Symplectic integrator for nonlinear high order Schrödinger equation with a trapped term

Abstract

In the manuscript, we discuss the symplectic integrator for the numerical solution of a kind of high order Schrödinger equation with trapped terms. The Hamiltonian formulism is discovered for it. We first discretize the Hamiltonian system in space to reduce it to a finite-dimensional one. Then the symplectic midpoint scheme is applied to the temporal discretization. The symplectic scheme we devise is of second order accuracy in time and 2lth order accuracy in space. It is proved that it preserves the charge of the original equation veraciously. The energy is not preserved explicitly for the exception of the linear case. However, after some computation, the energy transit formula in the temporal direction is obtained. In the numerical part, we compare our scheme with some existing schemes, including the leap frog scheme and the energy-preserving scheme. From the numerical evidence, we find that our numerical schemes are efficient and available.The numerical results are consistent with the theoretical analysis.