Computational and Numerical Solutions for (2+ 1)-Dimensional Integrable Schwarz–Korteweg–de Vries Equation with Miura Transform

)is paper investigates the analytical, semianalytical, and numerical solutions of the (2 + 1)–dimensional integrable Schwarz–Korteweg–de Vries (SKdV) equation. )e extended simplest equation method, the sech-tanh method, the Adomian decomposition method, and cubic spline scheme are employed to obtain distinct formulas of solitary waves that are employed to calculate the initial and boundary conditions. Consequently, the numerical solutions of this model can be investigated. Moreover, their stability properties are also analyzed. )e solutions obtained by means of these techniques are compared to unravel relations between them and their characteristics illustrated under the suitable choice of the parameter values.


Introduction
e Korteweg-de Vries (KdV) equation is a seminal model in fluid mechanics. is model was introduced by Boussinesq in 1877 and reintroduced by Diederik Korteweg and Gustav de Vries in 1895. e KdV has the following formula [1][2][3][4][5][6][7][8][9]: where U � U(x, t) characterizes the weakly nonlinear shallow water waves. Equation (1) can be written in many distinct forms and combined with other models. One of them is the Schwarz-Korteweg-de Vries (SKdV) equation given by where Q � Q(x, t) is the unknown function. e SKdV was derived by Krichever and Novikov [10] and Weiss [11,12].
If we adopt the wave transformation B(x, y, t) � B(Z), then we convert equation (7) into an ordinary differential equation (NLODE). e integration of the obtained NLODE with zero constant of integration leads to If we consider the substitution B ′ � F, then it results in 4cF + ρ 2 δF ″ + 6ρδF 2 � 0.
Having these ideas in mind, this paper is organized as follows: Section 2 presents the two methods and derives the solutions of the SKdV equation. Section 5 represents the solutions for several numerical values of the parameters. Additionally, their stability and properties are also discussed. Finally, Section 6 summarises the main conclusions.

Explicit Solutions
In this section, we apply two analytical techniques for deriving the solutions of the (2 + 1)-dimensional integrable SKdV model. We adopt the extended simplest equation method and the sech-tanh method to obtain various distinct formulas of solitary wave solutions of equation (3). For further details about the two methods, see [22][23][24][25][26].

Extended Simplest Equation.
According to the homogeneous balance rule between the highest derivative and the nonlinear term in equation (9), we obtain n � 2. us, the general solution of equation (10) is given by where a i (i � − 2, . . . , 2) are arbitrary constants. Additionally, C(Z) satisfies the following ordinary differential equation: where β, α, and μ are the arbitrary constants. Substituting equations (11) and (12) into (9) and collecting all terms of C i (Z) (i � − 4, − 3, . . . , 3, 4), we get a system of algebraic equations. Solving this system, we obtain two families of solutions.
Family 1 Family 2 From these two families, the solitary wave solutions of equation (7) can be obtained.
According to Family 1, we have the following expressions.

Complexity
According to Family 2, we have the following expressions.

Sech-Tanh Method.
e general solution of equation (10) according to the sech-tanh method and calculated value of balance is given by where a 0 , a 1 , a 2 , b 1 , and b 2 are the arbitrary constants. Substituting equation (29) into (10) and collecting all terms of sech(Z), sech 2 (Z), sech 3 (Z), tanh (Z), tanh(Z)sec h 2 (Z), and tanh(ξ)sech(Z), we obtain a system of algebraic equations. Solving this system, we obtain 4 Complexity Consequently, the explicit solution of equation (7) is given by

Stability Investigation
We now examine the stability property for (2 + 1)-dimensional integrable SKdV model with the Miura transformation by means of an Hamiltonian system. e momentum H in the Hamiltonian system is given by where B(Z) is the solution of the model. Consequently, the condition for stability of the solutions can be formulated as where c is the wave velocity. e momentum in the Hamiltonian system for equations (18) and (31) are given, respectively, by H � 1 72 4 100c + log e 7− 5c + e 5c + log e 23− 5c + e 5c − log e − 5c + e 5c+7 − log e − 5c + e 5c+23 − sech 2 5c + 7 2 − sech 2 5c + 23 2 + sech 2 7 2 − 5c + sech 2 23 2 − 5c .
And thus, We conclude that this solution is stable on the interval 5]. is result shows the ability of the solutions for their application. Using the same steps, we can check the stability property of all other obtained solutions.

Semianalytical and Numerical Solutions
is section applies semianalytical and numerical schemes for deriving the solutions of the (2 + 1)-dimensional integrable SKdV model. e Adomian decomposition method and cubic b-spline schemes are employed to the method to investigate the accuracy of the obtained analytical solutions. Also, this study aims to give a comparison between both used analytical schemes. For further details about the two methods, see [27][28][29][30].

Cubic B-Spline Scheme.
Employing the cubic B-spline scheme to evaluate the numerical solutions of equation (10). Using same initial and boundary conditions with respect to the obtained solutions (18) and (31), yields

Discussion
is section illustrates several of the results for F(x, y, t) to highlight the properties of the (2 + 1)-dimensional integrable SKdV model with Miura transformation. In the   Complexity follow-up, we fix the value of y to characterize these solutions and the interpretation is based on three different types of representations (three-and two-dimensional charts and contour plot). In the following steps, the physical interpretation of the represented figures is discussed: (i) Figure 1 shows the bright solitary for (18) in the three-dimensional plot (a) to illustrate the perspective view of the solution, the two-dimensional plot (b) to present the wave propagation pattern of the wave along xaxis, and the contour plot (c) to explain the overhead view of the solution when α � Figure 2 shows the dark solitary for (31) in the three-dimensional plot (a) to illustrate the perspective view of the solution, the two-dimensional plot (b) to present the wave propagation pattern of the wave along the x-axis, and the contour plot (c) to explain the overhead view of the solution when Complexity Figure 3 illustrates the exact and semianalytical obtained solutions, respectively, by the extended simplest equation method and Adomian decomposition method (iv) Figure 4 illustrates the exact and semianalytical obtained solutions, respectively, by the sech-tanh expansion method and Adomian decomposition method (v) Figure 5 illustrates the exact and numerical obtained solutions, respectively, via the sech-tanh expansion method and cubic B-spline scheme

Conclusion
In this paper, the extended simplest equation and sechtanh expansion methods have been successfully implemented on the (2 + 1)-dimensional integrable SKdV model with Miura transform. Moreover, the stability properties of the solutions have also been tackled. e Adomian decomposition method and cubic B-spline scheme have also employed to investigate the semianalytical and numerical solutions, and the two show the accuracy of the obtained analytical solutions. e rigor of the obtained solutions that have been obtained by the sech-tanh expansion method has been discussed. e solutions were represented by allowing a physical interpretation and better interpretation of their properties. In summary, this paper studied the SKdV and found relevant solutions that provide new interpretations of the realworld phenomena.

Data Availability
Data sharing is not applicable for this article as no datasets were generated or analyzed during the current study.

Conflicts of Interest
e authors declare that there are no conflicts of interest.   Tables 1 and 2.