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

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