This paper is based on the presentation done at the seminar Laurent Schwartz in January 2020. It describes and summarizes the results given in Rational normal forms and stability of small solutions to nonlinear Schrödinger equations, see [BFG20a], written with Joackim Bernier and Benoît Grébert, and published in Annals of PDE 6, article number: 14 (2020) 65p. We describe here the main arguments of the proof as well as the general strategy used in the Birkhoff normal form for Partial Differential Equations.
@article{SLSEDP_2019-2020____A11_0,
author = {Faou, Erwan},
title = {Resonances and genericity in {Birkhoff} normal forms},
journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
note = {talk:5},
pages = {1--10},
publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
year = {2019-2020},
doi = {10.5802/slsedp.145},
language = {en},
url = {http://www.numdam.org/articles/10.5802/slsedp.145/}
}
TY - JOUR AU - Faou, Erwan TI - Resonances and genericity in Birkhoff normal forms JO - Séminaire Laurent Schwartz — EDP et applications N1 - talk:5 PY - 2019-2020 SP - 1 EP - 10 PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique UR - http://www.numdam.org/articles/10.5802/slsedp.145/ DO - 10.5802/slsedp.145 LA - en ID - SLSEDP_2019-2020____A11_0 ER -
%0 Journal Article %A Faou, Erwan %T Resonances and genericity in Birkhoff normal forms %J Séminaire Laurent Schwartz — EDP et applications %Z talk:5 %D 2019-2020 %P 1-10 %I Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique %U http://www.numdam.org/articles/10.5802/slsedp.145/ %R 10.5802/slsedp.145 %G en %F SLSEDP_2019-2020____A11_0
Faou, Erwan. Resonances and genericity in Birkhoff normal forms. Séminaire Laurent Schwartz — EDP et applications (2019-2020), Exposé no. 5, 10 p. doi : 10.5802/slsedp.145. http://www.numdam.org/articles/10.5802/slsedp.145/
[BFG20a] J. Bernier, E. Faou and B. Grébert. Rational normal forms and stability of small solutions to nonlinear Schrödinger equations. Annals of PDE 6, article number: 14 (2020) 65p. | DOI | Zbl
[BFG20b] J. Bernier, E. Faou and B. Grébert. Long time behavior of the solutions of NLW on the d-dimensional torus Forum of Math. Sigma (2020) 26 pages. | DOI | MR | Zbl
[BG06] D. Bambusi and B. Grébert, Birkhoff normal form for PDE’s with tame modulus. Duke Math. J. 135 no. 3 (2006), 507–567. | DOI | MR | Zbl
[BMP20] L. Biasco, J. Massetti, and M. Procesi, An Abstract Birkhoff Normal Form Theorem and Exponential Type Stability of the 1d NLS. , Commun. Math. Phys. 375, 2089–2153 (2020). | DOI | MR | Zbl
[Bou00] J. Bourgain, On diffusion in high-dimensional Hamiltonian systems and PDE. J. Anal. Math., 80 (2000) 1–35. | DOI | MR | Zbl
[CF12] R. Carles and E. Faou, Energy cascades for NLS on the torus, Discrete Contin. Dyn. Syst. 32 (2012) 2063–2077. | DOI | MR | Zbl
[CKSTT10] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation. Invent. Math. 181 (2010), no. 1, 39–113 | DOI | Zbl
[FG13] E. Faou and B. Grébert, A Nekhoroshev-type theorem for the nonlinear Schrödinger equation on the torus. Analysis & PDE 6 (2013),1243–1262. | DOI | Zbl
[GT12] B. Grébert and L. Thomann. Resonant dynamics for the quintic non linear Schrödinger equation. Ann. I. H. Poincaré - AN, 29 (2012), 455–477. | DOI | Zbl
[KP96] S. B. Kuksin and J. Pöschel, Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation Annals of Math. 143 (1996), 149–179. | DOI | MR | Zbl
Cité par Sources :
