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We present the recent result in  concerning strong nonlinear instability and growth of Sobolev norms near quasiperiodic finite-gap solutions of the defocusing cubic nonlinear Schrödinger 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 construct solutions of the 2D cubic NLS that start arbitrarily close to such invariant tori in the topology (0 < s < 1) and whose norm can grow by any given factor.
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Marcel Guardia, Zaher Hani, Emanuele Haus, Alberto Maspero, Michela Procesi, A note on growth of Sobolev norms near quasiperiodic finite-gap tori for the 2D cubic NLS equation. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 30 (2019), no. 4, pp. 865–880DOI 10.4171/RLM/873