# American Institute of Mathematical Sciences

September  2014, 34(9): 3555-3574. doi: 10.3934/dcds.2014.34.3555

## An averaging theorem for nonlinear Schrödinger equations with small nonlinearities

 1 Centre Mathémathiques Laurent Schwartz, École Polytechnique, Palaiseau, 91125, France

Received  July 2013 Revised  December 2013 Published  March 2014

Consider nonlinear Schrödinger equations with small nonlinearities $\frac{d}{dt}u+i(-\triangle u+V(x)u)=\epsilon \mathcal{P}(\triangle u,\nabla u,u,x), x\in \mathbb{T}^d. (*)$ 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 $u(x) = \sum_{k≥1} v_k \zeta_k (x)$ and set $I_k(u)=\frac{1}{2}|v_k|^2$. Then for any solution $u(t,x)$ of the linear equation $(*)_{\epsilon=0}$ we have $I(u(t,\cdot))=const$. In this work it is proved that if $(*)$ is well posed on time-intervals $t≲\epsilon^{-1}$ 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.
Citation: Guan Huang. An averaging theorem for nonlinear Schrödinger equations with small nonlinearities. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3555-3574. doi: 10.3934/dcds.2014.34.3555
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