Matrix solutions of the cubic Szegő equation on the real line

2023/10/20 by Sun, Ruoci
#37K10 #47B35 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2310.13693

Abstract

This paper is dedicated to studying matrix solutions of the cubic Szegő equation on the line in Pocovnicu [arXiv:1001.4037, arXiv:1012.2943] and Gérard--Pushnitski [arXiv:2307.06734], leading to the following matrix Szegő equation on ℝ, i ∂_t U = Π_{≥ 0} (U U ^* U ), \widehat(Π_{≥ 0} U)(ξ)= 1_{ξ≥ 0}U(ξ)∈ ℂ^{M × N}. Inspired from the space-periodic matrix Szegő equation in Sun [arXiv:2309.12136], we establish its Lax pair structure via double Hankel operators and Toeplitz operators. Then the explicit formula in [arXiv:2307.06734] 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.

Related

• The matrix Szegő equation
• An explicit formula for the cubic Szegö equation
• A direct approach for solving the cubic Szegö equation
• Explicit formula for the solution of the Szegö equation on the real line and applications
• The cubic szego equation and hankel operators