C. J. Amick and J. F. Toland, Uniqueness and related analytic properties for the Benjamin-Ono equation-a nonlinear Neumann problem in the plane, Acta Math, vol.167, issue.12, pp.107-126, 1991.

C. J. Amick and J. F. Toland, Uniqueness of Benjamin's solitary-wave solution of the Benjamin-Ono equation, The Brooke Benjamin special issue, vol.46, issue.1-2, pp.21-28, 1989.

V. I. Arnold, Mathematical methods of classical mechanics, Graduate Texts in Mathematics, p.60, 1978.

P. Bizo?, B. Craps, O. Evnin, D. Hunik, V. Luyten et al., Conformal flow on S 3 and weak field integrability in AdS 4, Comm. Math. Phys, vol.353, issue.3, pp.1179-1199, 2017.

P. Bizo?, D. Hunik-kostyra, and D. Pelinovsky, Ground state of the conformal flow on S 3, 2017.

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Geometric and Functional Analysis, vol.3, issue.3, pp.209-262, 1993.
DOI : 10.1007/bf01896020

J. Bourgain, On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE, Internat. Math. Res. Notices, vol.72, issue.6, pp.277-304, 1996.

J. Bourgain, Problems in Hamiltonian PDE's, Geom. Funct. Anal., Special, pp.32-56, 1999.

H. Brezis and T. Gallouët, Nonlinear Schrödinger evolution equations, Nonlinear Anal, vol.4, issue.4, pp.677-681, 1980.

N. Burq, P. Gérard, and N. Tzvetkov, Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces, Invent. Math, vol.159, issue.1, pp.187-223, 2005.
DOI : 10.1007/s00222-004-0388-x

URL : http://arxiv.org/pdf/math/0308214

N. Burq, P. Gérard, and N. Tzvetkov, Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear schrödinger equations, Annales scientifiques de l'Ecole normale supérieure, vol.38, pp.255-301, 2005.

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, vol.181, issue.1, pp.39-113, 2010.

S. Demirbas, M. B. Erdo?an, and N. Tzirakis, Existence and uniqueness theory for the fractional Schrödinger equation on the torus, Some topics in harmonic analysis and applications, vol.34, pp.145-162, 2016.

B. Dodson, Global well-posedness and scattering for the defocusing, L 2-critical, nonlinear Schrödinger equation when d = 2, Duke Math. J, vol.165, issue.18, pp.3435-3516, 2016.
DOI : 10.1353/ajm.2016.0016

URL : http://arxiv.org/pdf/1006.1375

P. Gérard, P. Germain, and L. Thomann, On the cubic lowest Landau level equation, 2017.

P. Gérard and S. Grellier, The cubic Szeg? equation, Ann. Sci. Éc. Norm. Supér, vol.43, issue.4, pp.761-810, 2010.

P. Gérard and S. Grellier, Effective integrable dynamics for a certain nonlinear wave equation, Anal. PDE, vol.5, issue.5, pp.1139-1155, 2012.

P. Gérard and S. Grellier, Invariant tori for the cubic Szeg? equation, Invent. Math, vol.187, issue.3, pp.707-754, 2012.

P. Gérard and S. Grellier, On the growth of Sobolev norms for the cubic Szeg? equation. Séminaire Laurent Schwartz-EDP et applications, vol.20, p.p, 2014.

P. Gérard and S. Grellier, An explicit formula for the cubic Szeg? equation, Transactions of the American Mathematical Society, vol.367, issue.4, pp.2979-2995, 2015.

P. Gérard and S. Grellier, The cubic Szeg? equation and Hankel operators, Astérisque, issue.389, p.112, 2017.

P. Gérard and H. Koch, The cubic Szeg? flow at low regularity, Séminaire Laurent Schwartz-Équations aux dérivées partielles et applications. Année, 2016.

P. Gérard and E. Lenzmann, A Lax pair structure for the half-wave maps equation, Letters in Mathematical Physics, pp.1-14, 2018.

P. Gérard, E. Lenzmann, O. Pocovnicu, and P. Raphaël, A Two-Soliton with Transient Turbulent Regime for the Cubic Half-Wave Equation on the Real Line, Ann. PDE, vol.4, issue.1, 2018.

J. Ginibre, Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d'espace (d'après Bourgain), Exp. No. 796, vol.4, p.95, 1994.

G. Gottwald, R. Grimshaw, and B. A. Malomed, Stable two-dimensional parametric solitons in fluid systems, Physics Letters A, vol.248, issue.2, pp.208-218, 1998.
DOI : 10.1016/s0375-9601(98)00658-6

B. Grébert and T. Kappeler, The defocusing NLS equation and its normal form, EMS Series of Lectures in Mathematics, 2014.

B. Grébert and L. Thomann, Resonant dynamics for the quintic nonlinear Schrödinger equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, vol.29, issue.3, pp.455-477, 2012.

M. Guardia and V. Kaloshin, Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation, J. Eur. Math. Soc. (JEMS), vol.17, issue.1, pp.71-149, 2015.

M. Guardia and V. Kaloshin, Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation

J. Eur, Math. Soc. (JEMS), vol.19, issue.2, pp.601-602, 2017.

Z. Hani, Long-time instability and unbounded Sobolev orbits for some periodic nonlinear Schrödinger equations, Arch. Ration. Mech. Anal, vol.211, issue.3, pp.929-964, 2014.

Z. Hani, B. Pausader, N. Tzvetkov, and N. Visciglia, Modified scattering for the cubic Schrödinger equation on product spaces and applications, Forum Math. Pi, vol.3, p.4, 2015.

E. Haus and M. Procesi, Growth of Sobolev norms for the quintic NLS on T 2, Anal. PDE, vol.8, issue.4, pp.883-922, 2015.

E. Haus and M. Procesi, KAM for beating solutions of the quintic NLS, Comm. Math. Phys, vol.354, issue.3, pp.1101-1132, 2017.

N. Hayashi, T. Ozawa, and K. Tanaka, On a system of nonlinear Schrödinger equations with quadratic interaction. Annales de l'Institut Henri Poincaré / Analyse non linéaire, vol.30, pp.661-690, 2013.

R. Jost, Winkel-und Wirkungsvariable für allgemeine mechanische Systeme, Helvetica Physica Acta, vol.41, issue.6-7, pp.965-968, 1968.

T. Kappeler, J. Pöschel, &. Kdv, and . Kam, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas, vol.45

. Springer-verlag, , 2003.

T. Kappeler, B. Schaad, and P. Topalov, Scattering-like phenomena of the periodic defocusing NLS equation, Math. Res. Lett, vol.24, issue.3, pp.803-826, 2017.

C. E. Kenig and F. Merle, Scattering for ? H 1/2 bounded solutions to the cubic, defocusing NLS in 3 dimensions, Trans. Amer. Math. Soc, vol.362, issue.4, pp.1937-1962, 2010.

C. E. Kenig, G. Ponce, and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math, vol.46, issue.4, pp.527-620, 1993.

. S. Yu and . Kivshar, Bright and dark spatial solitons in non-kerr media, Optical and Quantum Electronics, vol.30, issue.7, pp.571-614, 1998.

H. Koch and D. Tataru, Conserved energies for the cubic NLS in 1, 2016.

P. D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math, vol.21, pp.467-490, 1968.

E. Lenzmann and A. Schikorra, On energy-critical half-wave maps into S 2. Inventiones mathematicae, 2018.

H. Lindblad and T. Tao, Asymptotic decay for a one-dimensional nonlinear wave equation, Anal. PDE, vol.5, issue.2, pp.411-422, 2012.

J. Liouville, Note sur l'intégration des équations différentielles de la dynamique, Journal de Mathématiques pures et appliquées, vol.20, pp.137-138, 1855.

A. J. Majda, D. W. Mclaughlin, and E. G. Tabak, A one-dimensional model for dispersive wave turbulence, J. Nonlinear Sci, vol.7, issue.1, pp.9-44, 1997.

H. Mineur, Réduction des systèmes mécaniques à n degrés de liberté admettant n intégrales premières uniformes en involution aux systèmes à variables séparées, J. Math. Pures Appl, vol.15, issue.9, pp.385-389, 1936.

Z. Nehari, On bounded bilinear forms, Annals of Mathematics, vol.65, pp.153-162, 1957.

T. Ozawa and N. Visciglia, An improvement on the Brezis-Gallouët technique for 2D NLS and 1D half-wave equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, vol.33, issue.4, pp.1069-1079, 2016.

V. Peller, Hankel operators and their applications, 2012.

F. Planchon, N. Tzvetkov, and N. Visciglia, On the growth of Sobolev norms for NLS on 2-and 3-dimensional manifolds, Anal. PDE, vol.10, issue.5, pp.1123-1147, 2017.

O. Pocovnicu, Explicit formula for the solution of the Szeg? equation on the real line and applications, Discrete Contin. Dyn. Syst, vol.31, issue.3, pp.607-649, 2011.

O. Pocovnicu, Traveling waves for the cubic Szeg? equation on the real line, Anal. PDE, vol.4, issue.3, pp.379-404, 2011.

O. Pocovnicu, Soliton interaction with small Toeplitz potentials for the Szegö equation on R, Dyn. Partial Differ. Equ, vol.9, issue.1, pp.1-27, 2012.

W. Rudin, Real and complex analysis, 1987.

E. Ryckman and M. Visan, Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in R 1+4, Amer. J. Math, vol.129, issue.1, pp.1-60, 2007.

G. Staffilani, On the growth of high Sobolev norms of solutions for KdV and Schrödinger equations, Duke Math. J, vol.86, issue.1, pp.109-142, 1997.

E. M. Stein, Harmonic analysis : real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol.43, 1993.

T. Tao, Nonlinear dispersive equations, CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, vol.106, 2006.

J. Thirouin, On the growth of Sobolev norms of solutions of the fractional defocusing NLS equation on the circle, Ann. Inst. H. Poincaré Anal. Non Linéaire, vol.34, issue.2, pp.509-531, 2017.

J. Thirouin, Optimal bounds for the growth of Sobolev norms of solutions of a quadratic Szeg? equation, 2017.

J. Thirouin, About the quadratic Szeg? hierarchy, 2018.

J. Thirouin, Classification of traveling waves for a quadratic Szeg? equation, 2018.

M. Tsutsumi, On smooth solutions to the initial-boundary value problem for the nonlinear Schrödinger equation in two space dimensions, Nonlinear Analysis : Theory, Methods & Applications, vol.13, pp.1051-1056, 1989.

Y. Wu, Simplicity and finiteness of discrete spectrum of the Benjamin-Ono scattering operator, SIAM J. Math. Anal, vol.48, issue.2, pp.1348-1367, 2016.

H. Xu, Large time blowup for a perturbation of the cubic Szeg? equation, Analysis & PDE, vol.7, issue.3, pp.717-731, 2014.

H. Xu, The cubic Szeg? equation with a linear perturbation, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01182835

H. Xu, Unbounded Sobolev trajectories and modified scattering theory for a wave guide nonlinear Schrödinger equation, Math. Z, vol.286, issue.1-2, pp.443-489, 2017.

V. E. Zakharov, P. Guyenne, A. N. Pushkarev, and F. Dias, Wave turbulence in onedimensional models. Phys. D, Advances in nonlinear mathematics and science, vol.152, pp.573-619, 2001.

V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, ?. Èksper. Teoret. Fiz, vol.61, issue.1, pp.118-134, 1971.

T. Zhou and M. Stone, Solitons in a continuous classical Haldane-Shastry spin chain, Phys. Lett. A, vol.379, pp.2817-2825, 2015.