, Volume 317, Issue 3, pp 563-591

On the Continuum Limit for Discrete NLS with Long-Range Lattice Interactions

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Abstract

We consider a general class of discrete nonlinear Schrödinger equations (DNLS) on the lattice ${h\mathbb{Z}}$ with mesh size h > 0. In the continuum limit when h → 0, we prove that the limiting dynamics are given by a nonlinear Schrödinger equation (NLS) on ${\mathbb{R}}$ with the fractional Laplacian (−Δ) α as dispersive symbol. In particular, we obtain that fractional powers ${\frac{1}{2} < \alpha < 1}$ arise from long-range lattice interactions when passing to the continuum limit, whereas the NLS with the usual Laplacian −Δ describes the dispersion in the continuum limit for short-range or quick-decaying 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.

Communicated by H.-T. Yau
K.K. was partially supported by NSF grants DMS-0703618, DMS-1106770 and OISE-0730136.
E.L. acknowledges support by a Steno fellowship from the Danish Research Council.
G.S. was partially supported by NSF grant DMS-1068815.