Weakly turbulent solution to the Schrödinger equation on the 2D torus with potential

Ambre Chabert (ENS Paris)

Jun 18. 2024, 15:00 — 15:15

In this talk, I will construct a smooth solution to the linear Schrödinger equation on the 2D torus, perturbed by a smooth real potential vanishing as time tends to infinity, while the H^1 Sobolev norm of the solution grows logarithmically and hence it blows up in infinite time. I will present how I have been able to adapt the well-known techniques of CKSTT 2010, which relies on nonlinear interactions between Fourier modes to exhibit (finite) norm growth in the cubic nonlinear Schrödinger equation on the 2D torus, to a similar linear setting; thus, I will argue how the nonlinear ideas yield a new perspective on the linear problem. Looking at the equation in Fourier modes, this problem can be approximated by a discrete resonant system of ODEs. Choosing carefully the frequencies on which it is supported, we can construct a special solution which can be seen as a sequence of finite-dimensional linear oscillators along which the energy propagates to higher and higher frequencies. This approach leads to straightforward and explicit constructions, thus enabling to control precisely the growth rate and understand the growth mechanism.

Further Information
Venue:
ESI Boltzmann Lecture Hall
Associated Event:
Nonlinear Waves and Relativity (Thematic Programme)
Organizer(s):
Roland Donninger (U of Vienna)
David Fajman (U of Vienna)
Birgit Schörkhuber (U of Innsbruck)