On the energy exchange between resonant modes in nonlinear Schrödinger equations

Abstract

We consider the nonlinear Schrödinger equation $$i{\psi }_t=-{\psi }_{xx}\pm 2\cos 2x|\psi {|}^2\psi ,x\in S^1,t\in \mathbb{R}$$ and we prove that the solution of this equation, with small initial datum $ψ_0=\e (\cos x+\sin x)$, 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.