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

Abstract

We construct global solutions of the periodic defocusing cubic half-wave equation whose $H^s$ norms grow exponentially as $t\to+\infty$ for every $1/2<s<3/2$. Such solutions exist at every positive mass. The construction starts from explicit concentrating trajectories of a linearly perturbed Szego equation and solves an infinite-time final-value problem for the full half-wave flow, including its negative-frequency component. We obtain the exact growth rate and conserved quantities. Finite Blaschke products allow us to prescribe finitely many concentration points and their relative widths, and we compute the limiting critical kinetic-energy measures. We also identify the endpoint Fourier decay of the constructed data and prove the sharp logarithmic regularity threshold on the resonant mass slice and for generic scalar parameters. In particular, 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.