Riemann-Hilbert Method For The Matrix-Type And Matrix-Type High Order NLS Equations

In this PhD thesis,Riemann-Hilbert(RH)method is developed for the initial value problem of matrix-type nonlinear Schr?dinger(NLS)equation and matrix-type high order nonlinear Schr?dinger(NLS)equation,respectively,under zero and non-zero boundary conditions,containing two cases:discrete spectrum as simple poles and double poles.Since the object of study is a matrix-type system,we will point out the similarities and differences between it and the RH method in the study of scalar and vector equations.In Chapter Ⅰ,we mainly introduce matrix-type NLS equation and matrix-type high order NLS equation,and briefly describe the origin and development of the RH method and other related contents.Finally,the selected topic and the main contents of this thesis are presented.In Chapter Ⅱ,we mainly study the matrix-type NLS equation with ZBCs and double poles by using the RH method.Firstly,the direct scattering problem of matrix-type NLS equation is discussed,including Jost solutions and scattering coefficients.And we then study the normalized constant and residue conditions with double poles in the case of rank(P(x,t,kn))=2,3,and give the symmetries of the normalized constant.Secondly,the inverse problem is formulated in terms of RH problem with double poles,and the reconstruction formula of the potential function is obtained by means of the eigenfunction and scattering data.Finally,we give N double poles solutions of the matrix-type NLS equation,and discuss some examples when taking the different ∑ andΩ.In Chapter Ⅲ,we mainly study the matrix-type NLS equation with NZBCs and double poles by using the RH method.First of all,we introduce single-valued parameters to avoid the multi-valued problem,and then discuss the RH problem on the complex plane of new spectral parameters.Secondly,we make a discussion for the direct scattering,and construct and solve the RH problem with double poles,and then the reconstruction formula of potential function is given.In the above process,it is necessary to discuss the normalized constant and the residue conditions in the case of double poles and rank(P(x,t,zn))=2,3,and to give the symmetries of the normalized constant.Finally,N double poles solution of matrix NLS equation is given under NZBCs.In Chapter Ⅳ,we mainly study the matrix-type high order NLS equation with ZBCs and simple and double poles by using the RH method.Firstly,the direct scattering is discussed,and the corresponding normalized constant and the residue conditions are given in the case of the different order poles(simple poles and double poles)and rank(P(x,t,kn))=2,3.The inverse scattering problem is formulated and solved by means of RH problems with simple poles and double poles,and the reconstruction formulae is given.Finally,N soliton solutions and double-poles solutions of the matrixtype high order NLS equation are given,and several examples are presented when taking the different ∑ and Ω.In Chapter V,we mainly study the matrix-type high order NLS equation with NZBCs and simple and double poles by using RH method.The single-valued parameters are introduced to avoid multi-valued problems.Next,we discuss the direct scattering,and give the normalized constant and the residue conditions with simple and double poles in rank(P(x,t,zn))]2,3.And we then construct and solve the RH problem.Finally,the reconstruction formula of the potential function is obtained,and then N soliton solutions and double-poles solutions of the matrix-type high order NLS equation are given under NZBCs,and some example are presented.In Chapter Ⅵ,we make a summary for the main results of the paper,and briefly discuss the differences between the object of studies of this paper and the integrable vector equation and the integrable scalar equation when using RH method to study them.Finally,the extensions and applications of these results of the paper are discussed.