Painlevé Test and Some Exact Solutions for (2+1)-Dimensional Modified Korteweg–de Vries–burgers Equation

Abstract

In this paper, we investigate the solitary wave solutions for the two-dimensional modified Korteweg–de Vries–Burgers (mKdV-B) equation in shallow water model. Despite that Painlevé test fails to prove the integrability of the mKdV-B equation by using the WTC-Kruskal algorithm, the Bäcklund transformation is obtained via the truncation expansion. The exact solutions of the mKdV-B equation are found using factorization techniques, Exp-function and energy integral approach of the corresponding ordinary differential equation. We found that the dispersion relation of the linearized mKdV-B equation lies on the complex plane yielding a damping character. By keeping the water height relatively small, we illustrate the resulting solutions in several figures showing the shock and solitary wave nature in the flow. The stability for the mKdV-B equation is analyzed by using the phase plane method.

Some Explicit Solutions of Nonlinear Evolution Equations

“…Apart from physical importance, the closed-form solutions of nonlinear partial differential equations assist the numerical solvers to compare the correctness of their results and help them in the stability analysis. * In this paper, we employ the system technique to obtain exact solutions for nonlinear partial differential equations that contain exponential function based on a suitable choice of parameters through the Painlevé test [19,20,21,22]. The aim of this paper is to obtain more exact explicit solutions and to analyze the motions of exact solutions as the values of parameters and proper coefficients about the equal width wave equation and the (2+1)-dimensional Maccari's system.…”