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

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 $\vec k\in \mathcal G$thereisasolutionasymptoticallyclosetoaplanewave$Ae^{i\langle{ \vec{k}, \vec{x} }\rangle}$as$|\vec k|\to \infty $,given$A$ is sufficiently small.