Exact Travelling Wave Solutions to the (3+1)-Dimensional Kadomtsev-Petviashvili Equation

Abstract

Travelling waves, whether their solution expressions are in explicit or implicit forms, are very interesting from the point of view of applications. These types of waves will not change their shapes during propagation and are thus easy to detect. Of particular interest are three types of travelling waves: the solitary waves, which are localized travelling waves, asymptotically zero at large distances, the periodic waves, and the kink waves, which rise or descend from one asymptotic state to another. Recently, a unified algebraic method, called the mapping method [1–4], is proposed to obtain exact travelling wave solutions for a large variety of nonlinear partial differential equations (PDEs). This method includes several direct methods as special cases, such as tanh-function method, sech-function method, and Jacobi elliptic function method. Above all, by means of this method, the solitary wave, the periodic wave, and the kink wave (or the shock wave) solutions can, if they exist, be obtained simultaneously to the equation in question without extra ∗e-mail: yanzepeng@163.com