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Strong Sobolev instability of quasi-periodic solutions of the 2D cubic Schrà ¶dinger equation

Banff International Research Station for Mathematical Innovation and Discovery

We consider the defocusing cubic nonlinear Schrà ¶dinger equation (NLS) on the two-dimensional torus. This equation admits a special family of elliptic invariant quasiperiodic tori called finite-gap solutions. These solutions 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(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. In my talk, I will also say some words about the ongoing work concerning Sobolev instability of more general 2D quasi-periodic solutions. The subject of this talk is partly motivated by the problem of infinite energy cascade for 2D NLS, and it is a joint work with M. Guardia, Z. Hani, A. Maspero and M. Procesi.

Mathematics; Dynamical systems and ergodic theory; Partial differential equations

video/mp4

Author affiliation: Università di Napoli Federico II

2019-12-09

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/72597

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/