Computational methods for the dynamics of the nonlinear Schrodinger/Gross-Pitaevskii equations

TL;DR

This paper reviews and compares various numerical methods for solving the nonlinear Schrödinger and Gross-Pitaevskii equations, highlighting their properties, extensions, and applications to Bose-Einstein condensates and nonlinear optics.

Contribution

It provides a comprehensive comparison of numerical techniques for NLSE/GPE, including extensions for damping, rotation, and coupled systems, with application to vortex lattice dynamics.

Findings

Finite difference and spectral methods effectively solve NLSE/GPE.

Extensions handle damping, rotation, and coupled equations.

Simulation of vortex lattice dynamics in rotating BECs.

Abstract

In this paper, we begin with the nonlinear Schrodinger/Gross-Pitaevskii equation (NLSE/GPE) for modeling Bose-Einstein condensation (BEC) and nonlinear optics as well as other applications, and discuss their dynamical properties ranging from time reversible, time transverse invariant, mass and energy conservation, dispersion relation to soliton solutions. Then, we review and compare different numerical methods for solving the NLSE/GPE including finite difference time domain methods and time-splitting spectral method, and discuss different absorbing boundary conditions. In addition, these numerical methods are extended to the NLSE/GPE with damping terms and/or an angular momentum rotation term as well as coupled NLSEs/GPEs. Finally, applications to simulate a quantized vortex lattice dynamics in a rotating BEC are reported.