Localized waves and interactions for the high dimensional nonlinear evolution equations

Abstract

In this paper, we construct some localized waves for the Bogoyavlenskii–Kadomtsev–Petviashili (BKP) and (3+1)-dimensional KP equations with the aid of the Hirota bilinear method and long wave limit. The BKP equation, as a modification of the KP equation, describes the shallow long waves and the (3+1)-dimensional KP equation can be applied to describe the propagation of nonlinear waves in fluids. Breather and breather-type kink soliton for the BKP equation are obtained from the two-soliton solutions; interactions between breather and kink soliton or between two breathers are obtained from the three-soliton solutions; interactions among one breather and two kink solitons, among two breathers and one kink soliton, or among three breathers are obtained from the four-soliton solutions. Furthermore, based on the explicit solutions with appropriate parameters, breathers of the (3+1)-dimensional KP equation have different behaviors in the different planes, and lump solutions are constructed via the long wave limit method. Results obtained in this paper may be helpful for understanding the propagation of localized waves.