Cyclic Maya diagrams and rational solutions of higher order Painlevé systems. Issue 3 (22nd January 2020)

• Record Type: Journal Article
• Authors: Clarkson, Peter A.; Gómez‐Ullate, David; Grandati, Yves; Milson, Robert
• Is Part Of: Studies in applied mathematics. Volume 144:Issue 3(2020)
• Page Start: 357
• Page End: 385
• Publication Date: 2020-01-22
• Subjects: Darboux dressing chains -- Hermite polynomials -- Painlevé equations -- Maya diagrams -- Noumi‐Yamada systems -- rational solutions -- Wronskian determinants; Mathematics -- Periodicals; Mathématiques -- Périodiques; Mathématique; Mathématique appliquée; Ressource Internet (Descripteur de forme); Périodique électronique (Descripteur de forme); 519
• DOI: 10.1111/sapm.12300
• Languages: English
• ISSNs: 0022-2526

Abstract

Abstract: This paper focuses on the construction of rational solutions for the A 2 n ‐Painlevé system, also called the Noumi‐Yamada system, which are considered the higher order generalizations of PIV . In this even case, we introduce a method to construct the rational solutions based on cyclic dressing chains of Schrödinger operators with potentials in the class of rational extensions of the harmonic oscillator. Each potential in the chain can be indexed by a single Maya diagram and expressed in terms of a Wronskian determinant whose entries are Hermite polynomials. We introduce the notion of cyclic Maya diagrams and characterize them for any possible period, using the concepts of genus and interlacing. The resulting classes of solutions can be expressed in terms of special polynomials that generalize the families of generalized Hermite, generalized Okamoto, and Umemura polynomials, showing that they are particular cases of a larger family.