Research On Integrability And Nonlinear Waves Of Nonlinear Partial Differential Equations

This dissertation focuses on the integrability,nonlinear waves and their inter-action solutions of several kinds of nonlinear partial differential equations.These equations have a wide range of applications in mechanics,condensed matter physics,plasma physics and nonlinear optics.The main work is carried out in four aspects.First,the Lie group method is applied to study the exact solutions and conservation laws of nonlinear partial differential equations with variable coefficients.Second,Bell polynomials are used to study bilinear forms and the long wave limit method is used to construct nonlinear waves for a higher dimensional nonlinear partial d-iffereutial equation.Third,bilinear form of equations and the three wave method are developed to construct multi-soliton solutions for some nonlinear partial differ-ential equations.Last,Hirota bilinear arid function quasi methods are devoted to construct lump and interaction solutions for nonlinear partial differential equations.The details are as follows:The first chapter is the Introduction part.This chapter introduces the back-ground and development status of the symmetry theory,integrability theory,non-linear wave.The main research results of this dissertation are also elaborated.In chapter 2,based on the Lie symmetry analysis,two types of nonlinear par-tial differential equations with variable coefficients have been studied.Firstly,by introducing a suitable potential transformation,the coupled system of the(2+1)-dimensional nonlinear Schroodinger equation(NLSE)with variable coefficients are obtained.The corresponding veetor field and optimal system are obtained using Lie symmetry analysis.Based on similarity transformations,four types of(1+1)-dimensional nonlinear partial differential equations and a lot of explicit solutions to this equations are presented by means of auxiliary equation method and G'/G method.In addition,conservation laws are obtained by using the new conservation theorem proposed by Ibragimov.Secondly,Lie point symmetries and optimal sys-tem of the(1+1)-dimensional Ablowitz-Kaup-Newell-Segur(AKNS)equation with variable coefficients are obtained.By similarity transformations,five types of ordi-nary differential equations are obtained.Some new exact solutions such as power series solutions,travelling and non-traveling wave solutions are obtained by means of auxiliary equation method.In Chapter 3,based on the binary Bell polynomial and Hirota bilinear methods,we study the bilinear form,nonlinear waves and interaction solutions of the(3+1)-dimensional B-type Kadomtsev-Petviashvili(BKP)equation.The Hirota bilinear equation are obtained via Bell polynomial theory.By introducing the transformation technique,two kinds of bilinear Backlund transformations and Lax pairs of the BKP equation are studied.The N-soliton solutions of the BKP equation are constructed by using the obtained bilinear equation.With the aid of the N-soliton solutions and long wave limit methods,breathers,lumps,rogue waves and interaction solutions are presented.In Chapter 4,based on Hirota bilinear,extended homoclinic test and three wave methods,multi-soliton solutions of the(3+1)-dimensional nonlinear evolution(NEE)equation and BKP equation are studied.Applying extended homoclinic test method to the NEE equation,two kinds of kink breather soliton solutions are obtained.By means of three wave approach,the multi-soliton solutions of NEE equation and BKP equation are derived,including kink breather two-soliton solu-tion,kink periodic two-soliton solution,double periodic solitary wave solution and three-solitary solution.In Chapter 5,based on Hirota bilinear and function quasi methods,homo-clinic breather waves,rouge waves,lump and interaction solutions to the(3+1)-dimensional KdV-type equation and KP-Boussinesq-like equation are constructed.Applying homoclinic test approach to the KdV-type equation,homoclinic breather solutions are obtained.Meanwhile,applying the long wave limit method to the obtained breather wave solutions,the rouge wave solutions are derived.Then in-teraction solutions between lump and line soliton,kink soliton or periodic function of the KdV-type equation are derived by function quasi method.In addition,lump and interaction solutions between lump and line soliton or kink soliton of the KP-Boussinesq-like equation are derived.In Chapter 6,the summary for the results in this thesis are given,and the outlook of future research work is prospected.