Infinite cascades for the defocusing cubic half-wave equation on the torus

Abstract

We construct global solutions of the defocusing cubic half-wave equation on the torus whose $H^s$ norms grow exponentially for every $1/2<s<3/2$. Such solutions exist at every positive mass. To the best of our knowledge, this provides the first example of a genuine infinite cascade for a defocusing fractional nonlinear Schrödinger equation on the torus. In particular, our result resolves an open question posed by Gérard and Grellier concerning the existence of unbounded Sobolev trajectories for the defocusing cubic half-wave equation on the torus.

arXiv commentA more polished version compared to v1