The Study Of The Analytical Solutions For Several Types Nonlinear Partial Differential Equations In The Physical Background

Recently,the research on Nonlinear Evolution Equations with strong physical background is changing with each passing day and making continuous breakthrough-s.With its analytical solutions being found,more and more nonlinear phenomena are explained,which promotes the development of scientific research.This paper mainly introduces how to use different effective methods to solve the exact analytical solutions of the generalized(3+1)-dimensional nonlinear wave equation,the(3+1)-dimensional Hirota bilinear equation,the coupled Hirota systems,the general coupled nonlinear Schrodinger equation,the(2+1)-dimensional nonlinear schrodinger equation and the time-fractional Drinfeld-Sokolov-Wilson systems.Especially when we study the non-linear Schrodinger equation,we generalize the more generalized Darboux transforma-tion on the basis of the classical Darboux transformation,and get a new semi-rational solutionsIn the first chapter,we mainly introduce the research background of solving non-linear evolution equations,as well as the research contents and methods of this article.In the second chapter,we first calculate the Hirota bilinear form of the general-ized(3+1)-dimensional nonlinear wave equation,and then use the hyperbolic function method,the long wave limit method,and the KP reduction method to discuss sever-al special types of exact analytical solutions to this equation,including the bright and dark soliton solutions,rogue waves and lumps.In addition,we also discuss the dynamic characteristics of these solutions in space and the changes in the corresponding dynam-ic graph after changing the parameter values.It is important to point out that the rogue wave solution and lump solution can be differentiated from rational solutions.Further-more,the high-order strange wave solution obtained by the KP reduction method can generate different patterns,including basic patterns,triangular patterns,and circular patterns,as parameters are changed.In the third chapter,based on the irota bilinear form of the(3+1)-dimensional Hiro-ta bilinear equation,the high-order rogue wave solutions of this equation are obtained,which include one-,second-and third-order rogue wave solutions.It is worth noting that these rogue waves all have the property limx?±?u=u0,limy?±? u=u0 and limz??u=u0.In the fourth chapter,a new type of breath waves and rogue waves in the soliton background of the coupled Hirota systems are established.We construct a type of typical breath solutions and get the rogue wave solutions on this basis by using Darboux transformation.Finally,the dynamic analysis of these solutions in three-dimensional space is shown.In the fifth chapter,some new semi-rational solutions of the general coupled non-linear Schrodinger equation are investigated by using a generalized Darboux transfor-mation.We divide semi-understanding into two types:(1)type-? degenerate soliton solution;(2)type-? degenerate soliton solution.For the case(1),we discuss in de-tail the dynamic characteristics of the first-,second-and third-order regular solitons.For the case(2),we find that the solutions can be divided into a variety of different interactions such as general elastic interactions,special elastic interactions,inelastic interactions and bound states by selecting different parameters.Moreover,we draw a three-dimensional dynamic graph of these solutions.In the sixth chapter,the(2+1)-dimensional nonlinear Schrodinger equation is s-tudied by using bilinear transformation method.Based on the classic KP equation,theorem 6.1 is proposed,and some special analytical solutions of the equation are ex-pressed in the form of a determinant by using theorem 6.1,for example,the fundamen-tal parallel wave waves |u| in the(x,y)plane and the fundamental wave waves P in the(x,t)plane.The dynamic characteristics of these solutions are analyzed,and we can easy to know that,with the value of time t increasing,amplitude of the line rogue wave increases until reaches its maximum at t=0.Finally,these waves approach the constant background when t?0.Furthermore,we observed that the second-order rogue wave and the third-order rogue wave consist of two parallel line rogue waves and three parallel line rogue waves,respectively.It is worth noting that the Nth-orderrogue wave consists of N(N+1)/2 localized waves,which have the characteristics of(1+1)dimensional rogue waves.In the seventh chapter,we mainly introduce the Lie point symmetry and some ana-lytical solutions to the time-fractional DSW system.More specifically,some definitions and theorems regards Riemann-Liouville and Caputo fractional differential are given.We also present the Lie point symmetry with the help of the Lie symmetry method.Under the background of Lie symmetry analysis,the similarity reduction theorem has also been proven.Moreover,the exact power series solution is worked out.Besides,the convergence of the power series solution is also proved.In addition,we apply the q-homotopy analysis method to the DSW system for finding the approximate analytical solution.Finally,the Noether theorem is applied to shape the conservation laws.In the last chapter,the conclusions and prospects of our paper are introduced.