Study On Several Nonlinear Differential Equations Based On Generalized Bilinear Operator

It is a difficult problem to solve nonlinear differential equations in interdisciplinary disciplines.Hirota bilinear derivative method and generalized bilinear method are effective tools for solving nonlinear differential equations.Therefore,based on Hirota bilinear derivative method and generalized bilinear method,we have discussed the lump solutions,rational solutions,interaction phenomena and breather-wave solutions of several nonlinear differential equations in this paper.The specific research contents are as follows:In chapter 1,the theoretical knowledge required in this paper is briefly summarized,such as the development history and present situation of Hirota bilinear derivative method and generalized bilinear method.In chapter 2,Nizhnik equations are studied based on Hirota bilinear derivative method.And its lump solutions,two interaction phenomena(lump-soliton solutions and lump-exponential solutions)and breather-wave solutions are discussed with the help of Mathematica.In chapter 3,the Shallow-Water-Wave equation,Sawada-Kotera equation and Nizhnik equations are studied based on the generalized bilinear method.A generalized Shallow-Water-Wave-like equation corresponding to p = 3 is constructed,and four kinds of rational solutions of this equation are obtained.Two generalized SawadaKotera-like equations corresponding to p = 3 and p = 5 are constructed,and their rational solutions are given respectively.A generalized Nizhnik-like equation corresponding to p = 3 is constructed,its lump solutions,rational solutions,two interaction phenomena(lump-soliton solutions and lump-exponential solutions)and breather-wave solutions are given.In chapter 4,the summary and outlook of the research contents of this paper are given,and the contents of the next research are discussed.