# 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 and Continuous Dynamical Systems, 2014, 34 (9) : 3555-3574. doi: 10.3934/dcds.2014.34.3555
##### References:
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##### References:
 [1] D. Bambusi, Nekhoroshev theorem for small amplitude solutions in nonlinear Schrödinger equation, Math. Z., 230 (1999), 345-387. doi: 10.1007/PL00004696. [2] D. Bambusi, Galerkin averaging method and Poincaré normal form for some quasilnear PDEs, Ann. Scuola Norm. Sup. Pisa C1. Sci., IV (2005), 669-702. [3] D. Bambusi and B. Grebert, Forme normale pour NLS en dimension quelconque, Comptes Rendus Mathématique, 337 (2003), 409-414. doi: 10.1016/S1631-073X(03)00368-6. [4] V. Bogachev, Differentiable Measures and the Malliavin Calculus, American Mathematical Society, 2010. [5] V. Bogachev and I. Malofeev, On the absolute continuity of the distributions of smooth functions on infinite-dimensional spaces with measures, preprint, (2013). [6] J. Bourgain, On diffusion in high-dimensional Hamiltonian systems and PDEs, Journal d'Analyse Mathématique, 80 (2000), 1-35. doi: 10.1007/BF02791532. [7] G. Huang, An averaging theorem for a perturbed KdV equation, Nonlinearity, 26 (2013), 1599-1621. doi: 10.1088/0951-7715/26/6/1599. [8] T. Kappeler and S. Kuksin, Strong non-resonance of Schrédinger operators and an averaging theorem, Physica D, 86 (1995), 349-362. doi: 10.1016/0167-2789(95)00115-K. [9] S. Kuksin, Damped-driven KdV and effective equations for long-time behavior of its solutions, GAFA, 20 (2010), 1431-1463. doi: 10.1007/s00039-010-0103-6. [10] S. Kuksin, Weakly nonlinear stochastic CGL equations, Annales de l'insitut Henri Poincaré-Probabilité et Statistiques, 49 (2013), 915-1231. doi: 10.1214/11-AIHP482. [11] S. Kuksin and A. Piatnitski, Khasminskii-Whitham averaging for randomly perturbed KdV equation, J. Math. Pures Appl., 89 (2008), 400-428. doi: 10.1016/j.matpur.2007.12.003. [12] P. Lochak and C. Meunier, Multiphase Averaging for Classical Systems, Springer-Verlag, 1988. doi: 10.1007/978-1-4612-1044-3. [13] J. Pöschel, On Nekhoroshev estimates for a nonlinear Schrödinger equation and a theorem by Bambusi, Nonlinearity, 12 (1999), 1587-1600. doi: 10.1088/0951-7715/12/6/310. [14] T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, de Gruyter, 1996. doi: 10.1515/9783110812411. [15] H. Whitney, Differentiable even functions, Duke Math. Journal, 10 (1943), 159-160. doi: 10.1215/S0012-7094-43-01015-4.
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