Localized Waves For The Multi-Component Coupled Nonlinear Systems

The localized waves of the nonlinear systems are one of the research hotspot in the filed of integrable system.Based on the symbolic computation software Maple platform and by using the traditional Darboux transformation,the generalized Darboux transformation and the KP hierarchy reduction method,we study the localized waves and dynamics for some multi-component coupled nonlinear systems.The main work is classified into five aspects: the integrability of the coupled nonisospectral Gross-Pitaevskii(GP)system and the corresponding traditional Darboux transformation are investigated,and the nonautonomous soliton,breather and rogue wave are constructed,besides,the multi-component generalization of the GP system and the corresponding Darboux transformation are all given;utilizing the generalized Darboux transformation,the hybrid solutions in the twocomponent coupled nonlinear systems are studied,which include higher-order rogue wave pairs,higher-order rogue waves interacting with multi bright solitons,multi dark solitons and multi breathers;based on the generalized Darboux transformation,the mixed interactions of localized waves in the three-component coupled nonlinear systems are researched;the two-dimensional multi dark solitons in Gram type determinant forms for the(2+1)-dimensional multi-component Maccari system are constructed by KP hierarchy reduction;based on the symbolic computation software Maple platform,the software package SRSCNS1 is developed,which is used to construct the semi-rational solutions for the multi-component coupled nonlinear systems with Ablowitz-Kaup-Newell-Segur(AKNS)spectral problem.The main contents of this thesis are listed as follows:The first chapter is the introduction part of the thesis,we mainly introduce the background and current situation of the localized waves consisted of soliton,breather and rogue wave,the Darboux transformation method,the KP hierarchy reduction method and symbolic computation.The topic selection and the main contents of the dissertation are provided.In the second chapter,the Lax pair and infinitely-many conservation laws of the nonisospectral coupled GP system are investigated,and the corresponding traditional Darboux transformation is constructed.Starting form zero seed solution,we derive the nonautonomous multi solitons of the coupled GP system by Darboux transformation,besides,the amplitude and velocity of the nonautonomous soliton change over time.Beginning form non-zero seed solution,the breather on a curved background for the coupled GP system is given.Then,utilizing the limiting process in the breather solution,the rogue wave on a curved background is also derived.Finally,the two-component coupled GP system is generalized to N-component case and its corresponding Darboux transformation is also given.In the third chapter,the generalized Darboux transformations of some two-component coupled nonlinear systems are derived,and the interactions among rogue wave,bright(dark)soliton and breather are investigated.For the two-component coupled cubic-quintic nonlinear Schr¨odinger(CCQNLS)equations,through considering the characteristic equation of the matrix for the Lax pair possesses a double root,we construct the interactional solutions that higher-order rogue waves interact with multi bright(dark)solitons and multi breathers,respectively.By considering the characteristic equation of the matrix for the Lax pair possesses a triple root,the hybrid solutions between rogue wave and rogue wave,which is called higher-order rogue wave pairs,can be constructed.This kind of the firstorder rogue wave is consisted of two fundamental first-order rogue waves,and this kind of the secod-order rogue waves can include four or six fundamental rogue waves.Utilizing the dressing-Darboux transformation,the semi-rational solutions in compact determinant form for the coupled Fokas-Lenells(FL)equations are given,these kinds of solutions are greatly similar with the interactional solutions in the CCQNLS equations,namely higher-order rogue waves interact with multi bright,dark solitons and multi breathers.In the fourth chapter,the generalized Darboux transformation and the mixed interaction of localized waves for some three-component coupled nonlinear systems are investigated.In the three-component coupled Hirota equations and nonlinear Schr¨odinger(NLS)equations,we construct four types of mixed interactional solutions in the three components that higher-order rogue waves interact separately with multi bright(dark)solitons and multi breathers.For the three-component coupled derivative nonlinear Schr¨odinger(DNLS)equations,we also derive four types of mixed interactions of localized waves,which are similar with ones in the three-component Hirota equations and the three-component NLS equations.Owing to the derivative terms,dark solitons do not exist in the interactional solutions of the three-component DNLS equations,and they become amplitude-varying solitons.Besides,this kind of amplitude-varying soliton may provides some theoretical guidances for the prediction of rogue waves.In the fifth chapter,utilizing the KP hierarchy reduction method,we construct the two-dimensional multi dark solitons in Gram type determinant forms for the(2+1)-dimensional multi-component Maccari system.The dynamics of the dark-dark solitons for the two-component Maccari system are discussed in detail,which include single darkdark soliton,two dark-dark solitons and solitons bound states.In the sixth chapter,based on the Maple platform,we firstly develop the package SRSCNS1,which is used to construct the semi-rational solutions for the multi-component coupled nonlinear systems with AKNS spectral problem.For the two-component systems,this package gives the concrete expressions of the first-order and second-order semirational solutions,and the corresponding figures;For the three-component cases,this package only derives the first-order semi-rational solutions and some related figures.The validity and universality of the package are verified by some concrete examples.In the last chapter,the summary for the results in this thesis are given,and the outlook of future research work is discussed.