Numerical studies for nonlinear Schrödinger equations: the Schrödinger–Poisson-Xα model and Davey–Stewartson systems

Abstract

We study the numerical approximation and analysis of two different time dependent nonlinear Schrödinger equations (NLS) : the “Schrödinger–Poisson-Xα” equation, which plays a role in the modeling of quantum particle dynamics, and the Davey–Stewartson system, which is a 2-d equation modeling unidirectional water surface waves. For both equations we use a Time Splitting Spectral scheme, which had previously shown to be a good tool for numerical simulation of cubic NLS. This scheme is particularly useful for calculations in the “semi-classical regime”, where the scaled Planck constant is taken to be small. Extensive numerical results of position density and Wigner measures in 1d, 2d and 3d for the S-P-Xα model with/without an external potential are presented. These results give an insight to understand the interplay between the nonlocal (“weak”) and the local (“strong”) nonlinearity.

For the Davey–Stewartson system, we give a convergence analysis for the semi-discrete version of the scheme. Numerical results are presented for various blowup phenomena of the equation, including blowup of defocusing, elliptic-elliptic Davey–Stewartson systems and simultaneous blowup at multiple locations in the focusing elliptic-elliptic system. Also the modeling of exact soliton type solutions for the hyperbolic-elliptic (DS2) system is studied.