Direct separation approach and multi-valued localized excitation for (M+N)-dimensional nonlinear system

Abstract

In this paper, we propose a new variable separation method that does not need the Hirota’s bilinear form and directly gives the analytic form of the solution u instead of its potential uy. This new method covers the N-soliton solution obtained by the Hirota’s direct method and the multi-valued solution of the multi-linear variable separation approach(MLVSA), and is applicable to the (M+N)-dimensional nonlinear model. We would like to call it the “direct separation approach” (DSA). Taking the extended (3+1)-dimensional Kadomtsev–Petviashvili equation as an example, we introduce an entirely fresh variable separation ansatz and substitute it into the equation, resulting in a new variable separation solution. With the help of this variable separation solution, we construct two types of N-soliton solutions via the single-valued functions. Then, the rogue waves and four typical folded solitary waves are obtained by using the multi-valued functions. In addition, we study the head-on and chase-after collision between two, three, and four folded solitary waves and systematically analyze their dynamic behaviors, such as the phase shifts and their difference values of the interaction. Especially, we detect a multi-directional movement in the collision between four folded solitary waves, which significantly differs from the others. The obtained results may help us simulate complex folded appearance in real life and better understand the wave motion of the solitary waves.