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We consider the defocusing cubic nonlinear Schr\"odinger equation (NLS) onthe two-dimensional torus. The equation admits a special family of ellipticinvariant quasiperiodic tori called finite-gap solutions. These are inheritedfrom the integrable 1D model (cubic NLS on the circle) by considering solutionsthat depend only on one variable. We study the long-time stability of suchinvariant tori for the 2D NLS model and show that, under certain assumptionsand over sufficiently long timescales, they exhibit a strong form of transverseinstability in Sobolev spaces $H^s(\mathbb{T}^2)$ ($0<s<1$). More precisely, weconstruct solutions of the 2D cubic NLS that start arbitrarily close to suchinvariant tori in the $H^s$ topology and whose $H^s$ norm can grow by any givenfactor. This work is partly motivated by the problem of infinite energy cascadefor 2D NLS, and seems to be the first instance where (unstable) long-timenonlinear dynamics near (linearly stable) quasiperiodic tori is studied andconstructed.

Strong nonlinear instability and growth of Sobolev norms near quasiperiodic finite-gap tori for the 2D cubic NLS equation