DMLPG method for numerical simulation of soliton collisions in multi-dimensional coupled damped nonlinear Schrödinger system which arises from Bose–Einstein condensates

Abstract

In this paper, the direct meshless local Petrov–Galerkin (DMLPG) method is applied to find the numerical solution of coupled damped nonlinear Schrödinger system in one, two and three-dimensional spaces. The propagation properties of single soliton, double and triple solitons of coupled damped nonlinear Schrödinger system are simulated and the interactions between these solitons are studied numerically. The efficient time differencing Runge–Kutta method is utilized for the time discretization. DMLPG shifts the numerical integrations over low-degree polynomials rather than over complicated shape functions and this significantly increases the computational efficiency of DMLPG in comparison with the other meshless local weak form methods especially in two and three dimensions. The main aim of this paper is to show that the DMLPG method can be simply used for solving high-dimensional system of non-linear partial differential equations especially coupled damped nonlinear Schrödinger system. The numerical results confirm the good efficiency of the proposed method for solving our model.