Application Of Inverse Scattering Metnod To A Class Of Nonlinear Integrable Model

Because the solution of the nonlinear model can reflect many mathematical physics phenomena,solving the solution of the nonlinear model is of great significance.As one of the methods to solve the nonlinear integrable model,the main step of the inverse scattering transformation is to construct the Lax pair and Riemann-Hilbert problem of the corresponding equation,and then solve the analytical solution of the Riemann-Hilbert problem in turn,and then get the corresponding solution of the equation.In this thesis,a class of nonlinear integrable models are solved by using the inverse scattering transformation method.Firstly,the soliton solutions of the local Kundu-Eckhaus(KE)equation with zero boundary conditions are studied by using the inverse scattering transformation method.The Lax pair of the equation is given,and the Riemann-Hilbert problem of the corresponding equation is constructed.The exact soliton solutions formula for the case of N simple poles,one high-order pole and multiple high-order poles are studied by using the solutions of the Riemann-Hilbert problem.In addition,we give a graph of one soliton solution,two soliton solution and the interaction between a simple soliton and a second-order soliton,and give the soliton solution intuitively.Then,due to the limitation of the local equation,we give an integrable nonlocal extension of the KE equation.The soliton solutions of the nonlocal KE equation with nonzero boundary conditions are studied by using the inverse scattering transformation,and the bright and dark soliton figures are given.