Resonant dynamics for the quintic nonlinear Schrödinger equation
Grébert, Benoît ; Thomann, Laurent
Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012), p. 455-477 / Harvested from Numdam
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Nous considérons lʼéquation de Schrödinger non linéaire (NLS) quintique sur le cercle i∂ t u+∂ x 2 u=±ν|u| 4 u,ν≪1,x∈𝕊 1 ,t∈ℝ. Nous montrons quʼil existe des solutions issues dʼune condition initiale construite sur quatre modes de Fourier formant un ensemble résonant (voir définition 1.1) ont une dynamique non triviale mettant en jeu des échanges périodiques dʼénergie entre ces quatre modes initialement excités. Il est remarquable que ce phénomène non linéaire soit indépendant du choix de lʼensemble résonant.Le résultat dynamique est obtenu en mettant dʼabord sous forme normale résonante jusquʼà lʼordre 10 lʼHamiltonien de NLS quintique puis en isolant un terme effectif dʼordre 6. Il est à noter que ce phénomène ne peut pas se produire pour NLS cubique pour lequel les amplitudes des modes de Fourier sont des presque-actions et donc ne varient quasiment pas au cours du temps.

We consider the quintic nonlinear Schrödinger equation (NLS) on the circle i∂ t u+∂ x 2 u=±ν|u| 4 u,ν≪1,x∈𝕊 1 ,t∈ℝ. We prove that there exist solutions corresponding to an initial datum built on four Fourier modes which form a resonant set (see Definition 1.1), which have a nontrivial dynamic that involves periodic energy exchanges between the modes initially excited. It is noticeable that this nonlinear phenomenon does not depend on the choice of the resonant set.The dynamical result is obtained by calculating a resonant normal form up to order 10 of the Hamiltonian of the quintic NLS and then by isolating an effective term of order 6. Notice that this phenomenon cannot occur in the cubic NLS case for which the amplitudes of the Fourier modes are almost actions, i.e. they are almost constant.

Publié le : 2012-01-01
DOI : https://doi.org/10.1016/j.anihpc.2012.01.005
Classification:  37K45,  35Q55,  35B34,  35B35
Mots clés: Forme normale, Equation de Schrödinger non linéaire, Résonances, Échange dʼénergie 
@article{AIHPC_2012__29_3_455_0,
     author = {Gr\'ebert, Beno\^\i t and Thomann, Laurent},
     title = {Resonant dynamics for the quintic nonlinear Schr\"odinger equation},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {29},
     year = {2012},
     pages = {455-477},
     doi = {10.1016/j.anihpc.2012.01.005},
     mrnumber = {2926244},
     zbl = {1259.37045},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2012__29_3_455_0}
}

Grébert, Benoît; Thomann, Laurent. Resonant dynamics for the quintic nonlinear Schrödinger equation. Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) pp. 455-477. doi : 10.1016/j.anihpc.2012.01.005. http://gdmltest.u-ga.fr/item/AIHPC_2012__29_3_455_0/

[1] V. Arnold, Mathematical Methods of Classical Mechanics, Grad. Texts in Math. vol. 60, Springer-Verlag, New York (1989) | MR 997295

[2] D. Bambusi, B. Grébert, Birkhoff normal form for PDEs with tame modulus, Duke Math. J. 135 (2006), 507-567 | MR 2272975 | Zbl 1110.37057

[3] J. Bourgain, On invariant tori of full dimension for 1D periodic NLS, J. Funct. Anal. 229 no. 1 (2005), 62-94 | MR 2180074 | Zbl 1088.35061

[4] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation, Invent. Math. 181 no. 1 (2010), 39-113 | MR 2651381 | Zbl 1197.35265

[5] E. Faou, B. Grébert, A Nekhoroshev type theorem for the nonlinear Schrödinger equation on the torus, arXiv:1003.4845 | MR 3148054 | Zbl 1284.35067

[6] B. Grébert, Birkhoff normal form and Hamiltonian PDEs, Partial Differential Equations and Applications, Sémin. Congr. vol. 15, Soc. Math. France, Paris (2007), 1-46 | MR 2352816 | Zbl 1157.37019

[7] B. Grébert, T. Kappeler, Perturbations of the defocusing nonlinear Schrödinger equation, Milan J. Math. 71 (2003), 141-174 | MR 2120919 | Zbl 1048.37067

[8] B. Grébert, T. Kappeler, J. Pöschel, The defocusing NLS equation and its normal form, arXiv:0907.3938 | MR 3203027 | Zbl 1298.35002

[9] B. Grébert, C. Villegas-Blas, On the energy exchange between resonant modes in nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 28 no. 1 (2011), 127-134 | Numdam | MR 2765514 | Zbl 1216.35137

[10] T. Kappeler, P. Lohrmann, P. Topalov, N.T. Zung, Birkhoff coordinates for the focusing NLS equation, Comm. Math. Phys. 285 no. 3 (2009), 1087-1107 | MR 2470917 | Zbl 1176.35168

[11] S.B. Kuksin, J. Pöschel, Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation, Ann. of Math. 143 (1996), 149-179 | MR 1370761 | Zbl 0847.35130

[12] Z. Liang, J. You, Quasi-periodic solutions for 1D Schrödinger equations with higher order nonlinearity, SIAM J. Math. Anal. 36 no. 6 (2005), 1965-1990 | MR 2178228 | Zbl 1093.37030