On the continuum limit for discrete NLS with long-range lattice interactions

TL;DR

This paper rigorously derives the continuum limit of discrete nonlinear Schrödinger equations with long-range interactions, showing that fractional Laplacians naturally emerge in the limit, thus justifying models used in physics.

Contribution

The paper proves the emergence of fractional Laplacians in the continuum limit of DNLS with long-range interactions, extending the understanding of dispersive limits in lattice systems.

Findings

Fractional Laplacian arises for long-range interactions in the continuum limit.

Standard Laplacian describes short-range interaction limits.

Results justify fractional NLS models in physics literature.

Abstract

We consider a general class of discrete nonlinear Schroedinger equations (DNLS) on the lattice $h \mathbb{Z}$ with mesh size $h>0$. In the continuum limit when $h \to 0$, we prove that the limiting dynamics are given by a nonlinear Schroedinger equation (NLS) on $\mathbb{R}$ with the fractional Laplacian $(-\Delta)^\alpha$ as dispersive symbol. In particular, we obtain that fractional powers $1/2 < \alpha < 1$ arise from long-range lattice interactions when passing to the continuum limit, whereas NLS with the non-fractional Laplacian $-\Delta$ describes the dispersion in the continuum limit for short-range lattice interactions (e.g., nearest-neighbor interactions). Our results rigorously justify certain NLS model equations with fractional Laplacians proposed in the physics literature. Moreover, the arguments given in our paper can be also applied to discuss the continuum limit for other lattice systems with long-range interactions.