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