Nonlinear Schrodinger E Poisson Equation, The Time-splitting Sine Spectrum Approximation Method

In this paper we deduce in detail the time-splitting sine-spectral approximations for nonlinear Schr(?)dinger-Poisson equations in the semiclassical regimes, where the Planck constantεis small.The time-splitting spectral approximation under study is explicit,unconditionally stable.Extensive numerical tests are presented for weak/strong defocusing nonlinearities and weak O(ε) focusing nonlinearities. The tests are geared towards the understanding of admissible meshing strategies for obtaining "correct" physical observables in the semiclassical regimes.