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

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

2011-08-31

arxiv.org/abs/1108.6136v1

Mathematics

Analysis of PDEs

25 pages; no figures

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

On the continuum limit for discrete NLS with long-range lattice interactions does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with On the continuum limit for discrete NLS with long-range lattice interactions, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On the continuum limit for discrete NLS with long-range lattice interactions will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-625230