About the quadratic Szeg{\"o} hierarchy

Abstract

The purpose of this paper is to go further into the study of the quadratic Szeg{\"o} equation, which is the following Hamiltonian PDE : $i \partial\_t u = 2J\Pi(|u|^2)+\bar{J}u^2$, $u(0, \cdot)=u\_0$, where $\Pi$ is the Szeg{\"o} projector onto nonnegative modes, and $J = J(u)$ is the complex number given by $J=\int\_\mathbb{T}|u|^2u$. We exhibit an infinite set of new conservation laws $\{\ell\_k \}$ which are in involution. These laws give us a better understanding of the "turbulent" behavior of certain rational solutions of the equation : we show that if the orbit of a rational solution is unbounded in some $H^s$, $s > 1/2$, then one of the $\ell\_k$'s must be zero. As a consequence, we characterize growing solutions which can be written as the sum of two solitons.