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Motivated by the problem of long time stability vs. instability of KAM toriof the Nonlinear cubic Schr\"odinger equation (NLS) on the two dimensionaltorus $\mathbb T^2:= (\mathbb R/2\pi \mathbb Z)^2$, we consider aquasi-periodically forced NLS equation on $\mathbb T^2$ arising from thelinearization of the NLS at a KAM torus. We prove a reducibility result as wellas long time stability of the origin. The main novelty is to obtain the preciseasymptotic expansion of the frequencies which allows us to impose Melnikovconditions at arbitrary order.

Reducibility and nonlinear stability for a quasi-periodically forced NLS