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An averaging theorem for nonlinear Schrödinger equations with small nonlinearities
1. | Centre Mathémathiques Laurent Schwartz, École Polytechnique, Palaiseau, 91125, France |
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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