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.