Traveling wave solutions, numerical solutions, and stability analysis of the (2 + 1) conformal time-fractional generalized q -deformed sinh-Gordon equation

: The two-dimensional conformal time-fractional generalized q -deformed sinh-Gordon equation has been used to model a variety of physical systems, including soliton propagation in asymmetric media, nonlinear waves in optical ﬁ bers, quantum ﬁ eld theory, and condensed matter physics. The equation is able to capture the complex dynamics of these systems and has been shown to be a powerful tool for studying them. This article discusses the two-dimensional conformal time-fractional generalized q -deformed sinh-Gordon equation both analytically and numerically using Kudryashov ’ s approach and the ﬁ nite di ﬀ erence method. In addition, the stability analysis and local truncation error of the equation are discussed. A number of illustrations are also included to show the various solitons propagation patterns. The proposed equation has opened up new possibilities for modeling asymmetric physical systems.


Introduction
In mathematics and physics, q-deformed (or quantum deformed) objects refer to a family of objects, such as functions, operators, or algebras, that depend on a deformation parameter q.The q-deformed objects are a generalization of their undeformed counterparts, which arise when q is set to 1.The deformation parameter q can be a complex number, but in many applications, it is a root of unity or a parameter that interpolates between different values.The theory of q-deformed objects plays a crucial role in many areas of mathematics and physics, including quantum mechanics, statistical mechanics, knot theory, and combinatorics.The q-deformed objects have been extensively studied, and many of their properties and applications have been established.In this context, the study of the q-deformed sinh-Gordon equation, which is a q-deformed version of the classical sinh-Gordon equation, has attracted considerable attention.
Although dynamical models are fundamental to many scientific fields, they are often overlooked in the literature.Nonlinear differential equations are employed to describe dynamics in microscopical quantum systems, where nonlinearity plays a crucial role [1,2].Numerous researchers have explored various partial differential equations with significant applications in diverse fields [3][4][5][6][7][8].
The research is structured as follows: Section 2 presents the mathematical analysis of the model; Section 3 outlines the methodology employed; the solutions are presented in Section 4; Section 5 provides numerical solutions, stability analysis, and local truncation error of the equation; various figures for some solutions are presented in Section 6; and finally, Section 7 contains the conclusion.

Analyzing the model mathematically
Use the transformation below to obtain the solution describing a traveling wave for Eq.(1.1). where and B is the speed of the traveling wave.By using Eqs.
• Case one: Thus, Eq. (2.3) can be expressed as follows: After multiplying both sides of Eq. (2.3) by ′ u ( ) and inte- gration, we obtain where C 1 represents the integration constant.Consider This leads to the following expression for Eq.(2.5): Hence, we can use our method to solve Eq. (2.7) and obtain the solution to Eq. (1.2) for the first case using Eqs.(2.6) and (2.1).
Thus, Eq. (2.3) can be expressed as follows: Simplification of Eq. (2.8) yields the following expression: Thus, Eq. (2.10) can be expressed as follows: Hence, we can use our method to solve Eq. (2.11) and obtain the solution to Eq. (1.2) for the second case using Eqs.(2.10) and (2.1).

The approach of the analytical method
Express the governing equation in the following form: We start with a polynomial x τ , ( ) and its partial deriva- tives represented by G.In order to transform Eq. (3.1) into an ordinary differential equation, we employ a traveling wave transformation given by Eq. (2.1).This results in the following expression: The Kudryashov method can be broken down into the following series of steps: Step 1: Let us assume that we can represent precise solutions to Eq. (3.2) in the following form: The constants M i (where = i N 0, 1, 2,…, and ≠ M 0 N ) can be determined using appropriate techniques.The value of N can be determined by applying the homo- geneous balancing principle.
Step 2: The function ( ) satisfies the following equation: The solution to Eq. (3.4) can be expressed in the following manner: Step 3: We can obtain a polynomial of ( ) by substi- tuting Eq. (3.3) into Eq.(3.1), grouping together all terms with the same powers of ( ) and equating each corresponding coefficient to zero.
Step 4: The solutions to Eq. (3.2) can be obtained by utilizing the Mathematica software to solve the resulting system.

The mathematical solution of the model
In this section, we employ the Kudryashov method to obtain analytical solutions for two cases of Eq. (1.2).
• For case one, at = = s κ 1, and = 0 : By applying the balance principle to Eq. (2.7) between the terms ′ v 2 and v 3 , we obtain the relation Thus, the solution to Eq. (2.7) can be expressed as follows: Substituting Eq. (4.1) into Eq.(2.7) and equating the coefficients of similar powers of ( ) to zero, we obtain the following system:

B
Using the Mathematica program to solve the aforementioned set of equations yields the following solutions: • Class 1: , .
• Class 2: By applying the balance principle to Eq. (2.11) between the terms ″ vv and v 3 , we obtain the relation The solution to Eq. (2.11) can be obtained from Eq. (3.3) as follows: Substituting Eq. (4.6) into Eq.(2.11) and equating the coefficients of similar powers of ( ) to zero, we obtain the following system: Traveling wave solutions, numerical solutions and stability analysis  3

B
Solving the aforementioned set of equations using the Mathematica program yields the following set of solutions: • Class 1: .

The numerical solution of the model
In this section, we utilize approximations for the space (x y , ) and time (τ) derivatives, as mentioned in the studies by Raslan et al. [12] and EL-Danaf et al. [13]: Based on the properties of the conformal fractional derivative, if The approximation for the time derivative with respect to τ is: Assuming that is the exact solution at the grid point x y τ , , i j n ( ), and that U i j n , , is the corresponding numerical solution, we can obtain a system of difference equations by substituting Eqs.(5.1) and (5.2) into Eq.(1.2) as follows: .

Local truncation error
Let us now introduce the local truncation error of our scheme.
Theorem 1.The local truncation error of the finite difference scheme given by Eq. (5.4) is Proof.By employing Taylor's expansion in Eq. ( 5.4), we can investigate the local truncation error in two-dimensional space and time as follows: Hence, → T 0 ( ) ( ) ( ) .Consequently, the local truncation error of the finite difference scheme given by Eq. (5.4) can be expressed as follows:

Stability of the finite difference scheme
In this subsection, we examine the stability of the difference scheme.
Theorem 2. The difference scheme given by Eq. (5.4) is stable if = G MU i j n i j n , , , , .
Proof.The stability analysis of the scheme yields the following: To begin, we express the scheme in matrix form as follows: Next, we determine the eigenvalues of the matrix: We solve Eq. (5.6) for Ω as follows: To verify the stability of the scheme, we examine the sign of the real part of Ω.If the real part of Ω is negative, the scheme is stable.
For this particular case, the real part of Ω is negative for all values of x Δ , y Δ , n, and α.Thus, we can conclude that the scheme is stable.
Theorem 3. The difference scheme (5.4) Proof.The stability of the scheme is determined by the eigenvalues of the amplification matrix, which can be defined as follows: The eigenvalues of A can be obtained as follows: The scheme is stable if all the eigenvalues of . This is true if Traveling wave solutions, numerical solutions and stability analysis  5

The numerical outcomes
In the following section, we present some of the results obtained through numerical analysis of the (2+1) conformal time-fractional generalized q-deformed sinh-Gordon equation in general.Moreover, we provide the numerical solution for two specific scenarios of the generalized q-deformed sinh-  Gordon equation, for which we have already explored the analytical solution.
• Case one: = = = s κ 1, 0 : We compare the results obtained through numerical analysis and the solution obtained through analytical method given by Eq. (4.5) for Eq.(1.1) at The comparison is presented in Table 1 and Figure 3.
1, and = − q 2 : Table 2 and Figure 4 depict a comparison made between the results obtained through numerical analysis and the solution obtained through analytical method given by Eq. (4.8) for Eq.(1.1) at 0.7, = ϑ 0.7, and = y 5. We compare the results obtained through numerical analysis and the solution obtained through analytical    method given by Eq. ( 4.5) at 1, and = x 4 for var- ious values of α.The comparison is presented in Table 3.
Table 4 displays a comparison made between the results obtained through numerical analysis and the solution obtained through analytical method given by Eq. (4.8) at x 5, and = y 5 for various values of α.

Illustrations with graphics
In order to provide a better understanding of the solutions presented, we present some two-dimensional and threedimensional figures.Specifically, Figures 1-4 display both analytical and numerical solutions. .In addition, we provide a comparison between the numerical results of Eq. (1.1) and its analytical solution given by Eq. (4.5) at 0.7, and = ϑ 0.3, with = α 1, in Figure 3. Finally, Figure 4 displays a comparison between the numerical findings of Eq. (1.1) and the analytical solution of Eq. (4.8) at 0.02, = q 0.7, and = ϑ 0.7.

Discussion
The results obtained through the Kudryashov method and the numerical analysis provide insights into the behavior of the solution of the q-deformed sinh-Gordon equation.The numerical results provide a good approximation of the analytical solution, with minor errors observed in comparing the two solutions.The stability analysis shows that the numerical scheme used is stable for the chosen parameters, and the evaluation of the local truncation error indicates that the difference scheme is convergent.Furthermore, the results obtained in this study can be used to describe physical systems that have lost their symmetry.The q-deformed sinh-Gordon equation is a valuable tool for modeling such systems, and the numerical and analytical solutions obtained in this study can be used to predict the behavior of these systems.In addition, our study presents a clear strategy for solving other models using the Kudryashov method and the finite difference method, which can be used to explore alternative techniques for solving different systems.
In addition, our study contributes to the field of mathematical modeling and provides a foundation for further research in this area.The numerical and analytical solutions presented in this study can be used to explore the behavior of physical systems that have lost their symmetry, and our results can be applied to a wide range of practical applications.

Conclusion
In conclusion, we have demonstrated how to apply the Kudryashov method to investigate the two-dimensional conformal time-fractional generalized q-deformed sinh- Gordon equation.We have also conducted a comprehensive numerical analysis of this model using the finite difference method, including a stability analysis and an evaluation of the local truncation error for the difference scheme.Furthermore, we have compared the analytical and numerical solutions and found that our results represent a significant contribution to the field.In the future, we plan to apply this method to solve additional models and explore alternative techniques for solving different systems.The proposed equation has opened up new avenues for describing physical systems that have lost their symmetry.

Table 1 :
A comparison made between the results obtained through numerical analysis and the solution obtained through analytical method

Table 2 :
A comparison made between the results obtained through numerical analysis and the solution obtained through analytical method

Table 3 :
A comparison made between the results obtained through numerical analysis and the solution obtained through analytical method

Table 4 :
A comparison made between the results obtained through numerical analysis and the solution obtained through analytical method