An averaging theorem for nonlinear Schrödinger equations with small nonlinearities

Abstract

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 $\{{\zeta }_1(x),{\zeta }_2(x),\dots \}$ be the $L_2$-basis formed by eigenfunctions of the operator $-△+V(x)$. For any complex function $u(x)$, write it as \mbox{$u(x)={\sum }_{k⩾1}{v}_k{\zeta }_k(x)$} and set $I_k(u)=\frac{1}{2}|v_k{|}^2$. Then for any solution $u(t,x)$ of the linear equation $(\ast {)}_{\epsilon =0}$ we have $I(u(t,\cdot ))=const$. In this work it is proved that if $(\ast )$ 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 $\epsilon \to 0$, can be uniquely characterized by solutions of a certain well-posed effective equation.