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
References:
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D. Bambusi, Galerkin averaging method and Poincaré normal form for some quasilnear PDEs,, Ann. Scuola Norm. Sup. Pisa C1. Sci., IV (2005), 669.   Google Scholar

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V. Bogachev and I. Malofeev, On the absolute continuity of the distributions of smooth functions on infinite-dimensional spaces with measures,, preprint, (2013).   Google Scholar

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J. Bourgain, On diffusion in high-dimensional Hamiltonian systems and PDEs,, Journal d'Analyse Mathématique, 80 (2000), 1.  doi: 10.1007/BF02791532.  Google Scholar

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G. Huang, An averaging theorem for a perturbed KdV equation,, Nonlinearity, 26 (2013), 1599.  doi: 10.1088/0951-7715/26/6/1599.  Google Scholar

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S. Kuksin, Damped-driven KdV and effective equations for long-time behavior of its solutions,, GAFA, 20 (2010), 1431.  doi: 10.1007/s00039-010-0103-6.  Google Scholar

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S. Kuksin and A. Piatnitski, Khasminskii-Whitham averaging for randomly perturbed KdV equation,, J. Math. Pures Appl., 89 (2008), 400.  doi: 10.1016/j.matpur.2007.12.003.  Google Scholar

[12]

P. Lochak and C. Meunier, Multiphase Averaging for Classical Systems,, Springer-Verlag, (1988).  doi: 10.1007/978-1-4612-1044-3.  Google Scholar

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J. Pöschel, On Nekhoroshev estimates for a nonlinear Schrödinger equation and a theorem by Bambusi,, Nonlinearity, 12 (1999), 1587.  doi: 10.1088/0951-7715/12/6/310.  Google Scholar

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T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations,, de Gruyter, (1996).  doi: 10.1515/9783110812411.  Google Scholar

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H. Whitney, Differentiable even functions,, Duke Math. Journal, 10 (1943), 159.  doi: 10.1215/S0012-7094-43-01015-4.  Google Scholar

show all references

References:
[1]

D. Bambusi, Nekhoroshev theorem for small amplitude solutions in nonlinear Schrödinger equation,, Math. Z., 230 (1999), 345.  doi: 10.1007/PL00004696.  Google Scholar

[2]

D. Bambusi, Galerkin averaging method and Poincaré normal form for some quasilnear PDEs,, Ann. Scuola Norm. Sup. Pisa C1. Sci., IV (2005), 669.   Google Scholar

[3]

D. Bambusi and B. Grebert, Forme normale pour NLS en dimension quelconque,, Comptes Rendus Mathématique, 337 (2003), 409.  doi: 10.1016/S1631-073X(03)00368-6.  Google Scholar

[4]

V. Bogachev, Differentiable Measures and the Malliavin Calculus,, American Mathematical Society, (2010).   Google Scholar

[5]

V. Bogachev and I. Malofeev, On the absolute continuity of the distributions of smooth functions on infinite-dimensional spaces with measures,, preprint, (2013).   Google Scholar

[6]

J. Bourgain, On diffusion in high-dimensional Hamiltonian systems and PDEs,, Journal d'Analyse Mathématique, 80 (2000), 1.  doi: 10.1007/BF02791532.  Google Scholar

[7]

G. Huang, An averaging theorem for a perturbed KdV equation,, Nonlinearity, 26 (2013), 1599.  doi: 10.1088/0951-7715/26/6/1599.  Google Scholar

[8]

T. Kappeler and S. Kuksin, Strong non-resonance of Schrédinger operators and an averaging theorem,, Physica D, 86 (1995), 349.  doi: 10.1016/0167-2789(95)00115-K.  Google Scholar

[9]

S. Kuksin, Damped-driven KdV and effective equations for long-time behavior of its solutions,, GAFA, 20 (2010), 1431.  doi: 10.1007/s00039-010-0103-6.  Google Scholar

[10]

S. Kuksin, Weakly nonlinear stochastic CGL equations,, Annales de l'insitut Henri Poincaré-Probabilité et Statistiques, 49 (2013), 915.  doi: 10.1214/11-AIHP482.  Google Scholar

[11]

S. Kuksin and A. Piatnitski, Khasminskii-Whitham averaging for randomly perturbed KdV equation,, J. Math. Pures Appl., 89 (2008), 400.  doi: 10.1016/j.matpur.2007.12.003.  Google Scholar

[12]

P. Lochak and C. Meunier, Multiphase Averaging for Classical Systems,, Springer-Verlag, (1988).  doi: 10.1007/978-1-4612-1044-3.  Google Scholar

[13]

J. Pöschel, On Nekhoroshev estimates for a nonlinear Schrödinger equation and a theorem by Bambusi,, Nonlinearity, 12 (1999), 1587.  doi: 10.1088/0951-7715/12/6/310.  Google Scholar

[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.  Google Scholar

[15]

H. Whitney, Differentiable even functions,, Duke Math. Journal, 10 (1943), 159.  doi: 10.1215/S0012-7094-43-01015-4.  Google Scholar

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