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

TL;DR

This paper investigates quasi-periodic solutions to a nonlinear polyharmonic equation, including the 2D Gross-Pitaevskii equation, establishing the existence of extensive non-resonant sets where solutions resemble plane waves at high frequencies.

Contribution

It proves the existence of a large non-resonant set of wave vectors for which solutions approximate plane waves in the 2D Gross-Pitaevskii equation, extending understanding of its solution structure.

Findings

Existence of a large non-resonant set ${ mf extbf G}$ in $ ^n$.

Solutions asymptotically close to plane waves for large wave vectors.

Application to the 2D Gross-Pitaevskii equation with small amplitude.

Abstract

Quasi-periodic solutions of a nonlinear polyharmonic equation for the case $4l>n+1$ in $\R^n$, $n>1$, are studied. This includes Gross-Pitaevskii equation in dimension two ($l=1,n=2$). It is proven that there is an extensive "non-resonant" set ${\mathcal G}\subset \R^n$ 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.