Explicit Symplectic Methods for the Nonlinear Schrödinger Equation (Communications in Computational Physics, 2009)
By performing a particular spatial discretization to the nonlinearSchrödinger equation (NLSE), we obtain a non-integrable Hamiltonian system whichcan be decomposed into three integrable parts (L-L-N splitting). We integrate each partby calculating its phase flow, and develop explicit symplectic integrators of differentorders for the original Hamiltonian by composing the phase flows. A 2nd-order reversibleconstructed symplectic scheme is employed to simulate solitons motion andinvariants behavior of the NLSE. The simulation results are compared with a 3rd-ordernon-symplectic implicit Runge-Kutta method, and the convergence of the formal energyof this symplectic integrator is also verified. The numerical results indicate thatthe explicit symplectic scheme obtained via L-L-N splitting is an effective numericaltool for solving the NLSE.