Simulation of two-dimensional nonlinear problem with solitary wave based on split-step finite pointset method

Abstract

In this paper, a split-step finite pointset method (SS-FPM) is proposed and applied to the simulation of the nonlinear Schrödinger/Gross-Pitaevskii equation (NLSE/GPE) with solitary wave solution. The motivation and main idea of SS-FPMisas follows. 1) The nonlinear Schrödinger equation is first divided into the linear derivative term and the nonlinear term based on the time-splitting method. 2) The finite pointset method (FPM) based on Taylor expansion and weighted least square method is adopted, and the linear derivative term is numerically discretized with the help of Wendland weight function. Then the two-dimensional (2D) nonlinear Schrödinger equation with Dirichlet and periodic boundary conditions is simulated, and the numerical solution is compared with the analytical one. The numerical results show that the presented SS-FPM has second-order accuracy even if in the case of non-uniform particle distribution, and is easily implemented compared with the FDM, and its computational error is smaller than those in the existed corrected SPH methods. Finally, the 2D NLS equation with periodic boundary and the two-component GP equation with Dirichlet boundary and outer rotation BEC, neither of which has an analytical solution, are numerically predicted by the proposed SS-FPM. Compared with other numerical results, our numerical results show that the SS-FPM can accurately display the nonlinear solitary wave singularity phenomenon and quantized vortex process.