Fokas Method For Initial-Boundary Value Problem Of The Integrable Systems

In this PhD thesis,the research object is a nonlinear integrable evolution equation.This research focus on the initial-boundary value problem of the integrable evolution equation on the half-line region ?={(x,t)|0<x<?,0<t<T}.By utilizing the Fokas unified transformation method,the initial-boundary value problem of the integrable evolution equation on the half-line is transformed into the corresponding matrix Riemann-Hilbert problem.The first chapter introduces the background of the four different research methods,which are related to the Riemann-Hilbert problem.These methods are known as the inverse scattering transformation(IST)method,the Fokas unified transformation method,the dressing transformation method,and the nonlinear steepest-descent method or Deift-Zhou method,which is introduced by Deift and Zhou.Furthermore,the main objective of this study and the present development of the dissertation is explored in this chapter.In chapter 2,it analyzes the initial-boundary value problem of the Kundu-Eckhaus equation related to the 2 × 2 matrix spectrum problem with attenuating properties on the half-line.By utilizing the Fokas unified transformation method,introducing the characteristic function spectrum analysis,as well as combining with the spectral problem The combination of eigenfunctions constructs the related Riemann-Hilbert problem,it gives the representation of the solution of the initial-boundary value problem and proves the uniqueness of its solution.In chapter 3,applied the Fokas unified transformation method,the initial-boundary value problem of two kinds of coupled nonlinear Schrodinger equations with attenuating properties on the half-line is analyzed in the 3 × 3 matrix spectral problem.Based on the eigenfunctions of the Lax pairs and the analytical properties of the spectral functions,the initial-boundary value problems of the two coupled nonlinear Schrodinger equations are reduced to the 3 × 3 matrix Riemann-Hilbert problem,and then given The formal representation of the solution of the Riemann-Hilbert problem for the solution of the initial boundary value problem,and proves the uniqueness of the solution of the corresponding matrix Riemann-Hilbert problem.Remarkably,the distinct integrable equations of other 3 × 3 matrices,with jump curves of these two types of integrable equations enjoy two symmetric hyperbola,while the other equations jump curves are straight lines.Similar to the chapter 3,chapter 4 analyses the two-component mKdV equations related to the 4 × 4 matrix spectrum problem and the initial-boundary value problem of the coherently coupled nonlinear Schrodinger equation on the half-line.It is proved that the solution of the initial value of the two-component mKdV equation and the coherently coupled nonlinear Schrodinger equation can be represented by the solution of the 4 × 4 matrix Riemann-Hilbert problem.It is worth noting that these two classes are analyzed.When the initial-boundary value problem of the integrable evolution equation of the 4 × 4 matrix Lax pair can be decomposed into the 2 × 2 order Lax pair block matrix form,in fact,it can also be similar to the second part of the analysis of these two types of equations,the results consistent with this article can be obtained.Therefore,these two types of equations are generalizations of the more general form of the Fokas unified transformation method for solving the initial-boundary value problem of integrable evolution equations.