Wannier functions analysis of the nonlinear Schr\"{o}dinger equation with a periodic potential

TL;DR

This paper uses Wannier functions to approximate the nonlinear Schrödinger equation with a periodic potential, revealing it as a vector lattice with long-range interactions and analyzing the tight-binding approximation's validity.

Contribution

It introduces a Wannier function-based lattice approximation for the nonlinear Schrödinger equation with periodic potentials, highlighting the long-range interactions and assessing the tight-binding approximation.

Findings

The nonlinear Schrödinger equation with a periodic potential is equivalent to a vector lattice with long-range interactions.

The tight-binding approximation's validity is analyzed for cosine potentials.

Results are applicable to Bose-Einstein condensates and nonlinear photonic crystals.

Abstract

In the present Letter we use the Wannier function basis to construct lattice approximations of the nonlinear Schr\"{o}dinger equation with a periodic potential. We show that the nonlinear Schr\"{o}dinger equation with a periodic potential is equivalent to a vector lattice with long-range interactions. For the case-example of the cosine potential we study the validity of the so-called tight-binding approximation i.e., the approximation when nearest neighbor interactions are dominant. The results are relevant to Bose-Einstein condensate theory as well as to other physical systems like, for example, electromagnetic wave propagation in nonlinear photonic crystals.