Higher Order Soliton Solutions And Long-time Asymptotic Behavior Of Integrable Systems

This article uses the Riemann-Hilbert（RH）method to investigate the soliton solu-tions,high-order soliton solutions,and their dynamic behavior of the higher-order Kaup-Newell（KN）equation;the long-time asymptotic behavior of the AB system with initial values belonging to the weighted Sobolev space is studied using the Dbar descent method;the local wave solutions and corresponding inverse problems of the nonlocal mKdV equa-tion are studied by means of integrable deep learning.The specific contents are as follows:In Chapter 1,the research background and current situation of the inverse scattering theory,RH method,Dbar descent method and integrable deep learning are first intro-duced,followed by a brief description of the main work of this thesis is given.In Chapter 2,the soliton solutions and higher-order soliton solutions of the high-order KN equation are studied using the RH method.Firstly,through spectral analysis,the analyticity,symmetry,and asymptotic behavior of the Jost solution and scattering matrix are studied,and the corresponding RH problem is constructed.Secondly,the potential q（x,t）is recovered at the spectral parameterζ→0 to avoid the appearance of implicit functions.Under the no-reflection condition,the general soliton solution matrix for simple zeros is given,and using this result,the properties of single and double soliton solutions are discussed in detail,and the elastic interaction of double soliton solutions is proved.Finally,the higher-order soliton solution with multiple zeros is studied by means of limit technique,and the dynamics behavior of the higher-order soliton solution is given by numerical simulation.In Chapter 3,the long-time asymptotic behavior of solutions with initial values be-longing to the weighted Sobolev space of the AB system is studied using the Dbar descent method.Based on the spectral analysis of the Lax pair of the AB system,the Cauchy prob-lem is transformed into a RH problem,and then a series of deformations are made to the RHP using the Dbar descent method to transform it into a solvable RH model.Finally,it is proved that the long-time asymptotic behavior of the initial value problem solution of the AB system in any fixed time cone C can be described by the soliton solution N（I）on the discrete spectrum,the O(t^-1/2)on the continuous spectrum,and the error term O(t^-3/4).In Chapter 4,the data-driven solutions and inverse problem of the nonlocal mKdV equation are studied using integrable deep learning.Firstly,based on the Riccati-type equation,the infinite conservation laws and conserved quantities for the nonlocal mKdV equation are given.Secondly,based on physical information neural network（PINN）,the data driven solutions of the nonlocal mKdV equation under zero and non-zero bound-aries are studied,including real solution,complex solution,kinks,dark soliton,anti-dark soliton and rational solution.Finally,PINN is used to study the inverse problem of the nonlocal mKdV equation,and it is found that adding moderate noise to the network can improve the learning accuracy of the parameters.In Chapter 5,a brief summary of the entire thesis is given,and further prospects for future research work are discussed.