Solutions of Gross–Pitaevskii equation with periodic potential in dimension three

Abstract

Quasiperiodic solutions of the Gross–Pitaevskii equation with a periodic potential in dimension three are studied. It is proved that there is an extensive “nonresonant” set G ⊂ R 3 \mathcal {G}\subset \mathbb {R}^3 such that for every \vv k\in \mathcal {G} there is a solution asymptotically close to a plane wave Ae^{i\langle \vv {k},\vv {x}\rangle } as |\vv k|\to \infty , given A A is sufficiently small.