Conservative Schemes For Nonlinear Schr?dinger Equations

This paper mainly studies the structure-preserving algorithms of the nonlinear fourth-order Schr?dinger equation with cubic nonlinear term and the nonlinear Schr?dinger equation with wave operator.Firstly, combining symplectic algorithm, splitting technique and compact method, a compact splitting symplectic scheme is proposed to solve the fourth-order Schr?dinger equation with cubic nonlinear term. The scheme has fourth-order accuracy in space and second-order accuracy in time. The symplectic conservation and charge conservation of the scheme are analyzed in theory. Secondly, using the combination method, a local energy conservation scheme and a local momentum conservation scheme are constructed for the fourth-order Schr?dinger equation with cubic nonlinear term. The two schemes can preserve discrete global energy and overall momentum conservation, respectively, and also preserve the discrete charge conservation law. We also prove that the two schemes are stable by linearization method.By adopting the idea of combined construction method and discrete Leibniz rules, a local energy conservation scheme and a local momentum conservation scheme are constructed to solve the nonlinear Schr?dinger equations with the wave operator. And the discrete conservation properties of the two schemes maintained are proved in the theory.In this paper, the feasibility and effectiveness of the proposed numerical schemes are verified by using Matlab software to programming, the numerical results are in conformity with the theory.