A Study Of Lump-like Solutions To Nonlinear Partial Differential Equations

In the field of soliton theory,integrable system is one of the main research topics of soliton theory.Because of the role of nonlinear partial differential equations and their exact solutions in describing natural phenomena,the research on the construction of integrable families and their exact solutions has always been the focus of scholars' attention.In this paper,on the basis of the Hirota bilinear equation and its extension equation,we mainly study the lump solutions of four(3+1)-dimensional nonlinear partial differential equations.The structure is as follows:In chapter one,the origin and development of the soliton theory,the related theoretical basis and the introduction of the research results of this paper are briefly introduced.In chapter 2,based on the Hirota bilinear equation,a(3+1)-dimensional equation with lump-type solution is obtained by using the positive quadratic function method,and the reduction equation with lump solution is studied.In Chapter 3,applying the extended positive quadratic function method,the periodical and rouge wave solutions of the(3+1)-dimensional BKP-like equation with the same bilinear type as the standard(3+1)-dimensional B-type Kadomtse-Petviashvili(BKP)equation are obtained.A(3+1)-dimensional Kd V equation with higher order rouge waves are solved.Meanwhile,the research results of chapter two and three have been published.In chapter 4,the multi-lump solutions of a(3+1)-dimensional Kd V equation and the extended Jimbo-Miwa(JM)equation are studied by using the long-wave limit method.At the end of the paper,the research methods and research results are summarized and prospected.