About the quadratic Szeg{\"o} hierarchy

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 $J=\int \mathrm{\_}\mathbb{T}\mathrm{\mid }u{\mathrm{\mid }}^{2}u$. We exhibit an infinite set of new conservation laws $\{\mathrm{\ell }\mathrm{\_}k\}$ 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 $H^s$, $s>1\mathrm{/}2$, then one of the $\mathrm{\ell }\mathrm{\_}k$’s must be zero. As a consequence, we characterize growing solutions which can be written as the sum of two solitons.