Solutions of Gross-Pitaevskii Equation with Periodic Potential in Dimension Three

TL;DR

This paper investigates quasi-periodic solutions to the three-dimensional Gross-Pitaevskii equation with a periodic potential, establishing the existence of a large set of solutions close to plane waves for large wave vectors.

Contribution

It proves the existence of a broad non-resonant set of solutions asymptotically near plane waves in three dimensions.

Findings

Existence of a large non-resonant set ${ m f G}$ in $ r^3$.

Solutions asymptotically close to plane waves for large $|f k|$.

Applicable for small amplitude $A$.

Abstract

Quasi-periodic solutions of the Gross-Pitaevskii equation with a periodic potential in dimension three are studied. It is proven that there is an extensive "non-resonant" set ${\mathcal G} \subset \mathbb{R}^3$ such that for every $\vec k\in \mathcal G$ there is a solution asymptotically close to a plane wave $Ae^{i\langle{ \vec{k}, \vec{x} }\rangle}$ as $|\vec k|\to \infty $, given $A$ is sufficiently small.