On the energy exchange between resonant modes in nonlinear Schrödinger equations

Benoît Grébert (1) , Carlos Villegas-Blas (2)

1 Equations aux dérivées partielles
2 Insituto de Matematicas, Unidad Cuernavaca

We consider the nonlinear Schrödinger equation $$ i\psi_t= -\psi_{xx}\pm 2\cos 2x \ |\psi|^2\psi,\quad x\in S^1,\ t\in \R$$ and we prove that the solution of this equation, with small initial datum $\psi_0=\e (\cos x+\sin x)$, will periodically exchange energy between the Fourier modes $e^{ix}$ and $e^{-ix}$. This beating effect is described up to time of order $\e^{-9/4}$ while the frequency is of order $\e^2$. We also discuss some generalizations.

Mots clés: Beating effect; Resonances; Nonlinear Schrödinger equation; Normal form

Domaines: Systèmes dynamiques [math.DS]; Equations aux dérivées partielles [math.AP]

Benoît Grébert, Carlos Villegas-Blas. On the energy exchange between resonant modes in nonlinear Schrödinger equations. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2011, 28, pp.127-134. ⟨10.1016/j.anihpc.2010.11.004⟩. ⟨hal-00505201⟩