Study On Dynamic Characteristics Of Localized Waves For Two Kinds Of Nonlinear Evolution Equations

With the development of nonlinear science,many natural phenomena can be described by nonlinear evolution equations.By studying nonlinear evolution equations and their solutions,the nature and laws of nonlinear phenomena in related fields can be found.As an important nonlinear evolution equation,nonlinear Schr¨odinger equation is widely used in nonlinear optics,fluid mechanics,finance and other fields.Its solutions can be used to study localized waves such as solitons,rogue waves and breathers in nonlinear systems.Considering the complexity and diversity of nonlinear systems,the study of coupled nonlinear evolution equations with localized waves can show more abundant structures and characteristics.In this paper,the generalized Darboux transformation is used to study the higher-order localized waves of two kinds of coupled nonlinear Schr¨odinger equations,in order to provide rich theoretical basis for the development of localized waves and expand the application value of localized waves in practical engineering.The first chapter is the introduction part of this paper.The research background,significance and current situation of nonlinear evolution equation,three kinds of localized waves and generalized Darboux transformation are introduced,and the main work of this paper is briefly described.In chapter 2,the coupled nonlinear Schr¨odinger equation with coherent coupling term is studied,which can describe the propagation of electromagnetic waves in nonlinear optical fibers.Starting from the seed solution of the equation and based on the first-order rogue wave solutions,the symmetric and asymmetric second-order rogue wave solutions of the coupled nonlinear Schr¨odinger equation are derived by the generalized Darboux transformation.Different parameter values are selected for numerical simulation,the contour plots of the second-order rogue wave are obtained,and the influence of parameters on the dynamic characteristics of the second-order rogue wave is analyzed.In chapter 3,the coupled nonlinear Schr¨odinger equation with derivative term is studied,which can describe the propagation of short pulses in the femtosecond or picosecond regions of birefringent optical fibers.Based on the two linearly independent solutions of the Lax pair,the two-soliton and three-soliton solutions of the nonlinear Schr¨odinger equation are obtained by using the generalized Darboux transformation.The dynamic characteristics of elastic collision,inelastic collision and bound state among solitons are further analyzed.Some novel interaction structures among solitons are presented by numerical analysis.In chapter 4,the localized wave solutions of the coupled nonlinear Schr¨odinger equation with derivative term are studied.The specific solution of Lax pair is solved by a method similar to the undetermined coefficient method,and the Nth-order iteration formulas of the localized wave solution for the nonlinear Schr¨odinger equation are constructed by using the generalized Darboux transformation.The dynamic characteristics of the first-and secondorder localized waves are analyzed by changing the parameters in the localized wave solution.The last chapter is the summary and the prospect of this paper.