Research On Solutions Of Nonlinear Schr(?)dinger Equation Via Riemann-Hilbert Approach

In recent decades,nonlinear partial differential equations play a significant role in mathematics and theoretical physics,which have attracted much attention in soliton theory and integrable system.Since exact solutions of integrable equations can describe and explain many natural phenomena,the study of integrable equations has become a hot topic.There are many techniques and transformations for finding exact solutions of soliton equations,such as the Darboux transformation method,the Backlund transformation method,the Hirota bilinear method,the homogeneous balance method and Wronskian technology.These methods have greatly promoted the development of soliton theory.In this paper,based on the Riemann-Hilbert(R-H)approach,N-soliton solutions of a class of nonlinear Schrodinger(NLS)equations are expressed explicitly,and the corresponding dynamic characteristics of equations can be visualized in three dimensions by selecting appropriate spectral parameter,the physical phenomena described by such solutions are revealed.This paper is structured as follows.In section 1,we summarize the background and current situation of the research.From the research background of soliton theory and development status of exact solutions for nonlinear partial differential equations,a brief description is made,in addition,the research status of R-H method based on the paper are described.Finally,the main work and structural arrangement of this paper are explained.In section 2,we mainly study nonlinear Schrodinger-type(NLST)equation.The Lax pair of NLST equation is obtained by using prolongation structure theory.The R-H problem of NLST equation is constructed by analyzing the matrix spectral problem.Through a specific R-H problem with the vanishing scattering coefficient,N-soliton solutions of NLST equation are expressed explicitly.In addition,the dynamic characteristics of soliton solutions are analyzed by selecting appropriate spectral parameters for single and two soliton solutions,respectively.In section 3,we mainly studies nonlinear Schrodinger-like(NLS-like)equation.We first construct a matrix spectral problem with NLS-like equation,and combine the spectral analysis to formulate a R-H problem.Then,we mainly use the involutory relationship of potential matrix Q to analyze the zeros of det P+ and det P-,the N-soliton solutions of NLS-like equation are expressed explicitly by a particular R-H problem with an unit jump matrix.Furthermore,the single-soliton solution and collisions of two-soliton solutions are analyzed,and the dynamic behaviors of the single-soliton solution and two-soliton solutions are shown graphically.Finally,on the basis of the R-H problem,the evolution equation of R-H data with perturbation is derived.