Localized Wave Solutions Of Three Kinds Of Nonlinear Wave Equations

With the progress of science and technology,complex nonlinear phenomena occur in many fields of science,such as ultra-short pulses in inhomogeneous optical fiber,rogue wave in the ocean,plasma theory,and Heisenberg ferromagnetic spin chain interactions.As a classical wave equation,the nonlinear Schr?dinger equation can be used to depict nonlinear phenomena mentioned above,and its analytical solution can be use to describe nonlinear localized waves such as soliton,rogue wave and breather in physics.In this paper,three kinds of nonlinear Schr?dinger equations are studied in theory.Localized wave solutions of equations are constructed by using similarity transformation and generalized Darboux transformation,and dynamic characteristics of localized wave are explored by numerical simulation.The main work is as follows:In the first chapter,the introduction,the bright and dark solitons,rogue wave and breather are introduced respectively.The status of research is analyzed at home and abroad.The similarity transformation and Darboux transformation are introduced,and finally the main work of this paper is expounded.In the second chapter,the nonlinear Schr?dinger equation in inhomogeneous optical fiber is investigated.It is transformed into a standard nonlinear Schr?dinger equation by using similarity transformation,and then soliton solution and rogue wave solution are obtained through the undermined coefficient method.Based on the expressions form of the solutions,different types of functions and corresponding parameters are selected for numerical simulation,and dynamic characteristics are analyzed.The results are helpful to further study the propagation of soliton in optical media.In the third chapter,a generalized inhomogeneous third-order nonlinear Schr?dingerequation from the Heisenberg ferromagnetic system is considered.Based on the integral condition Lax equation,the Nth-order rogue wave solution is constructed by generalized Darboux transformation.Starting from a seed solution,the first-,second-and third-order rogue wave solutions are derived by iterative formula and limit operation.Different parameter values are selected to obtain rogue wave evolution plots and contour plots.The new evolution plot of higher-order rogue wave enriches the theoretical research of Heisenberg ferromagnetic system greatly.In the fourth chapter,the two-component coupled nonlinear Schr?dinger equation is studied.Starting from the integral condition Lax equation,the Nth-order localized wave solution is obtained by the method of generalized Darboux transformation.The new form of seed solution is supposed,which is related to the normalized distance and retarded time.The first-to the third-order localized wave solutions are figured out by Maple.Three-dimensional corresponding plots are demonstrated by selecting appropriate parameters.Furthermore,the classification of Nth-order localized wave is given in the form of a table.These appealing results will lay a theoretical foundation for the follow-up studying of multi-component coupled nonlinear dynamical systems.In the fifth chapter,the work done is summarized,and the future research work and direction are given.