On the energy exchange between resonant modes in nonlinear Schrödinger equations

Grébert, Benoît; Villegas-Blas, Carlos

doi:10.1016/j.anihpc.2010.11.004

On the energy exchange between resonant modes in nonlinear Schrödinger equations
[Echange d'énergie entre modes résonants dans une équation de Schrödinger non linéaire cubique]

Grébert, Benoît ; Villegas-Blas, Carlos

Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 1, pp. 127-134

Abstract (VO)
Résumé

We consider the nonlinear Schrödinger equation

i ψ_{t} = - ψ_{x x} \pm 2 \cos 2 x {| ψ |}^{2} ψ, x \in S^{1}, t \in ℝ

and we prove that the solution of this equation, with small initial datum

ψ (0, x) = ϵ (A \exp (i x) + B \exp (- i x))

, will periodically exchange energy between the Fourier modes

e^{i x}

and

e^{- i x}

as soon as

A^{2} \neq B^{2}

. This beating effect is described up to time of order

ϵ^{- 5 / 2}

while the frequency is of order

ϵ^{2}

. We also discuss some generalizations.

Nous considérons l'équation de Schödinger non linéaire

i ψ_{t} = - ψ_{x x} \pm 2 \cos 2 x {| ψ |}^{2} ψ, x \in S^{1}, t \in ℝ

et nous montrons la solution de cette équation ayant pour donnée initiale

ψ (0, x) = ϵ (A \exp (i x) + B \exp (- i x))

avec ε petit, va échanger périodiquement de l'énergie entre les modes de Fourier

e^{i x}

et

e^{- i x}

dès que

A^{2} \neq B^{2}

. Cet effet de battement, dont la période est de l'ordre de

ϵ^{- 2}

, est mis en évidence pour des temps de l'ordre de

ϵ^{- 5 / 2}

. Nous présentons aussi quelques généralisations.

MR Zbl | 3 citations dans Numdam

DOI : 10.1016/j.anihpc.2010.11.004

Classification : 37K45, 35Q55, 35B34, 35B35
Keywords: Normal form, Nonlinear Schrödinger equation, Resonances, Beating effect
Mots-clés : Forme normale, Equation de Schrödinger non linéaire, Résonances, Échange d'énergie

@article{AIHPC_2011__28_1_127_0,
     author = {Gr\'ebert, Beno{\^\i}t and Villegas-Blas, Carlos},
     title = {On the energy exchange between resonant modes in nonlinear {Schr\"odinger} equations},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {127--134},
     year = {2011},
     publisher = {Elsevier},
     volume = {28},
     number = {1},
     doi = {10.1016/j.anihpc.2010.11.004},
     mrnumber = {2765514},
     zbl = {1216.35137},
     language = {en},
     url = {https://www.numdam.org/articles/10.1016/j.anihpc.2010.11.004/}
}

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Grébert, Benoît; Villegas-Blas, Carlos. On the energy exchange between resonant modes in nonlinear Schrödinger equations. Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 1, pp. 127-134. doi: 10.1016/j.anihpc.2010.11.004

Bibliographie
Cité par

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

[2] J. Bourgain, Global Solutions of Nonlinear Schrödinger Equations, Amer. Math. Soc. Colloq. Publ. vol. 46, Amer. Math. Soc., Providence, RI (1999) | Zbl

[3] 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 | Zbl

[4] J. Moser, Lectures on Hamiltonian systems, Mem. Amer. Math. Soc. 81 (1968), 1-60 | MR | Zbl

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