Multi-symplectic Runge-Kutta-Nystrom methods for nonlinear Schrodinger equations with variable coefficients

In this paper, we consider Runge-Kutta-Nystrom (RKN) methods applied to nonlinear Schrodinger equations with variable coefficients (NLSEvc). Concatenating symplectic Nystrom methods in spatial direction and symplectic Runge-Kutta methods in temporal direction for NLSEvc leads to multi-symplectic integrators, i.e. to numerical methods which preserve the multi-symplectic conservation law (MSCL), we present the corresponding discrete version of MSCL. It is shown that the multi-symplectic RKN methods preserve not only the global symplectic structure in time, but also local and global discrete charge conservation laws under periodic boundary conditions. We present a (4-order) multi-symplectic RKN method and use it in numerical simulation of quasi-periodically solitary waves for NLSEvc, and we compare the multi-symplectic RKN method with a non-multi-symplectic RKN method on the errors of numerical solutions, the numerical errors of discrete energy, discrete momentum and discrete charge. The precise conservation of discrete charge under the multi-symplectic RKN discretizations is attested numerically. Some numerical superiorities of the multi-symplectic RKN methods are revealed. (c) 2007 Elsevier Inc. All rights reserved.