About the quadratic Szeg{\"o} hierarchy

TL;DR

This paper advances the understanding of the quadratic Szeg{"o} equation by discovering new conservation laws, analyzing turbulent behaviors of solutions, and characterizing solutions that decompose into solitons.

Contribution

It introduces an infinite set of conservation laws in involution and links unbounded solution behavior to these laws, providing new insights into solution dynamics.

Findings

New conservation laws $\u03bb_k$ in involution.

Unbounded orbits imply some $\u03bb_k$ must be zero.

Characterization of solutions as sums of two solitons.

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.