On the energy exchange between resonant modes in nonlinear Schrödinger equations
Annales de l'I.H.P. Analyse non linéaire, Volume 28 (2011) no. 1, p. 127-134

We consider the nonlinear Schrödinger equation iψ t =-ψ xx ±2 cos 2x|ψ| 2 ψ,xS 1 ,t and we prove that the solution of this equation, with small initial datum ψ(0,x)=ϵ(A exp (ix)+B exp (-ix)), will periodically exchange energy between the Fourier modes e ix and e -ix as soon as A 2 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 iψ t =-ψ xx ±2 cos 2x|ψ| 2 ψ,xS 1 ,t et nous montrons la solution de cette équation ayant pour donnée initiale ψ(0,x)=ϵ(A exp (ix)+B exp (-ix)) avec ε petit, va échanger périodiquement de l'énergie entre les modes de Fourier e ix et e -ix dès que A 2 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.

DOI : https://doi.org/10.1016/j.anihpc.2010.11.004
Classification:  37K45,  35Q55,  35B34,  35B35
Keywords: Normal form, Nonlinear Schrödinger equation, Resonances, Beating effect
@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},
     publisher = {Elsevier},
     volume = {28},
     number = {1},
     year = {2011},
     pages = {127-134},
     doi = {10.1016/j.anihpc.2010.11.004},
     zbl = {1216.35137},
     mrnumber = {2765514},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2011__28_1_127_0}
}
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, Volume 28 (2011) no. 1, pp. 127-134. doi : 10.1016/j.anihpc.2010.11.004. http://www.numdam.org/item/AIHPC_2011__28_1_127_0/

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