# Mathematical Sciences Research Institute

Home » (Pre) Lunch with Hamilton: Growth of Sobolev norms for the cubic nonlinear Schrödinger equation near 1D quasi-periodic solutions

# Seminar

(Pre) Lunch with Hamilton: Growth of Sobolev norms for the cubic nonlinear Schrödinger equation near 1D quasi-periodic solutions October 31, 2018 (11:00 AM PDT - 12:00 PM PDT)
Parent Program: Hamiltonian systems, from topology to applications through analysis MSRI: Simons Auditorium
Speaker(s) Marcel Guardia (Polytechnical University of Cataluña (Barcelona) )
Description No Description
Video
Abstract/Media

The study of solutions of Hamiltonian PDEs undergoing growth

of Sobolev norms H^s (with s\neq 1) as time evolves has drawn

considerable attention in recent years. The importance of growth of

Sobolev norms is due to the fact that it implies that the solution

transfers energy to higher modes.

Consider the defocusing cubic nonlinear Schr\"odinger equation (NLS) on

the two-dimensional torus. The equation admits a special family of

invariant quasiperiodic tori. These are inherited from the 1D cubic NLS

(on the circle) by considering solutions that depend only on one

variable. We show that, under certain assumptions, these tori are

transversally unstable in Sobolev spaces $H^s$ ($0<s<1$). More

precisely, we construct  solutions of the 2D cubic NLS which start

arbitrarily close to such invariant tori in the $H^s$ topology and whose

$H^s$ norm can grow by any given factor. This is a joint work with Z.

Hani, E. Haus, A. Maspero and M. Procesi.