An averaging theorem for nonlinear Schrödinger equations with small nonlinearities

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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

Abstract
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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.

Keywords: Averaging, NLS, longtime dynamics..

Mathematics Subject Classification: Primary: 35Q55, 70K65; Secondary: 74H4.

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:

[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.

show all references

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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