{\displaystyle \frac{q}{m\left(m^2+n^2+p^2\right)}}}\chi \right).\hfill \end{array}$$

• Case II: If $\delta <0$ and $\tau =0$, we have

$$\begin{array}{ccc}\hfill v_{1,3}^{\pm }& =& \pm \left({\displaystyle \frac{\sqrt{-6}\sqrt{m^2+n^2+p^2}}{\sqrt{\gamma }}}\right)\sqrt{-{\displaystyle \frac{fgq}{m\left(m^2+n^2+p^2\right)}}}{\mathrm{Sec}}_{fg}\left(\sqrt{-{\displaystyle \frac{q}{m\left(m^2+n^2+p^2\right)}}}\chi \right),\hfill \\ \hfill v_{1,4}& =& \pm \left({\displaystyle \frac{\sqrt{-6}\sqrt{m^2+n^2+p^2}}{\sqrt{\gamma }}}\right)\sqrt{-{\displaystyle \frac{fgq}{m\left(m^2+n^2+p^2\right)}}}{\mathrm{Csc}}_{fg}\left(\sqrt{-{\displaystyle \frac{q}{m\left(m^2+n^2+p^2\right)}}}\chi \right).\hfill \end{array}$$

• Case III: If $\delta <0$ and $\tau ={\displaystyle \frac{{\delta }^2}{4}}$, we have:

$$\begin{array}{ccc}\hfill v_{1,5}^{\pm }& =& \pm \left({\displaystyle \frac{\sqrt{-6(m^2+n^2+p^2)}}{\sqrt{\gamma }}}\right)\sqrt{{\displaystyle \frac{-q}{2m\left(m^2+n^2+p^2\right)}}}{tanh}_{fg}\left(\sqrt{{\displaystyle \frac{-q}{2m\left(m^2+n^2+p^2\right)}}}\chi \right),\hfill \\ \hfill v_{1,6}^{\pm }& =& \pm \left({\displaystyle \frac{\sqrt{-6(m^2+n^2+p^2)}}{\sqrt{\gamma }}}\right)\sqrt{{\displaystyle \frac{-q}{2m\left(m^2+n^2+p^2\right)}}}{cot}_{fg}\left(\sqrt{{\displaystyle \frac{-q}{2m\left(m^2+n^2+p^2\right)}}}\chi \right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill v_{1,7}^{\pm }=\pm \left({\displaystyle \frac{\sqrt{-6(m^2+n^2+p^2)}}{\sqrt{\gamma }}}\right)[& \sqrt{{\displaystyle \frac{-q}{2m(m^2+n^2+p^2)}}}({tanh}_{fg}\left(\sqrt{{\displaystyle \frac{-2q}{m(m^2+n^2+p^2)}}}\chi \right)\hfill \\ \hfill & \pm i\sqrt{fg}{\sec \mathrm{h}}_{fg}\left(\sqrt{-{\displaystyle \frac{2q}{m(m^2+n^2+p^2)}}}\chi \right)\left)\right]\hfill \\ \hfill v_{1,8}^{\pm }=\pm \left({\displaystyle \frac{\sqrt{-6(m^2+n^2+p^2)}}{\sqrt{\gamma }}}\right)[& \sqrt{{\displaystyle \frac{-q}{2m\left(m^2+n^2+p^2\right)}}}({coth}_{fg}\left(\sqrt{{\displaystyle \frac{-2q}{m\left(m^2+n^2+p^2\right)}}}\chi \right)\hfill \\ \hfill & \pm \sqrt{fg}{\csc \mathrm{h}}_{fg}\left(\sqrt{{\displaystyle \frac{-2q}{m\left(m^2+n^2+p^2\right)}}}\chi \right)\left)\right],\hfill \\ \hfill v_{1,9}^{\pm }=\pm \left({\displaystyle \frac{\sqrt{-6(m^2+n^2+p^2)}}{\sqrt{\gamma }}}\right)[& {\displaystyle \frac{\sqrt{\frac{-q}{m\left(m^2+n^2+p^2\right)}}}{2\sqrt{2}}}({tanh}_{fg}\left({\displaystyle \frac{\sqrt{-\frac{q}{m\left(m^2+n^2+p^2\right)}}}{2\sqrt{2}}}\chi \right)\hfill \\ \hfill & +{coth}_{fg}\left({\displaystyle \frac{\sqrt{-\frac{q}{m\left(m^2+n^2+p^2\right)}}}{2\sqrt{2}}}\chi \right)\left)\right].\hfill \end{array}$$

• Case IV: If $\delta >0$ and $\tau ={\displaystyle \frac{{\delta }^2}{4}}$, we have

$$\begin{array}{ccc}\hfill v_{1,10}^{\pm }& =& \pm \left({\displaystyle \frac{\sqrt{-6(m^2+n^2+p^2)}}{\sqrt{\gamma }}}\right)\sqrt{{\displaystyle \frac{q}{2m\left(m^2+n^2+p^2\right)}}}{tan}_{fg}\left(\sqrt{{\displaystyle \frac{q}{2m\left(m^2+n^2+p^2\right)}}}\chi \right),\hfill \\ \hfill v_{1,11}^{\pm }& =& \pm \left({\displaystyle \frac{\sqrt{-6(m^2+n^2+p^2)}}{\sqrt{\gamma }}}\right)\sqrt{{\displaystyle \frac{q}{2m\left(m^2+n^2+p^2\right)}}}{cot}_{fg}\left(\sqrt{{\displaystyle \frac{q}{2m\left(m^2+n^2+p^2\right)}}}\chi \right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill v_{1,12}^{\pm }=\pm \left({\displaystyle \frac{\sqrt{-6(m^2+n^2+p^2)}}{\sqrt{\gamma }}}\right)[& \sqrt{{\displaystyle \frac{q}{2m\left(m^2+n^2+p^2\right)}}}({tan}_{fg}\left(\sqrt{{\displaystyle \frac{2q}{m\left(m^2+n^2+p^2\right)}}}\chi \right)\hfill \\ \hfill & \pm \sqrt{fg}{\sec }_{fg}\left(\sqrt{{\displaystyle \frac{2q}{m\left(m^2+n^2+p^2\right)}}}\chi \right)\left)\right],\hfill \\ \hfill v_{1,13}^{\pm }=\pm \left({\displaystyle \frac{\sqrt{-6(m^2+n^2+p^2)}}{\sqrt{\gamma }}}\right)[& \sqrt{{\displaystyle \frac{q}{2m\left(m^2+n^2+p^2\right)}}}({cot}_{fg}\left(\sqrt{{\displaystyle \frac{2q}{m\left(m^2+n^2+p^2\right)}}}\chi \right)\hfill \\ \hfill & \pm \sqrt{fg}{\csc }_{fg}\left(\sqrt{{\displaystyle \frac{2q}{m\left(m^2+n^2+p^2\right)}}}\chi \right)\left)\right],\hfill \\ \hfill v_{1,14}^{\pm }=\pm \left({\displaystyle \frac{\sqrt{-6(m^2+n^2+p^2)}}{\sqrt{\gamma }}}\right)[& {\displaystyle \frac{\sqrt{\frac{q}{m\left(m^2+n^2+p^2\right)}}}{2\sqrt{2}}}({tan}_{fg}\left({\displaystyle \frac{\sqrt{\frac{q}{m\left(m^2+n^2+p^2\right)}}}{2\sqrt{2}}}\chi \right)\hfill \\ \hfill & +{cot}_{fg}\left({\displaystyle \frac{\sqrt{\frac{q}{m\left(m^2+n^2+p^2\right)}}}{2\sqrt{2}}}\chi \right)\left)\right].\hfill \end{array}$$

4. Simulation and Discussion

In this work, we provide a visualization of the solutions found in the form of a 3D surface and contours in order to see how the fractional parameter affects the behavior of the system as shown in Figure 1, Figure 2, Figure 3 and Figure 4. In particular, we study the fractional modified KdV–ZK equation in space–time with a hybrid of the techniques of Jumarie modified Riemann–Liouville derivative and the sub-equation method (SSEM). The method provides a variety of accurate solutions, which can be used to gain useful information about the influence of fractional-order effects on the properties of solutions. According to our graphical outcomes the evolution of solution profiles is very sensitive to changes in the fractional order. When this parameter is systematically varied, we see a wide array of behaviors—including small modulations of wave profiles to profound modification of soliton interactions. The discussion shows that the value of the fractional order significantly affects soliton interactions. When the fractional parameter is changed only slightly, soliton interactions are largely repulsive, so the solitons can preserve their unique identities. But, when the fractional order is larger than a particular value, the interaction will be attractive, resulting in the existence of bound states. Solitons combine in these states and transform into more complex solitary structures.

Results Comparison

Table 1. Comparison of the current study with Guner et al. [11].

Aspect	Current Study	Guner et al. [11]
Analytical Approach	Sardar sub-equation (SSEM); exact wave solutions	The exp-function method, the generalized Kudryashov method and $\left({\displaystyle \frac{G^{\prime }}{G}}\right)$-expansion method are applied.
Sensitivity Analysis	Includes Galilean-based sensitivity analysis	Sensitivity analysis are not discuss in Literature

contrasts this study with previous research, emphasizing its unique contributions.

5. Sensitivity Analysis

In this section, we investigate the sensitivity of the fractional modified KdV–ZK equation to varying initial conditions. To this end, we apply the Galilean transformation to Equation (9), yielding the following dynamical system [38]:

$$\left\{\begin{array}{cc}\hfill & {\displaystyle \frac{dH}{d\chi }}=\mathcal{G},\hfill \\ \hfill & {\displaystyle \frac{d\mathcal{G}}{d\chi }}={\Psi }_{0}H-{\Psi }_1{H}^3-{\Psi }_2,\hfill \\ \hfill & {\Psi }_0={\displaystyle \frac{q}{(m^3+m{n}^2+m{p}^2)}},{\Psi }_1={\displaystyle \frac{\gamma m}{3(m^3+m{n}^2+m{p}^2)}},{\Psi }_2={\displaystyle \frac{C}{(m^3+m{n}^2+m{p}^2)}}·\hfill \end{array}\right.$$

(12)

To investigate the sensitivity of the model, we consider four different sets of initial conditions. Figure 5a displays two solution trajectories arising from the initial conditions $(0.34, 0.03)$ and $(0.04, 0.03)$, represented by the red and green curves, respectively. Figure 5b illustrates two solution paths generated from the starting points $(0.34, 0.03)$ and $(0.14, 0.03)$, shown by the red and green trajectories, respectively. In Figure 5c, two solution trajectories originating from $(0.34, 0.03)$ and $(0.22, 0.03)$ are presented as the red and green curves, respectively. Finally, Figure 5d showcases three solution trajectories arising from the initial conditions $(0.32, 0.03)$, $(0.22, 0.03)$, and $(0.12, 0.03)$, depicted by the red, green, and blue curves, respectively. As observed, even slight variations in the initial conditions result in noticeable differences in the dynamics of system (12). In other words, no two solution curves coincide [39,40], highlighting that the system is sensitive to its initial states, although this sensitivity is not exceedingly strong.

6. Conclusions

The research on the optical soliton solutions for the (3+1)-dimensional space–time fractional Mkdv–ZK equation has generated substantial perceptions into the behavior of solitons in complex dimensions. Using SSEM, new analytical soliton solutions have been derived, showing varied possessions and dynamics in nonlinear wave phenomena. In addition, the combination of fractional calculus has exposed the effect of fractional order on these soliton solutions, and considered the complexity and exclusive physical appearance introduced by fractional derivatives. This study contributes to a better understanding of how fractional calculus affects soliton dynamics, offering useful insights for nonlinear science and optical communication applications. The combination of the Sardar sub-equation method (SSEM) and the study of fractional-order effects offers new options for future research into the complicated dynamics of optical solitons in complex physical systems. The system is extremely sensitive to changes in the starting conditions. The new solitary wave solutions and sensitivity analysis described here provide a better understanding of how fractional-order dynamics influence wave propagation, which is important for building and managing systems in domains like plasma confinement, nonlinear optics, and material sciences.

7. Future Scope of Study

The current paper shows how the Sardar sub-equation technique coupled with the modified fractional derivative approach due to Jumarie can be exploited to obtain exact solitary wave solutions of the (3+1)-dimensional fractional generalized KdV–ZK equation. Based on these findings, additional studies can be developed in a number of potentially interesting ways. To begin with, one can carry the method further to include numerical investigations and stability studies that will give further information on the stability and physical viability of the solutions found. Also, this method can be extended to other nonlinear fractional models in fluid dynamics, plasma physics, and nonlinear optics to further expand its applicability. The combination of quantitative sensitivity and bifurcation analysis would help better understand the solutions’ parameter-dependent behavior. Lastly, an interesting direction for developing the theoretical and practical features of fractional soliton dynamics is to extend the method to higher-dimensional systems and linked nonlinear equations. In future work, this can be extended to investigate its rich dynamics through exploring lump solutions, PINN-based approaches, and other advanced numerical methods. Such studies will deepen our understanding of its behavior and broaden its range of applications in nonlinear modeling and complex dynamical systems.