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

Abstract

Quasi-periodic solutions of a nonlinear polyharmonic equation for the case $4l>n+1$ in ${\mathbb{R}}^n$, $n>1$, 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 $\mathcal{G}\subset {\mathbb{R}}^n$ 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:1707.01872