Investigation of the analytical and numerical solutions with bifurcation analysis for the (2+1)-dimensional Bogoyavlenskii-Kadomtsev-Petviashvili equation

In this work, we investigate the solutions of the (2+1)-dimensional Bogoyavlenskii-Kadomtsev-Petviashvili (BKP) equation by three powerful analytical methods: the expa\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\exp _{a}$$\end{document} function method, the (G′G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\frac{G'}{G})$$\end{document}-expansion method, and the Sine-Gordon expansion method. This equation describes the nonlinear wave propagation in many applications like waves of evolutionary shallow water, electrical networks, and engineering devices. Moreover, we study the solutions numerically via the finite difference method. We analyze the bifurcation of dynamical system resulting from the BKP equation. Finally, the majority of our solutions are displayed graphically to present the strength of imposed methods.

Our goal in this work is concerned with the solution of the (2+1)-dimensional BKP equation Where, u(x, y, t) presents the wave amplitude. The BKP equation is assorted as the equation of BKP-I when = 1 , and the equation of BKP-II when = −1 . The BKP Eq. (1) is a protraction of the Bogoyavlenskii-Schiff (BS) equation and the Kadomtsev-Petviashvili (KP) Eq. Estevez and Hernaez (2000), If = 0 then (1) Fan and Zhou (2018). In this work, we use three efficient analytical techniques, the first one is the exp m function method Method Hosseini et al. (2017Hosseini et al. ( , 2019, the second one is the ( G � G )-expansion method Nisar et al. (1) u xxt + u xxxxy + 12u xx u xy + 8u x u xxy + 4u xxx u y = u yyy , = ±1 or (u xt + u xxxy + 8u x u xy + 4u xx u y ) x = u yyy .

Page 3 of 19 585
Our work is streaked as: we demonstrate the main steps of the present methods in Sect. 2. We exhibit the application of the suggested methods in Sect. 3. We present some graphs to illustrate our solutions in Sect. 4. The numerical solution is presented in Sect. 5 using the finite difference method. We study the bifurcation analysis in Sect. 6. Lastly, in Sect. 7, we give a compact conclusion.

The exp m -function method
Assume that we have the following nonlinear PDF where, u = u(x, y, t) is an unknown function to be determined, F is a polynomial in u and its first and higher order partial derivatives.
Step 1: Employing the traveling wave transformation to the PDE switch the PDE to an ODE where a, b are arbitrary constants and c is the wave speed. Inserting (3) into (2), we get the following ODE: where H is a polynomial in u( ) and its derivatives.
Step 2: Suppose that the solution of (4) is presented by the next form: where A and B (0 ≤ i ≤ N) are constants to be evaluated, N is a positive integer.
Step 3: By balancing the term with the highest order derivative and the highest power nonlinear term in (4), we determine the positive integer N.
Step 4: Substituting (5) into (4), we obtain the next polynomial Putting the coefficients p j (0 ⩽ j ⩽ r) equal to zero. We get a system of nonlinear algebraic equations which can be solved by using Mathematica program to achieve the solution of Eq. (1).

The
The ( G � G )-expansion method is given by the following steps: Step 1: Assume that the solution of Eq. (4), is presented by: where G = G( ) fulfill the next linear ODE: where f (i = 0, 1, 2, ..., N), b N ≠ 0, and , are constants to be studied.
Step 2: The positive integer N is calculated as mentioned before by the homogeneous balance technique.
Step 3: Here, we present three groups of solutions for (8) Step 4: Replacing (7) with (8) into (4), then gathering all terms with the same powers of ( G � G ) and equaling their coefficients to zero, we achieve a set of equations can be solved by using Mathematica program to obtain the exact solution of equation (1).

The sine-Gordon expansion method(SGEM)
The sine-Gordon equation: where m is a constant and u = u(x, t) . Suppose that the wave transformation u(x, t) = U( ), = x − ct in (12) which converted to the nonlinear ODE: where, U = U( ) , is the traveling waves amplitude and c is the traveling waves speed. By integrating (13) once and putting the integration constant equal zero, we obtain let w( ) = U 2 and a 2 = m 2 (1−c 2 ) , so (14) becomes: Set a = 1 in (15), we obtain: The solution of (16), can be become as follow: where p is non zero constant of integration. It is presumed that the solution U( ) of (4) can be given as: Use (17) and (18), we get: Computing the value of N by using the balance principle. Assuming that the coefficients of sin i (w) cos i (w) with like power are equal to zero, we acquire an algebraic system of equations that can be solved by Mathematica software.

Applications
Start by reducing the PDE (1) into the ODE via the wave transformation (3), we obtain: Integrating twice with respect to Determine the value of N by Balancing u (3) ( ) with u � ( ) 2 in (14), we obtain N + 3 = 2(N + 1) , then N = 1.

Solutions via the exp m -function method
The solution of (22)is assumed to be: Inserting (23) into (22) and setting the coefficients that have the same powers of m equal to zero, we acquire the nonlinear system: We get the following set of solution, using (3) and (23):

Solutions via the ( G � G )-expansion method
Presenting the solution of (22) by: inserting (26) into (22), gathering and putting the coefficients of terms that have the same power of G ′ G equal to zero, we obtain the next set of equations: The traveling wave solutions of (1) are presented by: Group 1: Hyperbolic function solutions, when 2 − 4 > 0, Group 2: Trigonometric function solutions, when 2 − 4 < 0, u(x, y, t) Group 3: Rational function solutions, when 2 − 4 = 0,

Solutions via the Sine-Gordon expansion method(SGEM)
Using (20), we introduce the solution of (22) by the next form: substitute from (32) in (22), and putting the coefficients of trigonometric functions that have like powers equal to zero, we acquire the following algebraic set of equations: By the help of mathematica program, we attain the following groups of solutions: Group 1: Group 2:
Put the governing conditions to solve Eq. (1) We obtain the numerical solution ũ(x, y, t) equivalent to the exact solution u(x, y, t) by constructing finite difference scheme for Eq. (1) where, and its first and higher order partial derivatives As before, we formulate system of difference equations as follows can be solved using the conditions (40). We show the accuracy of the presented numerical scheme by calculating error norm Page 13 of 19 585 Table 1 presents error norms using L 2 and L ∞ formulas (45) at different time levels and Fig. 8(a, b) display the numerical and exact solutions at time level t = 0.5 with step size = = 1.0 and Δt = 0.01, 0.001 for Eq. (25), respectively. Figure 9 shows the maximum absolute error at different time levels t = 1, 3, 5 using the analytical solution (25) with step size = = 0.5 and Δt = 0.001. Table 2 exhibits L 2 and L ∞ error norms at different time levels and Fig. 10(a, b) compare between the numerical and exact solutions at t = 0.1 with = = 1.0, 0.5 and Δt = 0.01, 0.001 for Eq. (29), respectively. The maximum absolute error at different time levels t = 3, 5, 10 appear as in Fig. 11 using the analytical solution (29) with = = 1.0 and Δt = 0.001. Table 3 shows L 2 and L ∞ at different time levels and Fig. 12(a, b) display the analytical and numerical solutions at time level t = 0.5 with = = 1.0 and Δt = 0.01 for Eqs. (37) and (39), respectively. (45)

Bifurcation analysis
In this section, we discuss the bifurcation analysis of the (2+1)-dimensional BKP Eq. (1). For this analysis, we use the bifurcation theory Guckenheimer and Holmes (1983) and formulate equation (22) as a three-dimensional dynamical system as follows: ⟩ is a vector field with .F = u u + f f + g g = 0 , Then the dynamical system (46) is conservative. The dynamical system (46) with three parameters

Conclusion
In this work, we introduced new solutions for the (2+1)-dimensional BKP Eq. (1) which characterize the wave phenomena in scientific applications as fluid mechanics and others. The introduced solutions are gained by three powerful and simple analytical methods, namely: the exp m -function method, the ( G � G )-expansion method, the Sine-Gordon expansion method. Furthermore, we exhibited the acquired results graphically. We successfully attain solutions in different forms: trigonometric, exponential, hyperbolic, and rational functions which presented different types of waves as kink, cusp, and periodic wave solutions. The aforementioned results demonstrate that the suggested techniques are straightforward, effective, and good tools that give conclusive results when solving the proposed model and possibly be applied to different kinds of NPDE. We applied a numerical scheme via finite difference method of the BKP Eq. (1). Tables 1, 2, 3 and Figs. 8,9,10,11,12 showed efficiency and  accuracy of the imposed numerical scheme and compatible it with the corresponding analytical solutions. We analyzed the traveling wave solutions of the dynamical system (46) consisting of the BKP equation (1) and found all equilibrium points are nonhyperbolic equilibrium points. In future work, we will be concerned about searching for other types of wave solutions to the presented BKP equation (1) by trying other analytical and numerical techniques.