  Consider nonlinear Schr\"odinger equations with small nonlinearities
  \[\frac{d}{dt}u+i(-\triangle u+V(x)u)=\epsilon \mathcal{P}(\triangle
u,u,x),\quad x\in \mathbb{T}^d.\eqno{(*)}\]
  Let $\{\zeta_1(x),\zeta_2(x),\dots\}$ be the $L_2$-basis formed by
eigenfunctions of the operator $-\triangle +V(x)$. For any complex function
$u(x)$, write it as \mbox{$u(x)=\sum_{k\geqslant1}v_k\zeta_k(x)$} and set
$I_k(u)=\frac{1}{2}|v_k|^2$. Then for any solution $u(t,x)$ of the linear
equation $(*)_{\epsilon=0}$ we have $I(u(t,\cdot))=const$. In this work it is
proved that if $(*)$ is well posed on time-intervals $t\lesssim \epsilon^{-1}$
and satisfies there some mild a-priori assumptions, then for any its solution
$u^{\epsilon}(t,x)$, the limiting behavior of the curve
$I(u^{\epsilon}(t,\cdot))$ on time intervals of order $\epsilon^{-1}$, as
$\epsilon\to0$, can be uniquely characterized by solutions of a certain
well-posed effective equation.
