On the energy exchange between resonant modes in nonlinear Schr\"odinger equations

TL;DR

This paper analyzes energy exchange between resonant modes in a nonlinear Schrödinger equation on a circle, demonstrating a periodic beating effect between specific Fourier modes for small initial data over a long time scale.

Contribution

It proves the existence of a long-time periodic energy exchange between Fourier modes in a specific nonlinear Schrödinger equation with small initial data.

Findings

Energy periodically exchanges between modes e^{ix} and e^{-ix}.

Beating effect persists up to time of order ^{-9/4}.

Frequency of energy exchange is of order ^2.

Abstract

We consider the nonlinear Schr\"odinger 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.