Resonant dynamics for the quintic nonlinear Schrodinger equation

We consider the quintic nonlinear Schrodinger equation (NLS) on the circle i partial derivative(t)tu + partial derivative(2)(x)u = +/- nu vertical bar u vertical bar(4)u, nu << I. x is an element of S-1, 1 is an element of R. We prove that there exist solutions corresponding to an initial datum built on four Fourier modes which form a resonant set (see Definition 1.1), which have a nontrivial dynamic that involves periodic energy exchanges between the modes initially excited. It is noticeable that this nonlinear phenomenon 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 phenomenon cannot occur in the cubic NLS case for which the amplitudes of the Fourier modes are almost actions, i.e. they are almost constant. (C) 2012 Elsevier Masson SAS. All rights reserved.