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

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 $A{e}^{i⟨\vec{k},\vec{x}⟩}$ as $|\vec{k}|\to \infty$, given $A$ is sufficiently small.

arXiv admin note: substantial text overlap with arXiv:1805.03974, arXiv:1707.01872