Multi-pole Soliton Solutions For Higher Order Nonlinear Schr(?)dinger Equations

This thesis studies the multi-pole soliton solutions for three kinds of higher order nonlinear Schr (?)dinger equations by Riemann-Hilbert method.First,by spectral analysis for Lax pair of the equation,the eigenfunctions and scattering matrix are obtained,and their symmetry,analyticity and asymptotic properties are also given.Then the inverse scattering problem is expressed as a solvable Riemann-Hilbert problem.By using the relationship between the solution of equation and the solution of Riemann-Hilbert problem,the multi-pole soliton solutions of the equation are obtained.According to the involved initial boundary conditions and the types of soliton solutions,this thesis is divided into the following four parts:In Chapter 2,we study the two component Kundu-Eckhaus equation(a type of nonlinear Schr (?)dinger equations)under the zero boundary conditions.Firstly,based on the spectral analysis of Lax pair,we introduce a matrix function (,)to regularize the asymptotic behavior of the Jost function and establish the Riemann-Hilbert problem with simple-pole.Then,by making an appropriate transformation,the non-regular RiemannHilbert problem is converted into a regular one.Finally,using the Plemelj formula,we obtain the simple-pole soliton solution of the equation.In Chapter 3,we study the simple-pole soliton solution and double-pole soliton solution of the mixed Chen-Lee-Liu derivative nonlinear Schr (?)dinger equation under the nonzero boundary conditions.Unlike Chapter 2,we need to deal with nonzero boundary conditions.In order to avoid the complexity of Riemann surfaces and the multivaluedness of eigenvalues,it is necessary to introduce the uniformizing variable = + 6).Based on the symmetry,analyticity,and asymptotics of the Jost function and scattering matrix,Riemann-Hilbert problems with simple-pole and double-pole are constructed respectively.In addition,when dealing with Riemann-Hilbert problem with double-pole,we need to consider a more complex discrete spectrum problem.In Chapter 4,we study the double-pole solutions of the modified nonlinear Schr (?)dinger equation under the zero boundary conditions or nonzero boundary conditions respectively.Based on the spectral analysis of Lax pair,a related Riemann-Hilbert problems with double-pole are constructed.The inverse scattering problem is formulated as a Riemann-Hilbert problem.By using the Plemelj formula,the double-pole solutions of the equation are obtained.In Chapter 5,we further study multi-pole soliton solutions for the modified nonlinear Schr (?)dinger equation under the zero boundary conditions.Unlike in Chapter 4,when considering the discrete spectrum of scattering problems,we consider the cases of a single higher-order pole and multiple higher-order poles.By using the generalization of the residue theorem,the Riemann-Hilbert problem with multiple higher-order poles is solved.Finally,the -pole soliton solutions of the equation are obtained.