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 \( \overrightarrow{k} \epsilon \mathcal G \) there is a solution asymptotically close to a plane wave \( \mathcal Ae^ \mathit i \langle \overrightarrow{k},\overrightarrow{x} \rangle as |\overrightarrow{k}| \rightarrow \infty \) , given A is sufficiently small.