Researches On The Darboux Transformation And Dynamics For The Nonlocal Integrable System

The nonlocal integrable nonlinear evolution equations are one of the research hotspot in the filed of integrable system.Via using Mathematica software with symbolic computation,a study of several nonlocal nonlinear models is presented.Our thesis focuses on the following three main aspects: The Darboux matrix for several important nonlocal nonlinear integrable equations are constructed via a Darboux reduction technique,and the corresponding analytic solutions are also derived for those systems,which include the bright soliton solution,the high-order soliton solutions,the(1+1)-dimensional high-order rogue waves,the(1+2)-dimensional multi-and high-order line rogue wave solutions;Dynamics behavior for these exact solutions is further analysed,some novel behaviors are discussed,which includes the finite-time blowing-ups,the long-time asymptotic of solutions and the nonlinear interactions between solutions.Based on the Darboux's algorithm scheme,the NonlocSolve 1.0 program package is developed,and it can be used to calculate the analytic solutions for several nonlocal integrable equations.The main contents of this thesis are as follows:In the first chapter,which is the introduction part of the whole thesis,we briefly introduced the background as well as the findings for the integrable nonlocal equations.The topic selection and the main content of the research are also elaborated.In the second chapter,the Darboux matrix for the partially and the fully PT-symmetric nonlocal DS equations are constructed for the first time.The multi-and the high-order rogue-wave solutions are derived.In both of these two nonlocal systems,the fundamental rogue waves can develop finite-time singularity on a whole hyperbolic curve in the spatial plane when time approaches negative infinity.Meanwhile,the multi-rogue waves are found,which are generated from nonlinear interactions between several fundamental rogue waves.In this case,their singularities are usually appear in pairs or in an timeinterval.In addition,the hybrid-pattern rogue waves are found for the first time,which are generated from the interactions between fundamental rogue waves and the dark-antidark rational traveling waves.In the third chapter,the time-reverse nonlocal NLS and DS equations are investigated.Using Darboux transformation method,different types of rogue-wave solutions are constructed.Particularly,a unified Darboux transformation for the time-reverse nonlocal DS equations are found,which is quite different from the classical DS system.Dynamics behaviors for those rogue-wave solution are further discussed.The(1+1)-dimensional rogue waves are found in the nonlocal NLS equation.According to the range of the parameters,these solutions can be classified into two categories,one is the globally bounded,the other has finite-time bowing-ups.For the nonlocal DS system,the(2+1)-dimensional line rogue waves are found to be either bounded or develops finitetime singularities along the certain curve in the spatial plane.In addition,richer solution patterns are found in these multi-and high-order rogue-wave solutions,most of them have not been reported before in the local counterparts.In the fourth chapter,the high-order solitons in three recently-proposed nonlocal NLS-type equations are presented,which includes the PT-symmetric,the time-reversal and the space-time-reverse nonlocal NLS equation.In addition to different symmetry relations on the scattering data,general solitons in the three equations ca be reduced from the same Riemann-Hilbet solutions of the AKNS hierarchy.Furthermore,dynamics behaviors for those high-order solitons are further analysed.It is shown that the high-order fundamental solitons can be nonsingular or repeatedly collapsing.These solitons moves along different trajectories with nearly velocities.Moreover,the high-order multi-solitons and the high-order hybird pattern solitons are also analyzed,which reveal some novel and rich dynamic structures that are quite differ from the fundamental pattern solitons.In the fifth chapter,the NonlocSolve program package is developed on the platform of Mathematica software.This program is devoted to construct exact solutions for the NLS-type and the DS-type nonlocal integrable equations.For instance,the soltion solutions,the rogue-wave solutions and some rational solutions.The feasibility and the high efficiency of this program package are verified through several particular cases.In the final chapter,a general summary to this thesis is presented as well as the prospect to our future research work.