Long-time Asymptotics Of Solutions Of The Soliton Equations Associated With Higher-order Spectral Problems

The thesis mainly studies the long-time asymptotics of solutions of soliton equations associated with high-order matrix spectral problem with the aid of Deift-Zhou nonlinear steepest descent method.There are four soliton equations to be considered,two of them are the coupled derivative nonlinear Schrodinger equation and the generalized Sasa-Satsuma equation that are associated with 3 × 3 matrix spectral problems.The other two equations are the spin-1 GrossPitaevskii equation and the Hermitian symmetric space Fokas-Lenells equation,which are associated with 4 × 4 matrix spectral problems.In Chapter 2 and 3,the long-time asymptotics of solutions of the coupled derivative nonlinear Schrodinger equation and the generalized Sasa-Satsuma equation are studied in detail,respectively.The first step is constructing corresponding basic Riemann-Hilbert problems by using the inverse scattering method.Based on the basic Riemann-Hilbert problems,the nonlinear steepest descent method is applied and some proper transformations of the Riemann-Hilbert problems and strict error estimations are given.The long-time asymptotics of solutions of their initial problems are obtained with the help of model problems solved by the parabolic cylinder functions.They all have 3 × 3 Lax pairs,but there are two differences between them:(ⅰ)In the process of spectral analysis and the construction of the basic Riemann-Hilbert problem,a gauge transformation should be introduced for the coupled derivative nonlinear Schrodinger equation but the other one does not need;(ⅱ)From the phase functions in the jump matrix of the Riemann-Hilbert problems,there are three stationary points for the coupled derivative nonlinear Schrodinger equation and two for the generalized Sasa-Satsuma equation.So the transformations of the contours of the two cases are different.In Chapter 4 and 5,two equations,the spin-1 Gross-Pitaevskii equation and the Hermitian symmetric space Fokas-Lenells equation,associated with 4 × 4 matrix spectral problems are considered.On the basis of the spectral analysis of the 4 × 4 Lax pairs and the scattering matrix,the solutions to the Cauchy problem of the equations are transformed into the solutions to the corresponding RiemannHilbert problems.The Deift-Zhou nonlinear steepest descent method is extended to the Riemann-Hilbert problems,from which model Riemann-Hilbert problems are established with the help of distinct factorizations of the jump matrices for the Riemann-Hilbert problems and decompositions of the matrix-valued spectral functions.Finally,the leading-order asymptotics of the solutions are obtained.Compared with the previous two chapters,we take different block forms for the 4 × 4 matrices in these two chapters,so the computations and analysis are totally different.Secondly,the model Riemann-Hilbert problems in these two chapters are novel.Fortunately,we use the asymptotic properties of the corresponding functions and solve the model Riemann-Hilbert problems in terms of the standard parabolic cylinder functions.In addition,some differences appear when we study the equations in Chapter 4 and Chapter 5.On the one hand,spectral analysis is based on the Lax pair directly for the spin-1 Gross-Pitaevskii equation,but a gauge transformation still should be introduced for the Hermitian symmetric space Fokas-Lenells equation.On the other hand,the phase functions indicate that the spin-1 Gross-Pitaevskii equation has one stationary point and the Hermitian symmetric space Fokas-Lenells equation has four stationary points,which determine the different deformations of the contours.