Optimal bounds for the growth of Sobolev norms of solutions of a quadratic Szegő equation

Abstract:

In this paper, we study a quadratic equation on the one-dimensional torus \[ i \partial _t u = 2J\Pi (|u|^2)+\bar {J}u^2, \quad u(0, \cdot )=u_0,\] where $J=\int _\mathbb {T}|u|^2u \in \mathbb {C}$ has constant modulus, and $\Pi$ is the Szegő projector onto functions with nonnegative frequencies. Thanks to a Lax pair structure, we construct a flow on $BMO(\mathbb {T})\cap \mathrm {Im}\Pi$ which propagates $H^s$ regularity for any $s>0$, whereas the energy level corresponds to $s=1/2$. Then, for each $s>1/2$, we exhibit solutions whose $H^s$ norm goes to $+\infty$ exponentially fast, and we show that this growth is optimal.