Integrability Of Higher-order And Multicomponent Coupled Systems In Nonlinear Models

In order to reveal the principle and mechanism of nonlinear phenomena,researchers usually use some nonlinear models to describe these phenomena.By considering the analytical solutions of some nonlinear models,we can find more information about nonlinear phenomena and learn the essence of some nonlinear phenomena better.Thus,the investigation of the analytical solutions for some nonlinear models is the important subject in nonlinear science fields,especially in in the field of nonlinear mathematical physics.In this thesis,with the help of symbolic computation and by using Darboux transformation,variable separation technique,inverse scattering transformation method,nonlinear steepest descent method,we investigate the integrability of some higher-order and multicomponent coupled nonlinear models in the field of nonlinear Science.The specific research content is as follows:Based on Darboux transformation,we obtain the higher order rogue wave solutions for higher-order and multicomponent coupled nonlinear models which are related to spectral problems by using the variable separation technique,respectively,and a unified rogue wave solutions for Kundu-nonlinear Schr ¨odinger equation,coupled nonlinear Schr ¨odinger equation with four-wave mixing and three-component Manakov system are obtained.For Kundu-nonlinear Schr ¨odinger equation,the dynamic behaviors of firstorder,second-order and fourth-order rogue waves are discussed.For coupled nonlinear Schr ¨odinger equation with four-wave mixing,many novel rogue wave structures are discovered.For three-component Manakov system,the structures of rogue waves on multisoliton background are discovered,the interactions between rogue waves and bright-dark solitons are also discussed.We construct the Riemann-Hilbert problem and multi-soliton solutions for the 2n spectral problems of the vector modified Korteweg-de Vries equation for the first time.Firstly,based on its Lax pairs,we transform the vector modified Korteweg-de Vries equation into a model Riemann-Hilbert problem,then by solving the particular RiemannHilbert problem in the reflectionless cases,we can obtain the three types of multi-soliton solutions for the vector modified Korteweg-de Vries equation.Finally,the dynamic behaviors of these soliton solutions are discussed with some graphics.Based on the nonlinear steepest descent method,we construct the long-time asymptotics of solution for the coupled Sasa-Satsuma equation with decaying initial value.Because the spectral problem for the coupled Sasa-Satsuma equation is 5 × 5,we can change the higher order matrix into 2 × 2 form by using block matrix,and a 2 × 2 Riemann-Hilbert problem is also obtained.Based on the Deift-Zhou method and main transformations for the jump contours,we can change the model Riemann-Hilbert problem into a parabolic cylindrical equation.Utilizing the asymptotics of the obtained parabolic cylindrical equation,we finally obtain the long-time asymptotics for the coupled Sasa-Satsuma equation.We analyze the long-time asymptotics of solution of initial-boundary value problems for the focusing Kundu-Eckhaus equation with time-periodic boundary condition on the quarter plane.Firstly based on its Lax pair,we obtain the corresponding Riemann-Hilbert problem.Then we find the Riemann-Hilbert problem whose solution gives the solution of our initial-boundary-value problem.By analyzing the asymptotics of an associated matrix Riemann-Hilbert problem,we find that for a certain condition the solution of the initial-boundary value problem has different asymptotic behaviors in different regions.For the Zakharov-Manakov region,the solution takes the form of the Zakharov-Manakov vanishing asymptotics.For the elliptic wave region,the solution takes the form of a modulated elliptic wave.For the plane wave region,the solution takes the form of a plane wave.