Optical soliton wave profiles for the (2 + 1)-dimensional complex modified Korteweg–de Vries system with the impact of fractional derivative via analytical approach

Abstract

In this paper, we investigate exact analytical solutions of the (2+1)-dimensional complex modified Korteweg–de Vries (CmKdV) system using the truncated M-fractional derivative together with the Jacobi elliptic function expansion method. The CmKdV system plays a central role in modeling nonlinear wave propagation in optics, plasma physics, and fluid dynamics. By applying the truncated M-fractional derivative, the system is transformed into a more tractable form, enabling the effective use of the Jacobi elliptic function expansion method to construct exact soliton and periodic wave solutions. The results offer deeper insight into the system’s nonlinear dynamics and highlight the robustness of the proposed method. Graphical simulations generated in Mathematica illustrate the physical behavior of the obtained solutions across multiple dimensions, such as two-dimensional, three-dimensional, and contour, and the influence of time on the wave propagations. Overall, combining the Jacobi elliptic function expansion method and truncated M-fractional derivatives creates a strong framework for solving complex differential equations, leading to new possibilities. Opportunities for research and development exist. This research adds to our understanding of the (2+1)-dimensional complex modified Korteweg-de Vries (CmKdV) system and demonstrates how theoretical mathematics can be applied to real concerns. Mathematical modeling and computational visualization can significantly impact engineering and science, and our findings promote a multidisciplinary approach to research.

Introduction

Nonlinear partial differential equations play an important role in many scientific and engineering fields, including fluid dynamics, plasma physics^1,2, optical fiber communication³, and several other areas^4,5,6,7. These studies collectively demonstrate the richness of nonlinear wave structures and the effectiveness of analytical techniques in extracting exact solutions. Studying nonlinear partial differential equations (NLPDEs) is crucial for understanding a wide range of physical phenomena arising in science and engineering. The accurate understanding and physical interpretation of many natural and engineering phenomena strongly depend on the systematic investigation of NLPDEs^8,9,10,11,12. The (2+1)-dimensional complex modified Korteweg–de Vries (CmKdV) system is widely recognized for its ability to describe nonlinear wave propagation in many physical media. The (2+1)-dimensional complex modified Korteweg-de Vries (CmKdV) system is a recent extension of the classical mKdV equation. This improved model incorporates the truncated M-fractional derivative, which enhances its ability to describe memory and hereditary effects in complex systems. Solitons have found wide-ranging applications in communication system, networking, mathematical physics, mathematical biology, optical fiber systems, mechanics, hydrodynamics, and chemical physics, providing a powerful framework to explain complex behaviors in these fields^{13,14,15,16,17}. Due to the inherent nonlinearity and mathematical complexity of such equations, the development of efficient analytical and numerical methods for their solution is essential. This area includes a variety of innovative techniques. Examples of these strategies include the improved modified Sardar subequation method¹⁸, the improved generalized Riccati equation mapping method¹⁹, the Bernoulli subequation method²⁰, the Lie symmetry analysis²¹, the $(G'/G,\, 1/G)$-expansion method²², and additional solution methodologies are discussed in^{23,24,25,26,27}.

For several decades, the complex world of biological and physical systems has been analyzed using fractional derivatives, a fundamental mathematical concept. This concept, a generalized extension of the classical calculus derivative, redefines differentiation by allowing non-integer orders, thereby capturing memory and hereditary properties inherent in many natural processes. This sophisticated approach has transformed fields such as fluid mechanics and signal analysis by providing a more realistic framework for describing nonlinear systems and by enabling the investigation of complex dynamical systems of second order and beyond. Over time, various definitions of fractional derivatives have been introduced to improve modeling accuracy, among which the Caputo and Riemann–Liouville formulations have played a prominent role in describing real-world physical phenomena.

Over the past century, the Caputo fractional derivative has significantly enhanced the understanding of the stability and accuracy of singular solutions in nonlinear models. Important applications include the fractional analysis of the Caudrey–Dodd–Gibbon model²⁸, conformable space–time fractional formulations of the Fokas–Lenells equation²⁹, and the use of the Riemann–Liouville fractional integral in studying the time-fractional Kudryashov–Sinelshchikov equation³⁰. Further developments include conformable time-fractional approaches for Wick-fractional stochastic partial differential equations³¹, modified Riemann–Liouville schemes for the modified Camassa–Holm equation³², and fractional integral methods applied to nonlinear Fokas–Lenells and paraxial Schrödinger equations³³. These studies clearly demonstrate the depth and breadth of fractional calculus and emphasize its crucial role in analyzing and simulating complex phenomena across a wide range of scientific disciplines. Together, these advancements provide a strong foundation for future research in nonlinear dynamics and offer powerful analytical tools for emerging studies in physical and biological systems.A fundamental mathematical model for examining nonlinear wave phenomena and fluid dynamics is the complex modified Korteweg-de Vries (CmKdV) equation. It provides fundamental insight into how coherent structures are generated, evolve, and propagate in a wide variety of physical systems. In particular, the CmKdV equation is well known for admitting soliton solutions-localized, stable nonlinear waveforms that propagate without changing shape due to a balance between dispersion and nonlinearity. These solitons play a crucial role in describing several important physical phenomena, including rogue waves, nonlinear pulse propagation in optical fibers, and coherent structures in fluid dynamics. Owing to these remarkable properties and its broad range of applications, the CmKdV equation continues to attract significant attention in both theoretical and applied nonlinear science.rogue waves.

$$\begin{aligned} \left\{ \begin{aligned} D_{M,t}^{\alpha ,\Upsilon } v + D_{M,x}^{\alpha ,\Upsilon } D_{M,x}^{\alpha ,\Upsilon } D_{M,y}^{\alpha ,\Upsilon } v + i \mathcal {v} w + D_{M,x}^{\alpha ,\Upsilon } (v s)&= 0, \\ D_{M,x}^{\alpha ,\Upsilon } w + 2 i \sigma \left( v^*D_{M,x}^{\alpha ,\Upsilon } D_{M,y}^{\alpha ,\Upsilon } v - D_{M,x}^{\alpha ,\Upsilon } D_{M,y}^{\alpha ,\Upsilon } v^*\, v \right)&= 0, \\ D_{M,x}^{\alpha ,\Upsilon } s - 2 \sigma D_{M,y}^{\alpha ,\Upsilon } \left( |v|^2 \right)&= 0. \end{aligned} \right. \end{aligned}$$

(1)

Here, v(x, y, t) denotes a complex-valued function, and $v^{*}(x,y,t)$ represents its complex conjugate. The functions w(x, y, t) and s(x, y, t) are real-valued. The parameter $\sigma$ takes the values $\pm 1$, and partial derivatives with respect to the spatial variables x, y, and the temporal variable t are indicated by subscripts. Many researchers have investigated obtaining a wide variety of exact solutions, such as bright and dark solitons, periodic waves, and Jacobi elliptic function solutions using different analytical techniques^{34,35,36,37,38}. In particular, the Jacobi elliptic function expansion method has been successfully applied to the modified and improved modified Korteweg–de Vries equations to construct families of exact traveling-wave and optical soliton solutions³⁹. Recently, the combination of the Jacobi elliptic function expansion method with the truncated M-fractional derivative has emerged as a powerful framework for analyzing nonlinear fractional models, yielding periodic, solitary, and shock-type wave profiles in several physical systems^40,41.

Integrable nonlinear wave equations, including Kadomtsev–Petviashvili (KP)-based systems and nonlocal Schrödinger-type models, provide an effective framework for describing complex wave phenomena such as periodic line waves, breathers, and rogue waves. Exact and interaction solutions in $(2+1)$-dimensional KP-based systems and nonlocal Schrödinger equations have been widely studied, revealing rich localized wave structures^42,43,44. Analytical and semi-analytical techniques have been extensively developed for nonlinear differential equations, including numerical methods, transform-based approaches, and fractional-order analytical models^{45,46,47,48,49}. Compared with these techniques, the Jacobi elliptic function expansion method provides a direct and systematic approach for constructing exact closed-form solutions, yielding periodic solutions and their solitary-wave limits in a unified manner. Recent advances in fractional calculus further highlight the effectiveness of exact analytical methods in capturing memory effects and nonlinear dynamics in complex systems^50,51.

In this paper, we employ the Jacobi elliptic function expansion method together with the truncated M-fractional derivative to study the $(2+1)$-dimensional complex modified KdV-type system. This hybrid approach allows us to derive Jacobi elliptic function solutions, solitary wave solutions, and shock-wave solutions.The results provide new insight into how the fractional order and truncation parameters influence the amplitude, width, and propagation characteristics of nonlinear waves in such systems and complement existing classical (integer-order) solutions available in the literature.

The remainder of this paper is organized as follows: Section 1 provides a concise introduction; Section 2 presents the methodology of the proposed scheme; Section 3 investigates the application of the Jacobi elliptic function expansion method; Section 4 illustrates the graphical representation of the obtained solutions; and Section 5 concludes the study. Also the flow chart of the proposed work shown in Fig. 1.

Truncated M-fractional derivative: definition and its properties

In this subsection, we define some fundamental concepts and discuss the main properties of the truncated M-fractional derivative.

Definition 1

Let $u(\vartheta ):[ 0,\infty ) \rightarrow \mathbb {R}.$ The truncated M-fractional derivative of u of order $\alpha$ is defined by⁵² as:

$$\begin{aligned} D_{M,\theta }^{\alpha ,\Upsilon }u(\vartheta ) =\lim _{\tau \rightarrow 0} \frac{u\left( \vartheta \, E_\Upsilon \left( \tau \vartheta ^{1-\alpha }\right) \right) -u(\vartheta )}{\tau }, \quad 0<\alpha \le 1,\ \Upsilon >0. \end{aligned}$$

(2)

Here, $E_\Upsilon (\cdot )$ denotes the truncated Mittag-Leffler function with one parameter, defined by⁵³

$$\begin{aligned} E_\Upsilon (z)=\sum _{j=0}^{i}\frac{z^j}{\Gamma (\Upsilon j+1)}, \quad \Upsilon >0,\ z\in \mathbb {C}. \end{aligned}$$

(3)

The truncated M-fractional derivative satisfies the following properties:

$$\begin{aligned} D_{M,\theta }^{\alpha ,\Upsilon }\!\left( r g(\theta )+s f(\theta )\right) =r D_{M,\theta }^{\alpha ,\Upsilon } g(\theta ) +s D_{M,\theta }^{\alpha ,\Upsilon } f(\theta ), \end{aligned}$$

(4)

$$\begin{aligned} D_{M,\theta }^{\alpha ,\Upsilon }\!\left( g(\theta )f(\theta )\right) =g(\theta )D_{M,\theta }^{\alpha ,\Upsilon }f(\theta ) +f(\theta )D_{M,\theta }^{\alpha ,\Upsilon }g(\theta ), \end{aligned}$$

(5)

$$\begin{aligned} D_{M,\theta }^{\alpha ,\Upsilon }\!\left( \frac{g(\theta )}{f(\theta )}\right) =\frac{f(\theta )D_{M,\theta }^{\alpha ,\Upsilon }g(\theta ) -g(\theta )D_{M,\theta }^{\alpha ,\Upsilon }f(\theta )}{\big (f(\theta )\big )^2}, \end{aligned}$$

(6)

$$\begin{aligned} D_{M,\theta }^{\alpha ,\Upsilon }(A)=0, \quad \text {where}\, A\, \text {is a constant}, \end{aligned}$$

(7)

$$\begin{aligned} D_{M,\theta }^{\alpha ,\Upsilon }g(\theta ) =\frac{\theta ^{1-\alpha }}{\Gamma (1+\Upsilon )}\, \frac{d g(\theta )}{d\theta }. \end{aligned}$$

(8)

Fig. 1

Work flow chart.

Methodology

In this section, the principal steps of the proposed methodology are presented in an orderly manner. The solution procedure is implemented through the following sequence of steps.

Step 1: Consider a nonlinear partial differential equation involving two independent variables in the general form

$$\begin{aligned} F\!\left( u,\; D_{M,t}^{\alpha ,\Upsilon } u,\; D_{M,x}^{\gamma ,\Upsilon } u,\; D_{M,t}^{2\alpha ,\Upsilon } u,\; D_{M,x}^{2\gamma ,\Upsilon } u,\; \ldots \right) =0. \end{aligned}$$

(9)

Step 2: To reduce the governing equation into an ordinary differential equation, the traveling wave transformation is introduced as

$$\begin{aligned} u(x,t) ={u}(x,t)\, \exp \!\left( i\,\frac{\Gamma (1+\alpha )}{\alpha } \left( c x^\alpha + w t^\alpha \right) \right) . \end{aligned}$$

(10)

where w and c are nonzero constants.

Step 3: By applying the above transformation, the nonlinear partial differential equation is converted into an ordinary differential equation of integer order with respect to $\xi$.

Step 4: To enhance the probability of obtaining exact analytical solutions, an auxiliary ordinary differential equation belonging to the first category of the three-parameter Jacobian equation is employed. This auxiliary equation allows the construction of a wide class of Jacobian elliptic function solutions and also admits a geometric interpretation. The auxiliary equation is written as:

$$\begin{aligned} \left( F'(\xi )\right) ^2 = P F^4(\xi ) + Q F^2(\xi ) + R. \end{aligned}$$

(11)

where $F'=\dfrac{dF}{d\xi }$, P, Q, and R are real constants. The corresponding solutions are listed in Table 1. Here, $i^2=-1$, and the Jacobian elliptic functions are defined as $\textrm{sn}(\xi ,m)$, $\textrm{cn}(\xi ,m)$, and $\textrm{dn}(\xi ,m)$, with modulus $0<m<1$.

Table 1 Possible solutions of $F(\xi )$ in Eq. (5) for selected values of P, Q, and R.

Table 2 Limiting forms of Jacobi elliptic functions for $m \rightarrow 1$ and $m \rightarrow 0$.

As a consequence, periodic, hyperbolic, and trigonometric solutions can be constructed for the problem. The solution of the reduced ordinary differential equation is expressed as a finite series of Jacobian elliptic functions in the form

$$\begin{aligned} u(\xi ) = \sum _{i=0}^{n} a_i F^i(\xi ). \end{aligned}$$

(12)

where $a_i \ (i=0,1,2,\ldots ,n)$ are unknown constants to be determined. The highest order of the linear derivative term is given by:

$$\begin{aligned} O\left( \frac{\partial ^p u}{\partial \xi ^p}\right) = n + p, \qquad p=1,2,3,\ldots \end{aligned}$$

(13)

While the highest order of the nonlinear term is:

$$\begin{aligned} O\left( u^q\frac{\partial ^p u}{\partial \xi ^p}\right) = (q+1)n + p, \qquad q=0,1,2,\ldots ,\ p=1,2,3,\ldots \end{aligned}$$

(14)

Substituting the expansion (12) into the reduced equation and equating the coefficients of like powers of $F(\xi )$ to zero yields a system of nonlinear algebraic equations for the unknown parameters $a_i$. These equations are solved using the admissible values of P, Q, and R listed in Table 1. Finally, by inserting the obtained constants into the series solution and the auxiliary equation, exact analytical solutions of the original governing equation are obtained.

Implementation of the Jacobi elliptic function expansion method

For employing the Jacobi elliptic function expansion method, we reduce Eq. (1) to an ODE by using the transformation

$$\begin{aligned} u(x,t) = {g}(x,t)\, \exp \!\left( i\,\frac{\Gamma (1+\alpha )}{\alpha } \left( c x^\alpha + w t^\alpha \right) \right) . \end{aligned}$$

(15)

where $a_1,\, a_2,$ and $a_3$ are real constants. Using the above transformation, Eq. (1) reduces to

$$\begin{aligned} \begin{aligned}&D_{M,t}^{\alpha ,\Upsilon }\mathcal {g} - 2a_1 a_2 D_{M,x}^{\alpha ,\Upsilon } g - a_1^2 D_{M,y}^{\alpha ,\Upsilon } g + D_{M,x}^{\alpha ,\Upsilon } D_{M,x}^{\alpha ,\Upsilon } D_{M,y}^{\alpha ,\Upsilon }g+D_{M,x}^{\alpha ,\Upsilon }(g s) + N D_{M,x}^{\alpha ,\Upsilon } s \\&\quad + i\Big [ (a_3 - a_1^2 a_2) g + 2 D_{M,x}^{\alpha ,\Upsilon } D_{M,y}^{\alpha ,\Upsilon } g + a_2 D_{M,x}^{\alpha ,\Upsilon } D_{M,x}^{\alpha ,\Upsilon } g + a_1 g s + g w \Big ] = 0, \end{aligned} \end{aligned}$$

(16)

$$\begin{aligned} D_{M,x}^{\alpha ,\Upsilon } w - 4 \sigma \left( a_2 g D_{M,x}^{\alpha ,\Upsilon } g + a_1 g D_{M,y}^{\alpha ,\Upsilon } g \right) = 0, \end{aligned}$$

(17)

$$\begin{aligned} D_{M,x}^{\alpha ,\Upsilon } s - 2 \sigma D_{M,y}^{\alpha ,\Upsilon } \left( g^2\right) = 0. \end{aligned}$$

(18)

Substituting the wave transformation

$$\begin{aligned} v(x,y,t)&= g(\xi ) = g\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) , \\ w(x,y,t)&= w(\xi ) = w\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) , \\ s(x,y,t)&= s(\xi ) = s\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) . \end{aligned}$$

Into the system of Eqs. (16)–(18), we get

$$\begin{aligned} \begin{aligned}&(c-2a_1 a_2-a_1^2) g' + g^{\prime \prime \prime } + g' s + g s'+ i\Big [ (a_3-a_1^2 a_2) g + (2a_1+a_2) g^{\prime \prime } + a_1 g s + g s \Big ] = 0, \end{aligned} \end{aligned}$$

(19)

$$\begin{aligned} w' - 4\sigma (a_2+a_1) g g' = 0. \end{aligned}$$

(20)

$$\begin{aligned} s' - 2\sigma \left( g^2\right) ' = 0. \end{aligned}$$

(21)

Integrating Eqs. (20) and (21) once with respect to $\xi$ and taking the constants of integration to be zero, we obtain

$$\begin{aligned} w = 2\sigma (a_2 + a_1) g^2, \qquad s = 2\sigma g^2. \end{aligned}$$

(22)

Substituting Eq. (22) into Eq. (19), we obtain the following ODE:

$$\begin{aligned} \begin{aligned}&(c - 2a_1 a_2 - a_1^2) g' + g^{\prime \prime \prime } + 2\sigma \left( g^3\right) ' + i \Big [ (a_3 - a_1^2 a_2) g + (2a_1 + a_2) g^{\prime \prime } + 2\sigma (2a_1 + a_2) g^3 \Big ] = 0. \end{aligned} \end{aligned}$$

(23)

The prime denotes the derivative with respect to $\xi$. Now, separating the real and imaginary parts of Eq. (22), we have:

$$\begin{aligned} (c-2a_1a_2-a_1^2) g' + g^{\prime \prime \prime } + 2\sigma \left( g^3\right) ' = 0, \end{aligned}$$

(24)

$$\begin{aligned} \frac{a_3-a_1^2 a_2}{2a_1+a_2}\, g + g^{\prime \prime } + 2\sigma g^3 = 0. \end{aligned}$$

(25)

Taking the anti-derivative of Eq. (24) once with respect to $\xi$ and setting the constant of integration to zero, we obtain

$$\begin{aligned} (c-2a_1a_2-a_1^2) g + g^{\prime \prime } + 2\sigma g^3 = 0. \end{aligned}$$

(26)

The Eqs. (25) and (26) are the same if and only if the following constraint condition is satisfied:

$$\begin{aligned} c-2a_1a_2-a_1^2 = \frac{a_3-a_1^2 a_2}{2a_1-a_2}. \end{aligned}$$

(27)

Solving for c, we obtain

$$\begin{aligned} c = 2a_1 a_2 + a_1^2 + \frac{a_3-a_1^2 a_2}{2a_1-a_2}. \end{aligned}$$

(28)

We rewrite Eq. (25) as

$$\begin{aligned} g^{\prime \prime } + \frac{a_3-a_1^2 a_2}{2a_1-a_2}\, g + 2\sigma g^3 = 0. \end{aligned}$$

(29)

The Jacobi elliptic function expansion method is used to solve the transformed ODE in Eq. (29).

Exact solutions via the Jacobi elliptic function expansion method

Based on the technique, the solutions can be found using the series

$$\begin{aligned} g(\xi ) = \sum _{i=0}^{n} F^{i}(\xi ). \end{aligned}$$

(30)

By the balancing procedure, i.e., balancing the highest order nonlinear term and the highest order derivative term in Eq. (29), we obtain $n=1$, and hence from Eq. (30), we have

$$\begin{aligned} g(\xi ) = c_0 + c_1 F(\xi ). \end{aligned}$$

(31)

where $F(\xi )$ is the solution defined in Eq. (31) as

$$\begin{aligned} g'(\xi ) = c_1 F'(\xi ). \end{aligned}$$

(32)

Substituting Eq. (31) into Eq. (32), we get

$$\begin{aligned} g'(\xi ) = c_1 \left( P F^4(\xi ) + Q F^2(\xi ) + R \right) ^{\frac{1}{2}}. \end{aligned}$$

(33)

$$\begin{aligned} g''(\xi ) = c_1 F(\xi ) \left( 2P F^2(\xi ) + Q \right) . \end{aligned}$$

(34)

Now, substituting Eqs. (31) and (34) into Eq. (29), we obtain a set of algebraic equations. Solving this system with the help of Maple, we get the coefficients involved in the series (31) as

$$\begin{aligned} c_0 = 0, \qquad c_1 = \pm \sqrt{-\frac{m_2}{\sigma }}, \qquad a_3 = a_1^2 a_2 - 2m_1 a_1 - m_1 a_2 . \end{aligned}$$

(35)

Exact wave solutions via Jacobi elliptic function method

Using the data from Tables 1 and 2, the Jacobi elliptic function solutions of periodic type are given as follows:

$$\begin{aligned} v_1 = \pm \sqrt{-\frac{m^{2}}{\sigma }}\, \textrm{sn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \exp \!\left[ i\,\frac{\Gamma (1+\Upsilon )}{\alpha } \left( a_1 x^\alpha + a_2 y^\alpha + a_3 t^\alpha \right) \right] . \end{aligned}$$

(36)

$$\begin{aligned} w_1 = 2\sigma (a_2 + a_1) \left[ \pm \sqrt{-\frac{m^{2}}{\sigma }}\, \textrm{sn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \right] ^2. \end{aligned}$$

(37)

$$\begin{aligned} s_1 = 2\sigma \left[ \pm \sqrt{-\frac{m^{2}}{\sigma }}\, \textrm{sn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \right] ^2. \end{aligned}$$

(38)

$$\begin{aligned} v_2 = \pm \sqrt{-\frac{m^{2}}{\sigma }}\, \textrm{cd}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \exp \!\left[ i\,\frac{\Gamma (1+\Upsilon )}{\alpha } \left( a_1 x^\alpha + a_2 y^\alpha + a_3 t^\alpha \right) \right] , \end{aligned}$$

(39)

$$\begin{aligned} w_2 = 2\sigma (a_2 + a_1) \left[ \pm \sqrt{-\frac{m^{2}}{\sigma }}\, \textrm{cd}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \right] ^2, \end{aligned}$$

(40)

$$\begin{aligned} s_2 = 2\sigma \left[ \pm \sqrt{-\frac{m^{2}}{\sigma }}\, \textrm{cd}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \right] ^2, \end{aligned}$$

(41)

$$\begin{aligned} v_3 = \pm \sqrt{\frac{m^{2}}{\sigma }}\, \textrm{cn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \exp \!\left[ i\,\frac{\Gamma (1+\Upsilon )}{\alpha } \left( a_1 x^\alpha + a_2 y^\alpha + a_3 t^\alpha \right) \right] , \end{aligned}$$

(42)

$$\begin{aligned} w_3 = 2\sigma (a_2 + a_1) \left[ \pm \sqrt{\frac{m^{2}}{\sigma }}\, \textrm{cn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \right] ^2, \end{aligned}$$

(43)

$$\begin{aligned} s_3 = 2\sigma \left[ \pm \sqrt{\frac{m^{2}}{\sigma }}\, \textrm{cn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \right] ^2, \end{aligned}$$

(44)

$$\begin{aligned} v_4 = \pm \sqrt{\frac{1}{\sigma }}\, \textrm{dn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \exp \!\left[ i\,\frac{\Gamma (1+\Upsilon )}{\alpha } \left( a_1 x^\alpha + a_2 y^\alpha + a_3 t^\alpha \right) \right] , \end{aligned}$$

(45)

$$\begin{aligned} w_4 = 2\sigma (a_2 + a_1) \left[ \pm \sqrt{\frac{1}{\sigma }}\, \textrm{dn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \right] ^2. \end{aligned}$$

(46)

$$\begin{aligned} s_4 = 2\sigma \left[ \pm \sqrt{\frac{1}{\sigma }}\, \textrm{dn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \right] ^2. \end{aligned}$$

(47)

$$\begin{aligned} v_5 = \pm \sqrt{-\frac{1}{\sigma }}\, \textrm{ns}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \exp \!\left[ i\,\frac{\Gamma (1+\Upsilon )}{\alpha } \left( a_1 x^\alpha + a_2 y^\alpha + a_3 t^\alpha \right) \right] , \end{aligned}$$

(48)

$$\begin{aligned} w_5 = 2\sigma (a_2 + a_1) \left[ \pm \sqrt{-\frac{1}{\sigma }}\, \textrm{ns}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \right] ^2, \end{aligned}$$

(49)

$$\begin{aligned} s_5 = 2\sigma \left[ \pm \sqrt{-\frac{1}{\sigma }}\, \textrm{ns}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \right] ^2. \end{aligned}$$

(50)

$$\begin{aligned} v_6 = \pm \sqrt{\frac{1}{\sigma }}\, \textrm{dc}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \exp \!\left[ i\,\frac{\Gamma (1+\Upsilon )}{\alpha } \left( a_1 x^\alpha + a_2 y^\alpha + a_3 t^\alpha \right) \right] , \end{aligned}$$

(51)

$$\begin{aligned} w_6 = 2\sigma (a_2 + a_1) \left[ \pm \sqrt{\frac{1}{\sigma }}\, \textrm{dc}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \right] ^2, \end{aligned}$$

(52)

$$\begin{aligned} s_6 = 2\sigma \left[ \pm \sqrt{\frac{1}{\sigma }}\, \textrm{dc}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \right] ^2. \end{aligned}$$

(53)

$$\begin{aligned} v_7 = \pm \sqrt{\frac{1-m^2}{\sigma }}\, \textrm{nc}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \exp \!\left[ i\,\frac{\Gamma (1+\Upsilon )}{\alpha } \left( a_1 x^\alpha + a_2 y^\alpha + a_3 t^\alpha \right) \right] , \end{aligned}$$

(54)

$$\begin{aligned} w_7 = 2\sigma (a_2 + a_1) \left[ \pm \sqrt{\frac{1-m^2}{\sigma }}\, \textrm{nc}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \right] ^2, \end{aligned}$$

(55)

$$\begin{aligned} s_7 = 2\sigma \left[ \pm \sqrt{\frac{1-m^2}{\sigma }}\, \textrm{nc}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \right] ^2. \end{aligned}$$

(56)

$$\begin{aligned} v_8 = \pm \sqrt{\frac{1-m^2}{\sigma }}\, \textrm{sc}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \exp \!\left[ i\,\frac{\Gamma (1+\Upsilon )}{\alpha } \left( a_1 x^\alpha + a_2 y^\alpha + a_3 t^\alpha \right) \right] , \end{aligned}$$

(57)

$$\begin{aligned} w_8 = 2\sigma (a_2 + a_1) \left[ \pm \sqrt{\frac{1-m^2}{\sigma }}\, \textrm{sc}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \right] ^2, \end{aligned}$$

(58)

$$\begin{aligned} s_8 = 2\sigma \left[ \pm \sqrt{\frac{1-m^2}{\sigma }}\, \textrm{sc}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \right] ^2. \end{aligned}$$

(59)

$$\begin{aligned} v_{9} = \pm \sqrt{\frac{1}{\sigma }}\, \textrm{cs}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \exp \!\left[ i\,\frac{\Gamma (1+\Upsilon )}{\alpha } \left( a_1 x^\alpha + a_2 y^\alpha + a_3 t^\alpha \right) \right] , \end{aligned}$$

(60)

$$\begin{aligned} w_{9} = 2\sigma (a_2 + a_1) \left[ \pm \sqrt{\frac{1}{\sigma }}\, \textrm{cs}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \right] ^2, \end{aligned}$$

(61)

$$\begin{aligned} s_{9} = 2\sigma \left[ \pm \sqrt{\frac{1}{\sigma }}\, \textrm{cs}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \right] ^2. \end{aligned}$$

(62)

$$\begin{aligned} v_{10} = \pm \sqrt{\frac{1}{\sigma }}\, \textrm{ds}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \exp \!\left[ i\,\frac{\Gamma (1+\Upsilon )}{\alpha } \left( a_1 x^\alpha + a_2 y^\alpha + a_3 t^\alpha \right) \right] , \end{aligned}$$

(63)

$$\begin{aligned} w_{10} = 2\sigma (a_2 + a_1) \left[ \pm \sqrt{\frac{1}{\sigma }}\, \textrm{ds}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \right] ^2, \end{aligned}$$

(64)

$$\begin{aligned} s_{10} = 2\sigma \left[ \pm \sqrt{\frac{1}{\sigma }}\, \textrm{ds}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \right] ^2. \end{aligned}$$

(65)

$$\begin{aligned} \begin{aligned} v_{11}&= \pm \sqrt{\frac{1}{4\sigma }}\, \Bigg [ m\,\textrm{cn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \mp \textrm{dn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \Bigg ] \\&\quad \times \exp \!\left[ i\,\frac{\Gamma (1+\Upsilon )}{\alpha } \left( a_1 x^\alpha + a_2 y^\alpha + a_3 t^\alpha \right) \right] . \end{aligned} \end{aligned}$$

(66)

$$\begin{aligned} w_{11} = 2\sigma (a_2 + a_1) \left[ \pm \sqrt{\frac{1}{4\sigma }}\, \left( m\,\textrm{cn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha + y^\alpha + c t^\alpha ) \right) \mp \textrm{dn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha + y^\alpha + c t^\alpha ) \right) \right) \right] ^2 . \end{aligned}$$

(67)

$$\begin{aligned} s_{11} = 2\sigma \left[ \pm \sqrt{\frac{1}{4\sigma }}\, \left( m\,\textrm{cn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha + y^\alpha + c t^\alpha ) \right) \mp \textrm{dn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha + y^\alpha + c t^\alpha ) \right) \right) \right] ^2 . \end{aligned}$$

(68)

$$\begin{aligned} \begin{aligned} v_{12}&= \pm \sqrt{\frac{1}{4\sigma }} \Bigg [ \textrm{ns}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \mp \textrm{cs}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \Bigg ] \\&\quad \times \exp \!\left[ i\,\frac{\Gamma (1+\Upsilon )}{\alpha } \left( a_1 x^\alpha + a_2 y^\alpha + a_3 t^\alpha \right) \right] . \end{aligned} \end{aligned}$$

(69)

$$\begin{aligned} w_{12} = 2\sigma (a_2 + a_1) \left[ \pm \sqrt{\frac{1}{4\sigma }} \left( \textrm{ns}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha + y^\alpha + c t^\alpha ) \right) \mp \textrm{cs}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha + y^\alpha + c t^\alpha ) \right) \right) \right] ^2 . \end{aligned}$$

(70)

$$\begin{aligned} s_{12} = 2\sigma \left[ \pm \sqrt{\frac{1}{4\sigma }} \left( \textrm{ns}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha + y^\alpha + c t^\alpha ) \right) \mp \textrm{cs}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha + y^\alpha + c t^\alpha ) \right) \right) \right] ^2 . \end{aligned}$$

(71)

$$\begin{aligned} \begin{aligned} v_{13}&= \pm \sqrt{\frac{1}{4\sigma }} \Bigg [ \csc \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \mp \cot \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \Bigg ] \\&\quad \times \exp \!\left[ i\,\frac{\Gamma (1+\Upsilon )}{\alpha } \left( a_1 x^\alpha + a_2 y^\alpha + a_3 t^\alpha \right) \right] . \end{aligned} \end{aligned}$$

(72)

$$\begin{aligned} w_{13} = 2\sigma (a_2 + a_1) \left[ \pm \sqrt{\frac{1}{4\sigma }} \left( \csc \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha + y^\alpha + c t^\alpha ) \right) \mp \cot \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha + y^\alpha + c t^\alpha ) \right) \right) \right] ^2 . \end{aligned}$$

(73)

$$\begin{aligned} s_{13} = 2\sigma \left[ \pm \sqrt{\frac{1}{4\sigma }} \left( \csc \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha + y^\alpha + c t^\alpha ) \right) \mp \cot \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha + y^\alpha + c t^\alpha ) \right) \right) \right] ^2 . \end{aligned}$$

(74)

$$\begin{aligned} \begin{aligned} v_{14}&= \pm \sqrt{\frac{1-m^2}{4\sigma }} \Bigg [ \textrm{nc}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \mp \textrm{sc}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \Bigg ] \\&\quad \times \exp \!\left[ i\,\frac{\Gamma (1+\Upsilon )}{\alpha } \left( a_1 x^\alpha + a_2 y^\alpha + a_3 t^\alpha \right) \right] . \end{aligned} \end{aligned}$$

(75)

$$\begin{aligned} w_{14} = 2\sigma (a_2+a_1) \left[ \pm \sqrt{\frac{1-m^2}{4\sigma }} \left( \textrm{nc}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha ) \right) \mp \textrm{sc}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha ) \right) \right) \right] ^2 . \end{aligned}$$

(76)

$$\begin{aligned} s_{14} = 2\sigma \left[ \pm \sqrt{\frac{1-m^2}{4\sigma }} \left( \textrm{nc}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha ) \right) \mp \textrm{sc}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha ) \right) \right) \right] ^2 . \end{aligned}$$

(77)

$$\begin{aligned} \begin{aligned} v_{15}&= \pm \sqrt{\frac{1}{4\sigma }} \Bigg [ \textrm{ns}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \mp \textrm{ds}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \Bigg ] \\&\quad \times \exp \!\left[ i\,\frac{\Gamma (1+\Upsilon )}{\alpha } \left( a_1 x^\alpha + a_2 y^\alpha + a_3 t^\alpha \right) \right] . \end{aligned} \end{aligned}$$

(78)

$$\begin{aligned} w_{15} = 2\sigma (a_2+a_1) \left[ \pm \sqrt{\frac{1}{4\sigma }} \left( \textrm{ns}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha ) \right) \mp \textrm{ds}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha ) \right) \right) \right] ^2 . \end{aligned}$$

(79)

$$\begin{aligned} s_{15} = 2\sigma \left[ \pm \sqrt{\frac{1}{4\sigma }} \left( \textrm{ns}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha ) \right) \mp \textrm{ds}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha ) \right) \right) \right] ^2 . \end{aligned}$$

(80)

$$\begin{aligned} v_{16} = \pm \sqrt{\frac{m^2}{4\sigma }} \left[ \textrm{sn}\!\left( x+y+ct\right) \mp i\,\textrm{cn}\!\left( x+y+ct\right) \right] \exp \!\left[ i\,\frac{\Gamma (1+\Upsilon )}{\alpha } (a_1x^\alpha +a_2y^\alpha +a_3t^\alpha ) \right] . \end{aligned}$$

(81)

$$\begin{aligned} w_{16} = 2\sigma (a_2+a_1) \left[ \pm \sqrt{\frac{m^2}{4\sigma }} \left( \textrm{sn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha ) \right) \mp i\,\textrm{cn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha ) \right) \right) \right] ^2 . \end{aligned}$$

(82)

$$\begin{aligned} s_{16} = 2\sigma \left[ \pm \sqrt{\frac{m^2}{4\sigma }} \left( \textrm{sn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha ) \right) \mp i\,\textrm{cn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha ) \right) \right) \right] ^2 . \end{aligned}$$

(83)

$$\begin{aligned} \begin{aligned} v_{17}&= \pm \sqrt{\frac{m^2}{4\sigma }}\, \frac{ \textrm{sn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) }{ \sqrt{ 1 - m^2 \textrm{sn}^2\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \mp \textrm{cn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) } } \\&\quad \times \exp \!\left[ i\,\frac{\Gamma (1+\Upsilon )}{\alpha } \left( a_1 x^\alpha + a_2 y^\alpha + a_3 t^\alpha \right) \right] . \end{aligned} \end{aligned}$$

(84)

$$\begin{aligned} w_{17} = 2\sigma (a_2+a_1) \left[ \pm \sqrt{\frac{m^2}{4\sigma }}\, \frac{ \textrm{sn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha ) \right) }{ \sqrt{ 1 - m^2 \textrm{sn}^2\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha ) \right) \mp \textrm{cn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha ) \right) } } \right] ^2 . \end{aligned}$$

(85)

$$\begin{aligned} s_{17} = 2\sigma \left[ \pm \sqrt{\frac{m^2}{4\sigma }}\, \frac{ \textrm{sn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha ) \right) }{ \sqrt{ 1 - m^2 \textrm{sn}^2\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha ) \right) \mp \textrm{cn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha ) \right) } } \right] ^2 . \end{aligned}$$

(86)

$$\begin{aligned} \begin{aligned} v_{18}&= \pm \sqrt{\frac{1}{4\sigma }} \Bigg [ m\,\textrm{sn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \mp i\,\textrm{dn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \Bigg ] \\&\quad \times \exp \!\left[ i\,\frac{\Gamma (1+\Upsilon )}{\alpha } \left( a_1 x^\alpha + a_2 y^\alpha + a_3 t^\alpha \right) \right] . \end{aligned} \end{aligned}$$

(87)

$$\begin{aligned} w_{18} = 2\sigma (a_2+a_1) \left[ \pm \sqrt{\frac{1}{4\sigma }} \left( m\,\textrm{sn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha ) \right) \mp i\,\textrm{dn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha ) \right) \right) \right] ^2 . \end{aligned}$$

(88)

$$\begin{aligned} s_{18} = 2\sigma \left[ \pm \sqrt{\frac{1}{4\sigma }} \left( m\,\textrm{sn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha ) \right) \mp i\,\textrm{dn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha ) \right) \right) \right] ^2 . \end{aligned}$$

(89)

$$\begin{aligned} v_{19} = \pm \sqrt{\frac{1}{4\sigma }}\, \frac{ \textrm{sn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha ) \right) }{ 1 \mp \textrm{dn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha ) \right) } \exp \!\left[ i\,\frac{\Gamma (1+\Upsilon )}{\alpha } (a_1x^\alpha +a_2y^\alpha +a_3t^\alpha ) \right] . \end{aligned}$$

(90)

$$\begin{aligned} w_{19} = 2\sigma (a_2+a_1) \left[ \pm \sqrt{\frac{1}{4\sigma }} \frac{ \textrm{sn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) }{ 1 \mp \textrm{dn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) } \right] ^2 . \end{aligned}$$

(91)

$$\begin{aligned} s_{19} = 2\sigma \left[ \pm \sqrt{\frac{1}{4\sigma }} \frac{ \textrm{sn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) }{ 1 \mp \textrm{dn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) } \right] ^2 . \end{aligned}$$

(92)

$$\begin{aligned} v_{20} = \pm \sqrt{\frac{m^2}{4\sigma }} \frac{ \textrm{sn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) }{ 1 \mp \textrm{dn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) } \exp \!\left[ i\,\frac{\Gamma (1+\Upsilon )}{\alpha }(a_1x^\alpha +a_2y^\alpha +a_3t^\alpha ) \right] . \end{aligned}$$

(93)

$$\begin{aligned} w_{20} = 2\sigma (a_2+a_1) \left[ \pm \sqrt{\frac{m^2}{4\sigma }} \frac{ \textrm{sn}(x+y+ct) }{ 1 \mp \textrm{dn}(x+y+ct) } \right] ^2 . \end{aligned}$$

(94)

$$\begin{aligned} s_{20} = 2\sigma \left[ \pm \sqrt{\frac{m^2}{4\sigma }} \frac{ \textrm{sn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) }{ 1 \mp \textrm{dn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) } \right] ^2 . \end{aligned}$$

(95)

$$\begin{aligned} v_{21} = \pm \sqrt{\frac{1-m^2}{4\sigma }} \frac{ \textrm{cn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) }{ 1 \mp \textrm{sn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) } \exp \!\left[ i\,\frac{\Gamma (1+\Upsilon )}{\alpha }(a_1x^\alpha +a_2y^\alpha +a_3t^\alpha ) \right] . \end{aligned}$$

(96)

$$\begin{aligned} w_{21} = 2\sigma (a_2+a_1) \left[ \pm \sqrt{\frac{1-m^2}{4\sigma }} \frac{ \textrm{cn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) }{ 1 \mp \textrm{sn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) } \right] ^2 . \end{aligned}$$

(97)

$$\begin{aligned} s_{21} = 2\sigma \left[ \pm \sqrt{\frac{1-m^2}{4\sigma }} \frac{ \textrm{cn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) }{ 1 \mp \textrm{sn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) } \right] ^2 . \end{aligned}$$

(98)

$$\begin{aligned} \begin{aligned} v_{22}&= \pm \sqrt{-\frac{(1-m^2)^2}{4\sigma }} \frac{ \textrm{sn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) }{ \textrm{dn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) \mp \textrm{cn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) } \\&\quad \times \exp \!\left[ i\,\frac{\Gamma (1+\Upsilon )}{\alpha } \left( a_1x^\alpha +a_2y^\alpha +a_3t^\alpha \right) \right] . \end{aligned} \end{aligned}$$

(99)

$$\begin{aligned} w_{22} = 2\sigma (a_2+a_1) \left[ \pm \sqrt{-\frac{(1-m^2)^2}{4\sigma }} \frac{ \textrm{sn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) }{ \textrm{dn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) \mp \textrm{cn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) } \right] ^2 . \end{aligned}$$

(100)

$$\begin{aligned} s_{22} = 2\sigma \left[ \pm \sqrt{-\frac{(1-m^2)^2}{4\sigma }} \frac{ \textrm{sn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) }{ \textrm{dn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) \mp \textrm{cn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) } \right] ^2 . \end{aligned}$$

(101)

$$\begin{aligned} v_{23} = \pm \sqrt{\frac{m^4}{4\sigma }} \frac{ \textrm{cn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) }{ \sqrt{ 1 - m^2 \mp \textrm{dn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) } } \exp \!\left[ i\,\frac{\Gamma (1+\Upsilon )}{\alpha }(a_1x^\alpha +a_2y^\alpha +a_3t^\alpha ) \right] . \end{aligned}$$

(102)

$$\begin{aligned} w_{23} = 2\sigma (a_2+a_1) \left[ \pm \sqrt{\frac{m^4}{4\sigma }} \frac{ \textrm{cn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) }{ \sqrt{ 1 - m^2 \mp \textrm{dn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) } } \right] ^2 . \end{aligned}$$

(103)

$$\begin{aligned} s_{23} = 2\sigma \left[ \pm \sqrt{\frac{m^4}{4\sigma }} \frac{ \textrm{cn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) }{ \sqrt{ 1 - m^2 \mp \textrm{dn}\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) } } \right] ^2 . \end{aligned}$$

(104)

Solitary wave solutions

In the limiting case when $m \rightarrow 1$, the solitary wave solutions are obtained as follows:

$$\begin{aligned} v_{24} = \pm \sqrt{-\frac{1}{\sigma }}\, \tanh \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha ) \right) \exp \!\left[ i\,\frac{\Gamma (1+\Upsilon )}{\alpha } (a_1x^\alpha +a_2y^\alpha +a_3t^\alpha ) \right] . \end{aligned}$$

(105)

$$\begin{aligned} w_{24} = 2\sigma (a_2+a_1) \left[ \pm \sqrt{-\frac{1}{\sigma }}\, \tanh \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha ) \right) \right] ^2 . \end{aligned}$$

(106)

$$\begin{aligned} s_{24} = 2\sigma \left[ \pm \sqrt{-\frac{1}{\sigma }}\, \tanh \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha ) \right) \right] ^2 . \end{aligned}$$

(107)

$$\begin{aligned} v_{25} = \pm \sqrt{-\frac{1}{\sigma }}\, \cosh \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha ) \right) \exp \!\left[ i\,\frac{\Gamma (1+\Upsilon )}{\alpha } (a_1x^\alpha +a_2y^\alpha +a_3t^\alpha ) \right] . \end{aligned}$$

(108)

$$\begin{aligned} w_{25} = 2\sigma (a_2+a_1) \left[ \pm \sqrt{-\frac{1}{\sigma }}\, \cosh \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha ) \right) \right] ^2 . \end{aligned}$$

(109)

$$\begin{aligned} s_{25} = 2\sigma \left[ \pm \sqrt{-\frac{1}{\sigma }}\, \cosh \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha ) \right) \right] ^2 . \end{aligned}$$

(110)

$$\begin{aligned} v_{26} = \pm \sqrt{-\frac{1}{\sigma }}\, sech\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha ) \right) \exp \!\left[ i\,\frac{\Gamma (1+\Upsilon )}{\alpha } (a_1x^\alpha +a_2y^\alpha +a_3t^\alpha ) \right] . \end{aligned}$$

(111)

$$\begin{aligned} w_{26} = 2\sigma (a_2+a_1) \left[ \pm \sqrt{-\frac{1}{\sigma }}\, sech\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha ) \right) \right] ^2 . \end{aligned}$$

(112)

$$\begin{aligned} s_{26} = 2\sigma \left[ \pm \sqrt{-\frac{1}{\sigma }}\, sech\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha ) \right) \right] ^2 . \end{aligned}$$

(113)

$$\begin{aligned} v_{27} = \pm \sqrt{-\frac{1}{\sigma }}\, \coth \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha ) \right) \exp \!\left[ i\,\frac{\Gamma (1+\Upsilon )}{\alpha } (a_1x^\alpha +a_2y^\alpha +a_3t^\alpha ) \right] . \end{aligned}$$

(114)

$$\begin{aligned} w_{27} = 2\sigma (a_2+a_1) \left[ \pm \sqrt{-\frac{1}{\sigma }}\, \coth \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha ) \right) \right] ^2 . \end{aligned}$$

(115)

$$\begin{aligned} s_{27} = 2\sigma \left[ \pm \sqrt{-\frac{1}{\sigma }}\, \coth \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha ) \right) \right] ^2 . \end{aligned}$$

(116)

$$\begin{aligned} v_{28} = \pm \sqrt{\frac{1}{\sigma }}\, csch\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha ) \right) \exp \!\left[ i\,\frac{\Gamma (1+\Upsilon )}{\alpha } (a_1x^\alpha +a_2y^\alpha +a_3t^\alpha ) \right] . \end{aligned}$$

(117)

$$\begin{aligned} w_{28} = 2\sigma (a_2+a_1) \left[ \pm \sqrt{\frac{1}{\sigma }}\, csch\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha ) \right) \right] ^2 . \end{aligned}$$

(118)

$$\begin{aligned} s_{28} = 2\sigma \left[ \pm \sqrt{\frac{1}{\sigma }}\, csch\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha ) \right) \right] ^2 . \end{aligned}$$

(119)

$$\begin{aligned} \begin{aligned} v_{29}&= \pm \sqrt{\frac{1}{4\sigma }} \Bigg [ sech \! \left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \mp sech \! \left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \Bigg ] \\&\quad \times \exp \!\left[ i\,\frac{\Gamma (1+\Upsilon )}{\alpha } \left( a_1 x^\alpha + a_2 y^\alpha + a_3 t^\alpha \right) \right] . \end{aligned} \end{aligned}$$

(120)

$$\begin{aligned} w_{29} = 2\sigma (a_2+a_1) \left[ \pm \sqrt{\frac{1}{4\sigma }} \left( sech\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) \mp sech\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) \right) \right] ^2 . \end{aligned}$$

(121)

$$\begin{aligned} s_{29} = 2\sigma \left[ \pm \sqrt{\frac{1}{4\sigma }} \left( sech\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) \mp sech\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) \right) \right] ^2 . \end{aligned}$$

(122)

$$\begin{aligned} \begin{aligned} v_{30}&= \pm \sqrt{\frac{1}{4\sigma }} \Bigg [ \coth \! \left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \mp csch \! \left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \Bigg ] \\[6pt]&\quad \times \exp \! \left[ i\,\frac{\Gamma (1+\Upsilon )}{\alpha } \left( a_1 x^\alpha + a_2 y^\alpha + a_3 t^\alpha \right) \right] . \end{aligned} \end{aligned}$$

(123)

$$\begin{aligned} w_{30} = 2\sigma (a_2+a_1) \left[ \pm \sqrt{\frac{1}{4\sigma }} \left( \coth \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) \mp csch\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) \right) \right] ^2 . \end{aligned}$$

(124)

$$\begin{aligned} s_{30} = 2\sigma \left[ \pm \sqrt{\frac{1}{4\sigma }} \left( \coth \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) \mp csch\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) \right) \right] ^2 . \end{aligned}$$

(125)

$$\begin{aligned} v_{31} = \pm \sqrt{\frac{1}{4\sigma }} \Bigg [ \tanh \! \left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \mp i\, sech \! \left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \Bigg ] \exp \! \left[ i\,\frac{\Gamma (1+\Upsilon )}{\alpha } \left( a_1 x^\alpha + a_2 y^\alpha + a_3 t^\alpha \right) \right] . \end{aligned}$$

(126)

$$\begin{aligned} w_{31} = 2\sigma (a_2+a_1) \left[ \pm \sqrt{\frac{1}{4\sigma }} \left( \tanh \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) \mp i\,sech\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) \right) \right] ^2 . \end{aligned}$$

(127)

$$\begin{aligned} s_{31} = 2\sigma \left[ \pm \sqrt{\frac{1}{4\sigma }} \left( \tanh \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) \mp i\,sech\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) \right) \right] ^2 . \end{aligned}$$

(128)

$$\begin{aligned} v_{32} = \pm \sqrt{\frac{1}{4\sigma }} \frac{ \tanh \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) }{ 1 \mp sech\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) } \exp \!\left[ i\frac{\Gamma (1+\Upsilon )}{\alpha } (a_1x^\alpha +a_2y^\alpha +a_3t^\alpha ) \right] . \end{aligned}$$

(129)

$$\begin{aligned} w_{32} = 2\sigma (a_2+a_1) \left[ \pm \sqrt{\frac{1}{4\sigma }} \frac{ \tanh \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) }{ 1 \mp sech\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) } \right] ^2 . \end{aligned}$$

(130)

$$\begin{aligned} s_{32} = 2\sigma \left[ \pm \sqrt{\frac{1}{4\sigma }} \frac{ \tanh \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) }{ 1 \mp sech\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) } \right] ^2 . \end{aligned}$$

(131)

$$\begin{aligned} v_{33} = \pm \sqrt{\frac{1}{4\sigma }}\, \frac{ sech\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha ) \right) }{ \sqrt{ \pm sech\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha ) \right) } } \exp \!\left[ i\,\frac{\Gamma (1+\Upsilon )}{\alpha } (a_1x^\alpha +a_2y^\alpha +a_3t^\alpha ) \right] . \end{aligned}$$

(132)

$$\begin{aligned} w_{33} = 2\sigma (a_2+a_1) \left[ \pm \sqrt{\frac{1}{4\sigma }}\, \frac{sech \! \left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) }{\sqrt{ \pm sech\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha ) \right) }} \right] ^2 . \end{aligned}$$

(133)

$$\begin{aligned} s_{33} = 2\sigma \left[ \pm \sqrt{\frac{1}{4\sigma }}\, \frac{ sech\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha ) \right) }{ \sqrt{ \pm sech\!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha ) \right) } } \right] ^2 . \end{aligned}$$

(134)

Shock wave solutions

In the limiting case when $m \rightarrow 0$, the trigonometric function solutions are obtained as follows:

$$\begin{aligned} v_{34} = \pm \sqrt{-\frac{1}{\sigma }}\, \csc \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) \exp \!\left[ i\frac{\Gamma (1+\Upsilon )}{\alpha } (a_1x^\alpha +a_2y^\alpha +a_3t^\alpha )\right] . \end{aligned}$$

(135)

$$\begin{aligned} w_{34} = 2\sigma (a_2+a_1) \left[ \pm \sqrt{-\frac{1}{\sigma }}\, \csc \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) \right] ^2 . \end{aligned}$$

(136)

$$\begin{aligned} s_{34} = 2\sigma \left[ \pm \sqrt{-\frac{1}{\sigma }}\, \csc \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) \right] ^2 . \end{aligned}$$

(137)

$$\begin{aligned} v_{35} = \pm \sqrt{\frac{1}{\sigma }}\, \sec \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) \exp \!\left[ i\frac{\Gamma (1+\Upsilon )}{\alpha } (a_1x^\alpha +a_2y^\alpha +a_3t^\alpha )\right] . \end{aligned}$$

(138)

$$\begin{aligned} w_{35} = 2\sigma (a_2+a_1) \left[ \pm \sqrt{\frac{1}{\sigma }}\, \sec \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) \right] ^2 . \end{aligned}$$

(139)

$$\begin{aligned} s_{35} = 2\sigma \left[ \pm \sqrt{\frac{1}{\sigma }}\, \sec \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) \right] ^2 . \end{aligned}$$

(140)

$$\begin{aligned} v_{36} = \pm \sqrt{\frac{1}{\sigma }}\, \tan \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) \exp \!\left[ i\frac{\Gamma (1+\Upsilon )}{\alpha } (a_1x^\alpha +a_2y^\alpha +a_3t^\alpha )\right] . \end{aligned}$$

(141)

$$\begin{aligned} w_{36} = 2\sigma (a_2+a_1) \left[ \pm \sqrt{\frac{1}{\sigma }}\, \tan \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) \right] ^2 . \end{aligned}$$

(142)

$$\begin{aligned} s_{36} = 2\sigma \left[ \pm \sqrt{\frac{1}{\sigma }}\, \tan \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) \right] ^2 . \end{aligned}$$

(143)

$$\begin{aligned} \begin{aligned} v_{37}&= \pm \sqrt{\frac{1}{4\sigma }} \Bigg [ \sec \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \mp \tan \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha } \left( x^\alpha + y^\alpha + c t^\alpha \right) \right) \Bigg ] \\&\quad \times \exp \!\left[ i\,\frac{\Gamma (1+\Upsilon )}{\alpha } \left( a_1 x^\alpha + a_2 y^\alpha + a_3 t^\alpha \right) \right] . \end{aligned} \end{aligned}$$

(144)

$$\begin{aligned} w_{37} = 2\sigma (a_2+a_1) [\sec \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) \mp \tan \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) \Bigg ]^2 . \end{aligned}$$

(145)

$$\begin{aligned} s_{37} = 2\sigma [\sec \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) \mp \tan \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) \Bigg ]^2. \end{aligned}$$

(146)

$$\begin{aligned} v_{38} = \pm \sqrt{\frac{1}{\sigma }}\, \cot \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) \exp \!\left[ i\frac{\Gamma (1+\Upsilon )}{\alpha } (a_1x^\alpha +a_2y^\alpha +a_3t^\alpha )\right] . \end{aligned}$$

(147)

$$\begin{aligned} w_{38} = 2\sigma (a_2+a_1) \left[ \pm \sqrt{\frac{1}{\sigma }}\, \cot \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) \right] ^2 . \end{aligned}$$

(148)

$$\begin{aligned} s_{38} = 2\sigma \left[ \pm \sqrt{\frac{1}{\sigma }}\, \cot \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) \right] ^2 . \end{aligned}$$

(149)

$$\begin{aligned} v_{39} = \pm \sqrt{\frac{1}{4\sigma }} \frac{ \cos \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) }{ 1 \mp \sin \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) } \exp \!\left[ i\frac{\Gamma (1+\Upsilon )}{\alpha } (a_1x^\alpha +a_2y^\alpha +a_3t^\alpha )\right] . \end{aligned}$$

(150)

$$\begin{aligned} w_{39} = 2\sigma (a_2+a_1) [\frac{ \cos \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) }{ 1 \mp \sin \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) }\Big ]^2 . \end{aligned}$$

(151)

$$\begin{aligned} s_{39} = 2\sigma [\frac{ \cos \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) }{ 1 \mp \sin \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) }\Big ]^2 . \end{aligned}$$

(152)

$$\begin{aligned} v_{40} = \pm \sqrt{-\frac{1}{4\sigma }} \frac{ \sin \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) }{ 1 \mp \cos \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) } \exp \!\left[ i\frac{\Gamma (1+\Upsilon )}{\alpha } (a_1x^\alpha +a_2y^\alpha +a_3t^\alpha )\right] . \end{aligned}$$

(153)

$$\begin{aligned} w_{40} = 2\sigma (a_2+a_1) [\frac{ \sin \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) }{ 1 \mp \cos \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) }\Big ]^2 . \end{aligned}$$

(154)

$$\begin{aligned} s_{40} = 2\sigma [\frac{ \sin \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) }{ 1 \mp \cos \!\left( \frac{\Gamma (1+\Upsilon )}{\alpha }(x^\alpha +y^\alpha +ct^\alpha )\right) }\Big ]^2 . \end{aligned}$$

(155)

Graphical representation of the solutions

This section presents a detailed graphical analysis of the newly obtained exact solutions of the $(2+1)$-dimensional complex modified Korteweg–de Vries (CmKdV) system derived using the truncated M-fractional derivative and the Jacobi elliptic function expansion method. The graphical representations are designed not only to visualize the analytical solutions but also to provide a systematic dynamical and physical interpretation of the wave phenomena described by the model. Jacobi elliptic function solutions, solitary wave solutions, and shock-type wave solutions are illustrated through two-dimensional (2D), three-dimensional (3D), and contour plots generated using Mathematica for different values of the temporal variable t. As shown in Figs. 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 and 18, the solutions exhibit diverse and novel physical structures, including periodic waves, bright and dark solitons, singular bright and dark solitons, anti-kink waves, and kink waves. The 2D plots offer clear spatial cross-sections of the wave profiles, allowing direct observation of changes in amplitude and wavelength, while the 3D surface plots demonstrate the spatiotemporal evolution of wave amplitudes and the propagation characteristics of the solutions. Contour plots further highlight regions of strong oscillations and localized energy concentration, making the effects of nonlinear interactions more transparent. By varying key control parameters such as the wave number, fractional order, and time, the simulations reveal their direct influence on the amplitude, width, and oscillatory behavior of the solutions. In particular, the wave number plays a crucial role in controlling soliton propagation speed and waveform localization, providing a clear physical mechanism for tuning solution characteristics. These graphical results verify the correctness and stability of the analytical solutions obtained earlier and serve as an effective bridge between mathematical theory and physical interpretation. Moreover, the combined use of mathematical modeling and computational visualization enhances the applicability of the CmKdV system in multidisciplinary engineering and scientific research, including nonlinear wave propagation, fluid dynamics, optical pulse transmission, and signal processing. The ability to visually analyze and control solution behavior through key parameters demonstrates the practical relevance of the proposed model and analytical framework. Detailed explanations of each figure, including the physical roles of all parameters and constants, are provided in the subsequent discussion.

Fig. 2

Two-dimensional, three-dimensional, and contour visualizations of $|v_1|$, illustrating the effect of time on wave propagation for the chosen parameter values $m = 0.8,\quad \sigma = -1,\quad a_1 = 0.5,\quad a_2 = 0.5,\quad a_3 = 1,\quad y = 0,\quad \alpha = 1,\quad \gamma = 0.8,\quad c = 0.1.$.

Fig. 3

Two-dimensional, three-dimensional, and contour visualizations of $|v_1|$, illustrating the effect of time on wave propagation for the chosen parameter values $m = 0.8,\quad \sigma = -1,\quad a_1 = 0.5,\quad a_2 = 0.5,\quad a_3 = 1,\quad y = 0,\quad \alpha = 1,\quad \gamma = 0.8,\quad c = 1.5.$.

Fig. 4

Two-dimensional, three-dimensional, and contour visualizations of $|v_1|$, illustrating the effect of time on wave propagation for the chosen parameter values $m = 0.8,\quad \sigma = -1,\quad a_1 = 0.5,\quad a_2 = 0.5,\quad a_3 = 1,\quad y = 0,\quad \alpha = 1,\quad \gamma = 0.8,\quad c = 3.$.

Fig. 5

Two-dimensional, three-dimensional, and contour visualizations of $|v_4|$, illustrating the effect of time on wave propagation for the chosen parameter values $m = 0.8,\quad \sigma = -1,\quad a_1 = 0.5,\quad a_2 = 0.5,\quad a_3 = 1,\quad y = 0,\quad \alpha = 1,\quad \gamma = 0.8,\quad c = 3.$.

Fig. 6

Two-dimensional, three-dimensional, and contour visualizations of $|v_4|$, illustrating the effect of time on wave propagation for the chosen parameter values $m = 0.8,\quad \sigma = -1,\quad a_1 = 0.5,\quad a_2 = 0.5,\quad a_3 = 1,\quad y = 0,\quad \alpha = 1,\quad \gamma = 0.8,\quad c = 0.1.$.

Fig. 7

Two-dimensional, three-dimensional, and contour visualizations of $|v_4|$, illustrating the effect of time on wave propagation for the chosen parameter values $m = 0.8,\quad \sigma = -1,\quad a_1 = 0.5,\quad a_2 = 0.5,\quad a_3 = 1,\quad y = 0,\quad \alpha = 1,\quad \gamma = 0.8,\quad c = 3.$.

Fig. 8

Two-dimensional, three-dimensional, and contour visualizations of $|v_5|$, illustrating the effect of time on wave propagation for the chosen parameter values $m = 0.8,\quad \sigma = -1,\quad a_1 = 0.5,\quad a_2 = 0.5,\quad a_3 = 1,\quad y = 0,\quad \alpha = 1,\quad \gamma = 0.8,\quad c = 0.1.$.

Fig. 9

Two-dimensional, three-dimensional, and contour visualizations of $|v_5|$, illustrating the effect of time on wave propagation for the chosen parameter values $m = 0.8,\quad \sigma = -1,\quad a_1 = 0.5,\quad a_2 = 0.5,\quad a_3 = 1,\quad y = 0,\quad \alpha = 1,\quad \gamma = 0.8,\quad c = 1.5.$.

Fig. 10

Two-dimensional, three-dimensional, and contour visualizations of $|v_5|$, illustrating the effect of time on wave propagation for the chosen parameter values $m = 0.8,\quad \sigma = -1,\quad a_1 = 0.5,\quad a_2 = 0.5,\quad a_3 = 1,\quad y = 0,\quad \alpha = 1,\quad \gamma = 0.8,\quad c = 1.5.$.

Fig. 11

Two-dimensional, three-dimensional, and contour visualizations of $|v_7|$, illustrating the effect of time on wave propagation for the chosen parameter values $m = 0.8,\quad \sigma = -1,\quad a_1 = 0.5,\quad a_2 = 0.5,\quad a_3 = 1,\quad y = 0,\quad \alpha = 1,\quad \gamma = 0.8,\quad c = 0.1.$.

Fig. 12

Two-dimensional, three-dimensional, and contour visualizations of $|v_7|$, illustrating the effect of time on wave propagation for the chosen parameter values $m = 0.8,\quad \sigma = -1,\quad a_1 = 0.5,\quad a_2 = 0.5,\quad a_3 = 1,\quad y = 0,\quad \alpha = 1,\quad \gamma = 0.8,\quad c = 1.5.$.

Fig. 13

Two-dimensional, three-dimensional, and contour visualizations of $|v_7|$, illustrating the effect of time on wave propagation for the chosen parameter values $m = 0.8,\quad \sigma = -1,\quad a_1 = 0.5,\quad a_2 = 0.5,\quad a_3 = 1,\quad y = 0,\quad \alpha = 1,\quad \gamma = 0.8,\quad c = 3.$.

Fig. 14

Two-dimensional, three-dimensional, and contour visualizations of $|v_{11}|$, illustrating the effect of time on wave propagation for the chosen parameter values $m = 0.8,\quad \sigma = -1,\quad a_1 = 0.5,\quad a_2 = 0.5,\quad a_3 = 1,\quad y = 0,\quad \alpha = 1,\quad \gamma = 0.8,\quad c = 0.1.$.

Fig. 15

Two-dimensional, three-dimensional, and contour visualizations of $|v_{11}|$, illustrating the effect of time on wave propagation for the chosen parameter values $m = 0.8,\quad \sigma = -1,\quad a_1 = 0.5,\quad a_2 = 0.5,\quad a_3 = 1,\quad y = 0,\quad \alpha = 1,\quad \gamma = 0.8,\quad c = 1.5.$.

Fig. 16

Two-dimensional, three-dimensional, and contour visualizations of $|v_{11}|$, illustrating the effect of time on wave propagation for the chosen parameter values $m = 0.8,\quad \sigma = -1,\quad a_1 = 0.5,\quad a_2 = 0.5,\quad a_3 = 1,\quad y = 0,\quad \alpha = 1,\quad \gamma = 0.8,\quad c = 3.$.

Fig. 17

Two-dimensional, three-dimensional, and contour visualizations of $|v_{24}|$, illustrating the effect of time on wave propagation for the chosen parameter values $\sigma = -1,\quad a_1 = 0.5,\quad a_2 = 0.5,\quad a_3 = 1,\quad y = 0,\quad \alpha = 1,\quad \gamma = 0.8,\quad c = 0.12$.

Fig. 18

Two-dimensional, three-dimensional, and contour visualizations of $|v_{26}|$, illustrating the effect of time on wave propagation for the chosen parameter values $\sigma = -1,\ a_1 = 0.5,\ a_2 = 0.5,\ a_3 = 1,\ y = 0,\ \alpha = 0.7,\ \gamma = 0.8,\ c = 0.09$.

Conclusions

In this article, we present a novel analytical investigation of the $(2+1)$-dimensional modified Korteweg–de Vries (mKdV) system by combining the truncated M-fractional derivative with the Jacobi elliptic function expansion method. This combination represents the main originality of the work and enables the construction of new exact solution families, including Jacobi elliptic function solutions, solitary wave solutions, and shock-type wave solutions, which have not been previously reported for this fractional $(2+1)$-dimensional mKdV model. The obtained results demonstrate the strong capability of the proposed approach in capturing the intrinsic nonlinear characteristics and rich wave dynamics of the system, while the unified elliptic-function framework provides a smooth transition from periodic to solitary-wave structures. To enhance physical interpretation, Mathematica-based two-dimensional, three-dimensional, and contour plots were employed to clearly illustrate the evolution and structural features of the solutions. Furthermore, the exact analytical solutions derived in this study offer reliable benchmark data for future numerical and data-driven investigations, particularly for Physics-Informed Neural Networks (PINNs), where they can be used to generate accurate initial and boundary conditions and to evaluate model accuracy and convergence. Overall, this work advances the theoretical understanding of the fractional $(2+1)$-dimensional mKdV system and establishes a solid analytical and computational foundation for future research in nonlinear wave dynamics, fractional models, and engineering-related applications.