The matrix Szeg\H{o} equation

TL;DR

This paper investigates the matrix cubic Szeg ext{"o} equation on the torus, revealing its integrable structure and providing explicit solutions based on initial data through a two-Lax-pair framework.

Contribution

It introduces the matrix Szeg ext{"o} equation, analyzes its integrability via two-Lax-pair structure, and derives explicit solution formulas from initial conditions.

Findings

The matrix Szeg ext{"o} equation admits a two-Lax-pair structure.

Solutions can be explicitly expressed in terms of initial data and time.

The equation extends the scalar case to matrix-valued functions on the torus.

Abstract

This paper is dedicated to studying the matrix solutions to the cubic Szeg\H{o} equation, introduced in G\'erard--Grellier (2009) arXiv:0906.4540, leading to the cubic matrix Szeg\H{o} equation on the torus, \begin{equation*} i \partial_t U = \Pi_{\geq 0} \left(U U ^* U \right), \quad \Pi_{\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 $t \in \mathbb{R}$.