Matrix Solutions of the Cubic Szegő Equation on the Real Line

Abstract

This paper is dedicated to studying matrix solutions of the cubic Szegő equation on the real line, which is introduced in Pocovnicu [Anal PDE 4(3):379–404, 2011; Dyn Syst A 31(3):607–649, 2011] and Gérard–Pushnitski (Commun Math Phys 405:167, 2024), leading to the following cubic matrix Szegő equation on \({\mathbb {R}}\),

$$\begin{aligned} i \partial _t U = \Pi _{\ge 0} \left( U U ^* U \right) , \quad \widehat{\left( \Pi _{\ge 0} U\right) }(\xi )= {\textbf{1}}_{\xi \ge 0}{\hat{U}}(\xi )\in {\mathbb {C}}^{M \times N}. \end{aligned}$$

Inspired by the space-periodic case in Sun (The matrix Szegő equation, arXiv:2309.12136), we establish its Lax pair structure via double Hankel operators and Toeplitz operators. Then the explicit formula in Gérard–Pushnitski (Commun Math Phys 405:167, 2024) can be extended to two equivalent formulas in the matrix equation case, which both express every solution explicitly in terms of its initial datum and the time variable.