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

We consider the nonlinear Schr"odinger equation $$i{\psi }_t=-{\psi }_{xx}\pm 2\cos 2x\mathrm{\mid }\psi {\mathrm{\mid }}^2\psi ,x\in S^1,t\in \mathbb{R}$$ and we prove that the solution of this equation, with small initial datum $\psi_0=\e (\cos x+\sin x)$,willperiodicallyexchangeenergybetweentheFouriermodes$e^{ix}$and$e^{-ix}$\mathrm{.}Thisbeatingeffectisdescribeduptotimeoforder$\e^{-9/4}$whilethefrequencyisoforder$\e^2$. We also discuss some generalizations.