On the energy exchange between resonant modes in nonlinear Schrödinger equations

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..

The proof uses the classical Birkhoff normal form procedure (see for instance [4] in a finite dimensional setting or [3] in the infinite dimensional one).. Since the free

Gallo, The Cauchy problem for defocusing nonlinear Schrödinger equations with non-vanishing initial data at infinity, preprint..

We also would like to highlight the fact that global solutions of the type men- tioned above, i.e., infinite time blow-up and flattening, have been constructed in the case of the

Key Words: damped Schr¨ odinger equation, existence, uniqueness, finite time extinction, asymptotic behavior... The literature is too extensive to give an

We also use this abstract KAM theorem to prove the existence of small amplitude quasi-periodic solutions for the nonlinear wave equation on the circle without parameter (see [4]).

Gallo, The Cauchy Problem for defocusing Nonlinear Schr¨odinger equations with non- vanishing initial data at infinity, preprint..

We demonstrate how to set up a scattering framework for equations of this type, including appropriate decay estimates of the free time evolution and the construction of wave