V. I. Arnold, Mathematical methods in classical mechanics, 1989.

V. I. Arnold, V. V. Kozlov, and A. I. Neistadt, Mathematical Aspects of Classical and Celestial Mechanics, 2006.
DOI : 10.1007/978-3-642-61237-4

R. Aron and J. Cima, A theorem on holomorphic mappings into Banach spaces with basis. Proceeding of the, 1972.

D. Bambusi, Nekhoroshev theorem for small amplitude solutions in nonlinear Schrödinger equation, Math. Z, vol.130, pp.345-387, 1999.

D. Bambusi, An Averaging Theorem for Quasilinear Hamiltonian PDEs, Annales Henri Poincar??, vol.4, issue.4, pp.685-712, 2003.
DOI : 10.1007/s00023-003-0144-6

D. Bambusi, Galerkin averaging method and Poincaré normal form for some quasilnear pdes, Ann. Scuola Norm. Sup. Pisa C1. Sci, pp.669-702, 2005.

D. Bambusi and B. Grebert, Forme normale pour NLS en dimension quelconque, Comptes Rendus Mathematique, vol.337, issue.6, pp.409-414, 2003.
DOI : 10.1016/S1631-073X(03)00368-6

D. Bambusi and B. Grébert, Birkhoff normal form for partial differential equations with tame modulus, Duke Mathematical Journal, vol.135, issue.3, pp.507-567, 2006.
DOI : 10.1215/S0012-7094-06-13534-2

E. D. Belokolos, A. I. Bobenko, V. Z. Enolskii, A. R. Its, and V. B. Matveev, Algebro-geometric approach to nonlinear integrable equations, 1994.

A. Bobenko and S. Kuksin, Finite-gap periodic solutions of the KdV equation are non-degenerate, Physics Letters A, vol.161, issue.3, pp.274-276, 1991.
DOI : 10.1016/0375-9601(91)90016-2

V. Bogachev, Differentiable measures and the Malliavin calculus, 2010.

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

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Geometric and Functional Analysis, pp.209-262, 1993.

J. Bourgain, Global solutions of nonlinear Schrödinger equations, 1999.

J. Bourgain, On diffusion in high-dimensional Hamiltonian systems and PDE, Journal d'Analyse Math??matique, vol.86, issue.1, pp.1-35, 2000.
DOI : 10.1007/BF02791532

J. Boussinesq, Théorie de l'intumescence liquid appelée onde solitaire ou de translation, se propageant dans un canal rectangulaire, Comptes Rend. Acad. Sci (Paris), vol.72, pp.755-759, 1871.

H. Brézis and T. Gallouet, Nonlinear Schrödinger evolution equations. Nonlinear Analysis : Theory, Methods & Applications, vol.4, issue.4, pp.677-681, 1980.

N. Burq and N. Tzvetkov, Random data Cauchy theory for supercritical wave equations II: a global existence result, Inventiones mathematicae, vol.3, issue.3, pp.477-496, 2008.
DOI : 10.1007/s00222-008-0123-0

URL : https://hal.archives-ouvertes.fr/hal-00449548

J. Colliander, M. Keel, G. Staffiliani, and T. Tao, Symplectic nonsqueezing of the korteweg-de vries flow, Acta Mathematica, vol.195, issue.2, pp.197-252, 2005.
DOI : 10.1007/BF02588080

B. Dubrovin, Theta functions and non-linear equations, Russian Mathematical Surveys, vol.36, issue.2, pp.11-92, 1981.
DOI : 10.1070/RM1981v036n02ABEH002596

B. Dubrovin, V. Matveev, and S. Novikov, Nonlinear equations of Kortewegde Vries type, finite zone linear operators, and Abelian varieties, Russ. Math. Surv, issue.1, pp.3155-136, 1976.

R. M. Dudley, Real Analysis and Probability, 2002.
DOI : 10.1017/CBO9780511755347

H. Eliasson, Normal forms for Hamiltonian systems with Poisson commuting integrals???elliptic case, Commentarii Mathematici Helvetici, vol.65, issue.1, pp.4-35, 1990.
DOI : 10.1007/BF02566590

E. Faou, P. Germain, and Z. Hani, The weakly nonlinear large box limit of the 2d cubic nonlinear Schrödinger equation. preprint, 2013.

H. Flaschka and D. Mclaughlin, Canonically conjugate variables for the Koreweg-de Vries equation and the Toda lattice with periodic boundary conditions . Progress of Theoretical Physics, pp.438-457, 1976.

C. Gardner, Korteweg???de Vries Equation and Generalizations. IV. The Korteweg???de Vries Equation as a Hamiltonian System, Journal of Mathematical Physics, vol.12, issue.8, pp.1548-1551, 1971.
DOI : 10.1063/1.1665772

J. Garnett and E. Trubowitz, Gaps and bands of one dimensional periodic Schr??dinger operators, Commentarii Mathematici Helvetici, vol.59, issue.1, pp.258-312, 1984.
DOI : 10.1007/BF02566350

P. Gérard and S. Grellier, Effective integrable dynamics for a certain nonlinear wave equation, Analysis & PDE, vol.5, issue.5, pp.1139-1154, 2012.
DOI : 10.2140/apde.2012.5.1139

S. Herr, D. Tataru, and N. Tzvetkov, Global well-posedness of the energy-critical nonlinear Schr??dinger equation with small initial data in $H^1(\mathbb{T}^3)$, Duke Mathematical Journal, vol.159, issue.2, pp.329-349, 2011.
DOI : 10.1215/00127094-1415889

G. Huang, An averaging theorem for nonlinear Schrödinger equations with small nonlinearities. to appear in DCDS-A

G. Huang, An averaging theorem for a perturbed KdV equation, Nonlinearity, vol.26, pp.1599-1621, 2013.

G. Huang, On long time dynamics of perturbed KdV equations. preprint, 2013.
URL : https://hal.archives-ouvertes.fr/hal-01076604

G. Huang and S. Kuksin, KdV equation under periodic boundary conditions and its perturbations. preprint, 2013.
URL : https://hal.archives-ouvertes.fr/hal-01076598

H. Ito, Convergence of Birkhoff normal forms for integrable systems, Commentarii Mathematici Helvetici, vol.64, issue.1, pp.412-461, 1989.
DOI : 10.1007/BF02564686

A. Its and V. Matveev, Schrödinger operators with the finite-band spectrum and the N -soliton solutions of the Korteweg-de Vries equation, Teoret. Mat. Fiz, vol.23, pp.51-68, 1975.

T. Kappeler and S. Kuksin, Strong nonresonance of Schr??dinger operators and an averaging theorem, Physica D: Nonlinear Phenomena, vol.86, issue.3, pp.349-362, 1995.
DOI : 10.1016/0167-2789(95)00115-K

T. Kappeler, C. Möhr, and P. Topalov, Birkhoff coordinates for KdV on phase spaces of distributions, Selecta Mathematica, vol.11, issue.1, pp.37-98, 2005.
DOI : 10.1007/s00029-005-0009-6

T. Kappeler, B. Schaad, and P. Topalov, Asymptotics of spectral quantities of Schrödinger operators, Spectral geometry, Proc. Sympos. Pure Math, pp.243-284, 2012.

T. Kappeler, B. Schaad, and P. Topalov, Qualitative Features of Periodic Solutions of KdV, Communications in Partial Differential Equations, 2013.
DOI : 10.1007/s11006-006-0204-6

T. Kappeler and P. Topalov, Global wellposedness of KdV in H ?1 (T, R). Duke Math, J, vol.135, pp.327-360, 2006.

E. Korotyaev, Estimates for the Hill operator, II, Journal of Differential Equations, vol.223, issue.2, pp.229-260, 2006.
DOI : 10.1016/j.jde.2005.04.017

I. Krichever, Perturbation theory in periodic problems for two-dimensional integrable systems, Sov. Sci. Rev. C. Math. Phys, vol.9, pp.1-101, 1991.

M. Kruskal and N. Zabusky, Interaction of " solitons" in a collision-less plasma and the recurrence of initial states, Phys. Rev. Lett, vol.15, pp.240-243, 1965.

S. Kuksin, Perturbation theory for quasi-periodic solutions of infinite dimensional Hamiltonian systems, and its application to the Koreteweg-de Vries equation, Math. USSR Subornik, vol.67, pp.397-413, 1989.

S. Kuksin, Analysis of Hamiltonian PDEs, 2000.

S. Kuksin, Hamiltonian PDEs In Handbook of dynamical systems, pp.1087-1133, 2006.

S. Kuksin, Damped-Driven KdV and Effective Equations for Long-Time Behaviour of its Solutions, Geometric and Functional Analysis, vol.10, issue.6, pp.1431-1463, 2010.
DOI : 10.1007/s00039-010-0103-6

S. Kuksin, Weakly nonlinear stochastic CGL equations Annales de l'insitut Henri Poincaré-Probabilité et Statistiques, pp.1033-1056, 2013.

S. Kuksin and A. Maiocchi, Resonant averaging for weakly nonlinear stochastic Schrödinger equations. preprint, 2013.

S. Kuksin and G. Perelman, Vey theorem in infinite dimensions and its application to KdV, DCDS-A, vol.27, pp.2010-2011
URL : https://hal.archives-ouvertes.fr/hal-00832747

S. Kuksin and A. Piatnitski, Khasminskii???Whitham averaging for randomly perturbed KdV equation, Journal de Math??matiques Pures et Appliqu??es, vol.89, issue.4, pp.400-428, 2008.
DOI : 10.1016/j.matpur.2007.12.003

URL : https://hal.archives-ouvertes.fr/hal-00832698

S. Kuksin and J. Poschel, Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation, Annals of Mathematics, pp.149-179, 1996.

P. Lax, Integrals of nonlinear equations of evolution and solitary waves, Communications on Pure and Applied Mathematics, vol.15, issue.5, pp.467-490, 1968.
DOI : 10.1002/cpa.3160210503

P. Lax, Outline of a theory of the KdV equation, Lecture Notes in Mathematics, vol.30, pp.70-102, 1996.
DOI : 10.1007/BF01425567

P. Lochak and C. Meunier, Multiphase averaging for classical systems : with applications to adiabatic theorems, 1988.
DOI : 10.1007/978-1-4612-1044-3

V. Marcênko, Sturm-Liouville operators and applications, Birkhäuser, 1986.

H. Mckean and E. Trubowitz, Hill's operator and hyperelliptic function theory in the presence of infinitely many branch points, Communications on Pure and Applied Mathematics, vol.77, issue.2, pp.143-226, 1976.
DOI : 10.1002/cpa.3160290203

H. Mckean and E. Trubowitz, Hill's surfaces and their theta functions, Bulletin of the American Mathematical Society, vol.84, issue.6, pp.1042-1085, 1978.
DOI : 10.1090/S0002-9904-1978-14542-X

R. Miura, Korteweg???de Vries Equation and Generalizations. I. A Remarkable Explicit Nonlinear Transformation, Journal of Mathematical Physics, vol.9, issue.8, pp.1202-1204, 1968.
DOI : 10.1063/1.1664700

J. Moser, Lectures on Hamiltonian systems, Mem. Amer. Math. Soc, vol.81, pp.1-60, 1968.

J. Moser and C. L. Siegel, Lectures on celestial mechanics, 1971.

R. Muira, C. Gardner, and M. Kruskal, Korteweg???de Vries Equation and Generalizations. II. Existence of Conservation Laws and Constants of Motion, Journal of Mathematical Physics, vol.9, issue.8, pp.1204-1209, 1968.
DOI : 10.1063/1.1664701

S. Nazarenko, Wave turbulence, Contemporary Physics, vol.40, issue.3, 2011.
DOI : 10.1016/S0167-2789(03)00214-8

URL : https://hal.archives-ouvertes.fr/cea-01366996

A. I. Neishtadt, Averaging in multi-frequency systems, I and II, Soviet Phys, Doklady, vol.20, issue.212, pp.492-494, 1975.

L. Nirenberg, On Elliptic Partial Differential Equations, Ann. Sci. Norm. Sup. Pisa, vol.13, pp.115-162, 1959.
DOI : 10.1007/978-3-642-10926-3_1

T. Runst and W. Sickel, Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations, 1996.
DOI : 10.1515/9783110812411

J. Saut and R. Teman, Remarks on the Korteweg-de Vries equation, Israel Journal of Mathematics, vol.14, issue.1, pp.78-87, 1976.
DOI : 10.1007/BF02761431

N. Temirgaliev, A connection between inclusion theorems and the uniform convergence of multiple Fourier series, Mathematical Notes of the Academy of Sciences of the USSR, vol.5, issue.No. 2, pp.139-148, 1972.
DOI : 10.1007/BF01095009

J. Vey, Sur Certains Systemes Dynamiques Separables, American Journal of Mathematics, vol.100, issue.3, pp.591-614, 1978.
DOI : 10.2307/2373841

H. Whitney, Differentiable even functions, Duke Mathematical Journal, vol.10, issue.1, pp.159-160, 1942.
DOI : 10.1215/S0012-7094-43-01015-4

URL : http://projecteuclid.org/download/pdf_1/euclid.dmj/1077471799

V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskij, Theory of Solitons : The Inverse Scattering Method, 1984.

P. E. Zhidkov, Korteweg-de Vries and Nonlinear Schrödinger equations : qualitative theory, Lecture Notes in Mathematics, vol.1756, 2001.