Finite difference scheme for two-dimensional periodic nonlinear Schrödinger equations

Abstract

A nonlinear Schrödinger equation (NLS) on a periodic box can be discretized as a discrete nonlinear Schrödinger equation (DNLS) on a periodic cubic lattice, which is a system of finitely many ordinary differential equations. We show that in two spatial dimensions, solutions to the DNLS converge strongly in $L^2$ to those of the NLS as the grid size $h>0$ approaches zero. As a result, the effectiveness of the finite difference method (FDM) is justified for the two-dimensional periodic NLS.

25 pages

References in corpus (5)

• Asymptotic behavior of small solutions for the discrete nonlinear Schrödinger and Klein-Gordon equations
• On the uniqueness of solutions to the periodic 3D Gross-Pitaevskii hierarchy
• Strong Convergence for Discrete Nonlinear Schrödinger equations in the Continuum Limit
• The Klein-Gordon Equation on ${\mathbb{Z}}^2$ and the Quantum Harmonic Lattice
• Convergence rates for dispersive approximation schemes to nonlinear Schrödinger equations