Some Nonlinear Waves In Nonlinear Shallow Water Wave Models And Complex Modified Short Pulse System

In nature,nonlinear phenomena exist widely with rich forms and diverse contents,so it is an important and significant work to study nonlinear science.Studying the solutions of nonlinear partial differential equations is helpful for us to understand and explain the nonlinear phenomena in nature.Therefore,finding the solutions of nonlinear partial differential equations have become the main research task for many experts and scholars,and many research methods have also been put forward.In this paper,Darboux transformation method,the Hirota bilinear method and the long wave limit approach are used to solve some nonlinear shallow water wave equations and complex modified short pulse system with the help of mathematical software Mathematica.Multi-soliton solutions,multi-breather solutions,high-order lump solutions and mixed solutions of the nonlinear wave equations are constructed.The first chapter,as an introduction,which introduces the research background of nonlinear wave equations.The basic ideas and specific solving steps of the research methods which are used in this paper are given.The second chapter,Lax pairs and conversation laws of the coupled Burgers equation and the coupled KdV equation are constructed based on a 3×3 spectral matrix,and then soliton solutions of these two coupled equations are obtained by Darboux transformation method.We also use Darboux transformation method to obtain soliton solutions,breather soltuions and rogue wave solutions of a complex modified short pulse system.By introducing nonlocal transformation into the modified short pulse system,the modified short pulse system is reduced to nonlocal type,and the nonlocal soliton solutions of this equation are obtained.The third chapter,a(2+1)-dimensional Boussinesq equation and a(2+1)-dimensional potential YTSF equation are transformed into corresponding bilinear equations by the Hirota bilinear method,and multi-soliton solutions and multi-breather solutions of the(2+1)-dimensional Boussinesq equation and the(2+1)-dimensional potential YTSF equation are obtained by constructing auxiliary functions.On the basis of the multi-soliton solutions,we use the long wave limit approach to obtain the high-order lump solutions.In this paper,a general auxiliary function is given to construct the mixed solution of multi-soliton,multi-breather and high-order lump of nonlinear partial differential equations.The mixed solutions of the two equations are obtained by giving appropriate values to the auxiliary function parameters,and the dynamic characteristics of mixed waves are analyzed.Finally,the full text is summarized and the future research work is prospected.