Application Of Nonlinear Localized Waves And PINN Method In The Forward-Inverse Problems For The Variable Coefficient Systems

Nonlinear localized waves have always been an important research topic in integrable systems.In this paper,the solutions of derivative nonlinear Schr?dinger(DNLS)equation and inverse time-space DNLS equation in multi-period background are studied by using Darboux transform method,such as multi-solitons,higher-order mixed mode solitons,breathers,and higher-order rogue waves.Secondly,the extended nonlinear Schr?dinger(ENLS)equation with higher-order terms and the higher-order soliton solutions of variable coefficients Hirota equation are studied based on the Riemann-Hilbert(RH)method.Third,the physical information neural network(PINN)algorithm is used to study the data driven forward-inverse problems of variable coefficients Hirota equation.The main contents of this paper are as follows:Chapter 1 is the introduction,which mainly introduces the research background and current situation of nonlinear localized waves,variable coefficient equation,Darboux transform method,RH method and PINN method of integrable system,and expounds the topic selection and main work of this paper.In Chapter 2,the N-fold Darboux transformation of Kaup-Newell(KN)system is proposed,and the semi-degenerate,degenerate and generalized semi-degenerate and degenerate Darboux transformation are derived.Based on the Darboux transformation of KN system,the Darboux transformation of DNLS equation and inverse timespace DNLS equation can be reduced.For DNLS equation,soliton solutions on multiperiodic background are derived from N-fold Darboux transformation.The degenerate and semi-degenerate Darboux transformation are used to construct higher-order soliton solutions on the multi-period background.By using the generalized degenerate and semi-degenerate Darboux transformation,the higher-order mixed-mode soliton solutions on multi-period background are obtained.For the inverse time-space DNLS equation,the N-fold Darboux transformation is used to construct the breather solution on the double periodic background.The solution of higher-order rogue wave on double periodic background is given by semi-degenerate Darboux transformation.It is found that the interaction of multi-period waves will produce different amplitudes and peaks of different sizes.The soliton on the periodic background is intuitively similar to the shape of breather waves.The first-order rogue waves in the double periodic background have two peaks and four peaks.In Chapter 3,based on RH method and generalized Darboux transformation method,the formula of multi-soliton and high-order soliton matrix for ENLS equation with thirdorder and fourth-order dispersion terms are derived,and the collision dynamics behavior,asymptotic behavior of two-soliton solutions and the long-term asymptotic estimation of higher-order single soliton solutions are analyzed in detail.For the given spectral parameters,the propagation direction,velocity,wave width and other physical quantities of solitons can be controlled by adjusting the free parameters of ENLS equation.On the other hand,the variable coefficient form of ENLS equation(i.e.Hirota equation)whose highest order term is third-order term is studied,and a series of new soliton solutions of variable coefficient Hirota equation are obtained.In addition,through the dynamic analysis of the multi-soliton solutions and higher-order soliton solutions of the variable coefficients Hirota equation,the ”heart” type periodic wave and ”O” type periodic wave are found.In Chapter 4,the forward-inverse problems of variable coefficients Hirota equation are studied based on PINN algorithm.Firstly,the improved PINN(IPINN)algorithm with local adaptive activation function of neurons and slope recovery term is used to successfully learn the data-driven multi-soliton solutions and higher-order soliton solutions of the variable coefficients Hirota equation.Secondly,by introducing the parameter regularization strategy with appropriate weight coefficient into the IPINN algorithm,the prediction parameters under different noise intensity are trained stably and accurately.Thirdly,a PINNs algorithm with two neural networks is proposed,which successfully simulates the function discovery problem of variable coefficient Hirota equation under different noise intensity for the first time.Chapter 5 summarizes the whole paper and gives the further prospects.