Reducibility and nonlinear stability for a quasi-periodically forced NLS

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 ${\mathbb{T}}^2:=(\mathbb{R}\mathrm{/}2\pi \mathbb{Z}{)}^2$, 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.