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

In this paper, we propose a new variable separation method which does not need Hirota’s bilinear form and directly gives analytic form of solution u instead of its potential u y . This new method not only covers N-soliton solution obtained by Hirota’s direct method but also multi-valued solution of multi-linear variable separation approach(MLVSA), and applicable to (M+N)-dimensional nonlinear model. We would like to call it “direct separation approach” (DSA). Taking the extended (3+1)-dimensional KP equation as an example, we first construct two types of N-soliton solution. Then, by introducing multi-valued functions, rogue wave and four typical folded solitary waves are obtained. In addition, we study the head-on and chase-after collision between two, three, four folded solitary waves, and systematically analyze their dynamic behaviors. Many novel structures are obtained that may help simulate complex folded appearance in real life.