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We prove a Nekhoroshev type theorem for the nonlinear Schr\"odinger equation $$ iu_t=-\Delta u+V\star u+\partial_{\bar u}g(u,\bar u)\, \quad x\in \T^d, $$ where $V$ is a typical smooth potential and $g$ is analytic in both variables.
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L. Galgani and A. Giorgilli, Rigourous estimates for the series expansions of Hamiltonian perturbation theory , Celestial Mech. 37
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J. Pöschel, Nekhoroshev estimates for quasi-convex Hamiltonian systems , Math. Z. 213
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D. Bambusi, Birkhoff normal form for some nonlinear PDEs , Comm. Math. Physics 234
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D. Bambusi and B. Grebert, Forme normale pour NLS en dimension quelconque , Compt. Rendu. Acad. Sciences Paris 337
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J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Weakly turbulent solutions for the cubic defocusing nonlinear Schrödinger equation
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H. L. Eliasson, S. B. Kuksin, KAM for non-linear Schroedinger equation
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T. Cazenave, Semilinear Schrödinger equations
2003
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D. Bambusi and B. Grébert, Birkhoff normal form for PDE’s with tame modulus
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B. Grébert, Birkhoff normal form and Hamiltonian PDEs
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E. Faou, B. Grébert and E. Paturel, Birkhoff normal form for splitting methods applied to semi linear Hamiltonian PDEs. Part II: Abstract splitting
2010
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