Resonant dynamics for the quintic non linear Schr\"odinger equation

TL;DR

This paper demonstrates the existence of resonant solutions with periodic energy exchanges in the quintic nonlinear Schr"odinger equation on the circle, using a resonant normal form analysis up to order 10.

Contribution

It introduces a novel resonant normal form approach for the quintic NLS, revealing non-trivial dynamics independent of the specific resonant set.

Findings

Existence of solutions with periodic energy exchange among Fourier modes

Resonant phenomena do not occur in the cubic NLS case

Normal form analysis up to order 10 isolates effective resonant interactions

Abstract

We consider the quintic nonlinear Schr\"odinger equation (NLS) on the circle. We prove that there exist solutions corresponding to an initial datum built on four Fourier modes which form a resonant set, which have a non trivial dynamic that involves periodic energy exchanges between the modes initially excited. It is noticeable that this nonlinear phenomena does not depend on the choice of the resonant set. The dynamical result is obtained by calculating a resonant normal form up to order 10 of the Hamiltonian of the quintic NLS and then by isolating an effective term of order 6. Notice that this phenomena can not occur in the cubic NLS case for which the amplitudes of the Fourier modes are almost actions, i.e. they are almost constant.