The Study On The Propagation Of Light Pulses And Its Inverse Problem For The(2+1)-dimensional Complex Ginzburg-landau Equation

The Ginzburg-Landau equation is a differential equation model which can describe non-linear phenomena such as second-order phase transitions,superconductivity and Bose-Einstein condensate.The generated dissipative solitons have important application in the fields of laser processing,optical communication,biomedicine and so on.For the(2+1)-dimensional Ginzburg-Landau equation with dispersion,optical filtering,nonlinear gain and linear gain,a finite difference scheme with second-order convergence in space and time is presented.And the existence,convergence and stability of the solution are proved.On this basis,the inverse problems of parameter determination of dispersion coefficient,filter coefficient and nonlinear gain coefficient are proposed.Physics-informed neural networks(PINNs)and particle swarm optimization(PSO)are used to solve the problems.This article consists of the following sections:In chapter 1,the background and current research results of Ginzburg-Landau equation are introduced,as well as the main research contents and contributions of this paper.In chapter 2,for the(2+1)-dimensional nonlinear Ginzburg-Landau equation,a two-layer nonlinear difference scheme and a three-layer linearized difference scheme are given.The P-R(Peaceman-Rachford)alternating direction implicit(ADI)scheme corresponding to the two schemes is given.And the existence,convergence and stability of the solutions of the two schemes are proved respectively.Finally,the validity of ADI scheme to simulate soliton evo-lution is verified by numerical experiments.The effects of initial and boundary perturbations on soliton evolution are investigated.In chapter 3,we introduce the deep learning method of physics-informed neural networks,and present the inverse problem of parameter determination for(2+1)-dimensional nonlinear Ginzburg-Landau equation.The inverse problem is solved by PINNs and PSO methods.The results show that PINNs compute the inverse problems faster than PSO.Chapter 4 summarizes this paper and points out the following research direction.