{"title":"Matrix Solutions of the Cubic Szegő Equation on the Real Line","authors":"Ruoci Sun","doi":"10.1007/s11040-025-09500-8","DOIUrl":null,"url":null,"abstract":"<div><p>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 <span>\\({\\mathbb {R}}\\)</span>, </p><div><div><span>$$\\begin{aligned} i \\partial _t U = \\Pi _{\\ge 0} \\left( U U ^* U \\right) , \\quad \\widehat{\\left( \\Pi _{\\ge 0} U\\right) }(\\xi )= {\\textbf{1}}_{\\xi \\ge 0}{\\hat{U}}(\\xi )\\in {\\mathbb {C}}^{M \\times N}. \\end{aligned}$$</span></div></div><p>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.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"28 1","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2025-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Physics, Analysis and Geometry","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s11040-025-09500-8","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
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.
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