Energy-Preserving Methods Of The Triple-Coupled Nonlinear Schrodinger Equation

Based on the study of the nonlinear Schrodinger equation and the coupled nonlinear Schrodinger equa-tion,we proposed and constructed two finite difference schemes for the triple-coupled nonlinear Schrodinger equation(T-CNLS).These two schemes have two kinds of standard conserved quantities mass and energy.And the different solution unconditionally converges to the exact solution with second order in the L∞-norm or L2-norm.Comparing the numerical accuracy,convergence,conservation,and new algorithm with theoretical analysis and numerical experiments to prove the efficiency of the new algorithm.In the first chapter,the research background,related knowledge,some lemmas,and useful techniques are introduced.In the second chapter,firstly,a linearized conservative difference scheme for T-CNLS whose coeffi-cients of the nonlinear term are symmetric is studied.Then,some lemmas and useful techniques are used to analyze the different scheme,it’s shown that the difference solution unconditionally converges to the exact solution with second order in the maximum norm.Finally,the numerical examples are provided to illustrate the theoretical results.In the third chapter,firstly,we proposed an energy-preserving scheme for T-CNLS with symmetric coefficients in the nonlinear term,the scheme used Averaged Vector Field.T-CNLS can be rewritten into classical Hamiltonian system.The system is discretized by a central difference method in space.Next,an Averaged Vector Field(AVF)method in time is applied and results in an energy-preserving scheme.Some theoretical results such as convergence are investigated.Finally,numerical experiments show that the theoretical analysis is correct.In the fourth chapter,the content of this paper is briefly summarized,and the further study of the high order energy-preserving symplectic form of T-CNLS are proposed.