The matrix Szegő equation

This paper is dedicated to studying the matrix solutions to the cubic Szegő equation, introduced in Gérard–Grellier (2009) arXiv:0906.4540, leading to the cubic matrix Szegő equation on the torus, \begin{equation*} i \partial_t U = Π_{\geq 0} \left(U U ^* U \right), \quad Π_{\geq 0}: \sum_{n\in $\mathbb{Z}$}\hat{U}(n) e^{inx} \mapsto \sum_{n \geq 0} \hat{U}(n) e^{inx}. \end{equation*}This equation enjoys a two-Lax-pair structure, which allow every solution to be expressed explicitly in terms of the initial datum and the time $t\in \mathbb{R}$.