2018/11/22 by Peter A. Clarkson, Clarkson, Peter A., David Gómez-Ullate +6 · 2 citations
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Quantum Mechanics and Non-Hermitian Physics #math-ph #math.MP #nlin.SI
paper · pdf · doi:10.48550/arxiv.1811.09274
26 pages, 9 figures, typed in AMS-LaTeX
openalex publication_date 2018/11/22 · arxiv created 2019/09/14 · arxiv updated 2019/09/17 · openalex created_date 2022/08/01 · openalex updated_date 2026/07/28
This paper focuses on the construction of rational solutions for the A2n 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 we 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.