Reducibility and nonlinear stability for a quasi-periodically forced NLS. (English) Zbl 1541.35323
Summary: Motivated by the problem of long time stability vs. instability of KAM tori of the Nonlinear cubic Schrödinger equation (NLS) on the two dimensional torus \(\mathbb{T}^2:=(\mathbb{R}/2\pi\mathbb{Z})^2\), we consider a quasi-periodically forced NLS equation on \(\mathbb{T}^2\) arising from the linearization of the NLS at a KAM torus. We prove a reducibility result as well as long time stability of the origin. The main novelty is to obtain the precise asymptotic expansion of the frequencies which allows us to impose Melnikov conditions at arbitrary order.
MSC:
35P15 | Estimates of eigenvalues in context of PDEs |
35J10 | Schrödinger operator, Schrödinger equation |
35B35 | Stability in context of PDEs |
35Q55 | NLS equations (nonlinear Schrödinger equations) |
37K55 | Perturbations, KAM theory for infinite-dimensional Hamiltonian and Lagrangian systems |