Abundant Traveling Wave and Numerical Solutions of Weakly Dispersive Long Waves Model

: In this article, plenty of wave solutions of the (2 + 1)-dimensional Kadomtsev–Petviashvili– Benjamin–Bona–Mahony ((2 + 1)-D KP-BBM) model are constructed by employing two recent analytical schemes (a modiﬁed direct algebraic (MDA) method and modiﬁed Kudryashov (MK) method). From the point of view of group theory, the proposed analytical methods in our article are based on symmetry, and effectively solve those problems which actually possess explicit or implicit symmetry. This model is a vital model in shallow water phenomena where it demonstrates the wave surface propagating in both directions. The obtained analytical solutions are explained by plotting them through 3D, 2D, and contour sketches. These solutions’ accuracy is also tested by calculating the absolute error between them and evaluated numerical results by the Adomian decomposition (AD) method and variational iteration (VI) method. The considered numerical schemes were applied based on constructed initial and boundary conditions through the obtained analytical solutions via the MDA, and MK methods which show the synchronization between computational and numerical obtained solutions. This coincidence between the obtained solutions is explained through two-dimensional and distribution plots. The applied methods’ symmetry is shown through comparing their obtained results and showing the matching between both obtained solutions (analytical and numerical). sech-tanh expansion algebraic equation Adomian decomposition method, iteration method, Khater methods, B-spline


Introduction
Recently, the phenomenon of shallow water waves has attracted the attention of many researchers in different fields. The flow below the medium pressure surface of the fluid is one of their primary major interests [1,2]. A set of hyperbolic nonlinear evolution equations are the keyword driving this phenomenon [3]. Following Saint-Venard Adéma Jean-Claude Bar de Venat, the shallow water wave equation is named the Saint-Venat equation in bidirectional form [4]. Additionally, the well-known Navier-Stokes equation explains that the conservation of mass means that the vertical velocity scale of the fluid is smaller than the horizontal velocity scale when the horizontal length scale is much larger than the vertical length scale [5][6][7]. Many nonlinear evolution equations have been formulated to demonstrate the waves' dynamic behavior through shallow water waves. This phenomenon has many applications in engineering and science, such as plasma physics, cosmology, fluid dynamics, electromagnetic theory, acoustics, electrochemistry astrophysics, and so on [8][9][10][11][12][13]. These models have forced many mathematicians and physicists to find suitable tools for finding computational, semi-analytical, and numerical solutions. Distinct schemes have been derived such as the well-known Ψ Ψ -expansion methods, the auxiliary equation method, exponential expansion method, Kudryashov method, sech-tanh expansion method, direct algebraic equation method, Adomian decomposition method, iteration method, Khater methods, B-spline schemes and so on [14][15][16][17][18][19][20]. These techniques have been applied to several nonlinear evolution equations to construct the solutions. Still, there is no unified method that can be used for all models until now. In this scientific race to derive the most general computational technique that can apply to all nonlinear evolution equations, no one has stopped for a single second and asked about the accuracy of all models of the already derived computational schemes.
Sophus Lie has put forth several essential concepts and developed basic tools to study DEs' group properties. In applied mathematics, he achieved several tangible findings of enormous value. In particular, he established the maximum group of point (local) transformations accepted by the one-dimensional heat equation, discovering the Galilei group's projective representation. Only lately have these findings been uncovered. Lie's Theory of continuous groups is based on the well-known Noether theorem on conserved law. Nowadays, several discoveries from Lie are recognized and rediscovered in connection with the present evolution of mathematical and theoretical physics, and we can see the victory of the Lie Theory in all mathematical disciplines.
In this context, this paper studies the analytical and numerical solutions of the (2 + 1)-D KP-BBM equation. This model is given by [21][22][23] where r i , (i = 0, 1, 2, 3) are undetermined positive constants while B = B(ζ, t) is a spacetime function. This function explains the bidirectional propagating water wave surface. Handling Equation (1) through the next transformation B(ζ, t) = Y (℘), where c is the wave velocity which converts the PDE into ODE. Integrating the result ODE twice with respect to ℘, and with zero constants of the integration, obtains the next ODE Using the homogeneous balance principles and the following auxiliary equations for MDA and MK methods [24][25][26][27] for Equation (2), respectively, a are arbitrary constants to be constructed later; give n = 2. Thus, the general solutions of Equation (2) are formulated in the following forms where a −2 , . . . , a 2 are positive constants. The paper's remaining sections are given in the following order; Section 2 constructs novel and accurate solutions of the considered model through the suggested above-mentioned schemes. Section 3 explains the paper's novelty and contributions. Finally, Section 4 gives the conclusion of the whole paper.

Accuracy of Computational Solutions
Applying the MDA and MK methods to Equation (2) to construct traveling wave solutions of the (2 + 1)-D KP-BBM equation is conducted. Additionally, estimate the requested conditions for investigating the numerical solutions of considered model by applying the AD and VI methods as follows:

MDA Method's Solutions
Handling Equation (3), through the suggested analytical scheme' framework, calculates the parameters shown above in the following forms: Family I Family II Family III Family IV Consequently, the considered model's traveling solutions are evaluated in the following forms: For J 1 = 0, J 2 > 0, we obtain For J 1 = 0, J 2 < 0, we obtain For 4J 1 J 3 > J 2 2 , we obtain where ζ = x, y, ζ 1 = x + y.

1.
Applying AD method [28,29] for Equation (2) with the following initial and boundary gives the following solutions . (21) Thus, the semi-analytical solutions of the (2 + 1)-D KP-BBM equation is given by 2.

MK Method's Solutions
Handling Equation (3) through the suggested analytical scheme's framework allows calculation of the parameters shown above in the following forms: Family I Family II Consequently, the considered model's traveling solutions are evaluated by the following forms: Applying the AD method for Equation (2) with the following initial and boundary conditions Y (0) = 8e x (e x +1) 2 , Y (0) = 0 gives the following solutions Thus, the semi-analytical solutions of the (2 + 1)-D KP-BBM equation are given by 2.

Interpretation of Results
In this section the interpretation of the results and the paper's contribution are shown through comparing the obtained results with those that have been recently published for the considered model. Comparing our analytical solutions with those that have been obtained by [21][22][23] shows the novelty of our result, where all our solutions are completely different from those that have been obtained in those papers. Additionally, we explain the shown figures for more physical explanation of each of them and for demonstration of the flow's dynamical behavior. Figures 1, 2, 3, 4, 5, 6, 7, 8 show breather and kink wave in two and three-dimensions and the contour plot of Equations (4), (7), (25) and (26) Tables 1-4 the obtained solution via the MK method is more accurate than that obtained by the MDA method that is demonstrated in Figure 9.           Table 2. Absolute error between computational and the obtained semi-analytical through the VIM of the (2 + 1)-D KP-BBM equation with different values of t and x when y = 5. 3.092× 10 −9 3.092× 10 −9 3.092× 10 −9 3.092× 10 −9 3.092× 10 −9 3.092× 10 −9 3.092× 10 −9 3.092× 10 −9 3.0922× 10 −9 3.0923× 10   Table 4. Absolute error between computational and the obtained semi-analytical through the VIM of the (2 + 1)-D KP-BBM equation with different values of t and x when y = 5.

Conclusions
This manuscript successfully applied four analytical and numerical techniques to the (2 + 1)-D KP-BBM equation used as a shallow water wave model. Many accurate novel traveling wave solutions were obtained. The accuracy and novelty of the obtained solutions were investigated. The traveling obtained solutions were demonstrated by 2D, 3D, and contour 3D plots. The symmetry between analytical and numerical solutions is explained through the given tables and figures. Funding: This paper has not been funded by any outside entity.

Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.

Data Availability Statement:
The data that support the findings of this study are available from the corresponding author upon reasonable request.