An averaging theorem for nonlinear Schrödinger equations with small nonlinearities

Consider nonlinear Schrödinger equations with small nonlinearities \[\frac{d}{dt}u+i(-\triangle u+V(x)u)=ε\mathcal{P}(\triangle u,u,x),\quad x\in \mathbb{T}^d.\eqno{(*)}\] Let $\{ζ_1(x),ζ_2(x),\dots\}$ be the $L_2$-basis formed by eigenfunctions of the operator $-\triangle +V(x)$. For any complex function $u(x)$, write it as \mbox{$u(x)=\sum_{k\geqslant1}v_kζ_k(x)$} and set $I_k(u)=\frac{1}{2}|v_k|^2$. Then for any solution $u(t,x)$ of the linear equation $(*)_{ε=0}$ we have $I(u(t,\cdot))=const$. In this work it is proved that if $(*)$ is well posed on time-intervals $t\lesssim ε^{-1}$ and satisfies there some mild a-priori assumptions, then for any its solution $u^ε(t,x)$, the limiting behavior of the curve $I(u^ε(t,\cdot))$ on time intervals of order $ε^{-1}$, as $ε\to0$, can be uniquely characterized by solutions of a certain well-posed effective equation.

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