Multi-symplectic Runge–Kutta–Nyström methods for nonlinear Schrödinger equations with variable coefficients

In this paper, we consider Runge–Kutta–Nyström (RKN) methods applied to nonlinear Schrödinger equations with variable coefficients (NLSEvc). Concatenating symplectic Nyström 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.