Article overview

On the energy exchange between resonant modes in nonlinear Schr"odinger equations
Benoit Grebert ; Carlos Villegas-Blas ;
Date:	20 Aug 2010
Abstract:	We consider the nonlinear Schr"odinger equation $$ ipsi_t= -psi_{xx}pm 2cos 2x \|psi\|^2psi,quad xin S^1, tin R$$ 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.
Source:	arXiv, 1008.3527
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