On the continuum limit for the discrete Nonlinear Schr\"odinger equation on a large finite cubic lattice

TL;DR

This paper proves that solutions of the discrete nonlinear Schrödinger equation on large finite lattices converge to the continuous NLS in Euclidean space, providing precise bounds and leveraging lattice Strichartz estimates, especially addressing the 3D case.

Contribution

It establishes the continuum limit for the discrete NLS on large finite lattices in 2D and 3D, with explicit convergence rates and improved estimates as the domain expands.

Findings

Convergence of discrete NLS solutions to continuous NLS as lattice size increases

Global-in-time bounds for the rate of convergence

Reduced regularity loss in Strichartz estimates with domain expansion

Abstract

In this study, we consider the nonlinear Sch\"odinger equation (NLS) with the zero-boundary condition on a two- or three-dimensional large finite cubic lattice. We prove that its solution converges to that of the NLS on the entire Euclidean space with simultaneous reduction in the lattice distance and expansion of the domain. Moreover, we obtain a precise global-in-time bound for the rate of convergence. Our proof heavily relies on Strichartz estimates on a finite lattice. A key observation is that, compared to the case of a lattice with a fixed size [Y. Hong, C. Kwak, S. Nakamura, and C. Yang, \emph{Finite difference scheme for two-dimensional periodic nonlinear {S}chr\"{o}dinger equations}, Journal of Evolution Equations \textbf{21} (2021), no.~1, 391--418.], the loss of regularity in Strichartz estimates can be reduced as the domain expands, depending on the speed of expansion. This allows us to address the physically important three-dimensional case.