A KAM algorithm for the resonant non-linear Schrödinger equation

Abstract

Abstract. We prove, by applying a KAM algorithm, existence of large families of stable and unstable quasi periodic solutions for the NLS in any number of independent frequencies. The main tools are the existence of a non-degenerate integrable normal form proved in [18] and [20] and a suitable generalization of the quasi-Töplitz functions introduced in [24]

KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation

“…We now show that also the Sobolev norm of the solution h(t) in (5.39) does not grow in time. For each t ∈ R, A(ωt) and W (ωt) are transformations of the phase space H s x that depend quasi-periodically on time, and satisfy, by (3.69), (3.71), (4.9), 40) where the constant C(s) depends on u s+σ+β+s0 < +∞. Moreover, the transformation B is a quasi-periodic reparametrization of the time variable (see (2.25)), namely…”

Quasi-periodic solutions for fully nonlinear forced reversible Schrödinger equations

“…Indeed there are very few results on reducibility on tori. We mention Geng-You in [25] for the smoothing NLS, Eliasson-Kuksin in [22] for the NLS and Procesi-Procesi [37,38] for the resonant NLS. All the aforementioned papers, both using KAM or multiscale, are naturally on semi linear Pde's with no derivatives in the non linearity.…”

Quasi-periodic solutions of Schrodinger equations with quasi-periodic forcing in higher dimensional spaces

“…For the completely resonant cubic Schrödinger equation on a torus T d , the existence of quasi-periodic solutions were proved by Procesi and Procesi [23]. Later Geng-You [17] proved an infinite dimensional KAM theory.…”

“…In view of (2.20) and (2.22) we have 23) and from (2.19), we have for any fixed j ∈ Z d ,j ∈ Z d \ {0}, the limits limt →∞ f j+jt,ν and limt →∞ ∂ ω f j+jt,ν exist and…”