2020/09/30 by David Gómez‐Ullate, David Gómez-Ullate, Yves Grandati +4
Computer Science · Mathematics · Physics and Astronomy · #33C45 #34M35 #34M55 #37K10 #Advanced Differential Equations and Dynamical Systems #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Polynomial and algebraic computation #math-ph #math.MP #msc:33C45 #msc:34M35 #msc:34M55 #msc:37K10 #nlin.SI
paper · pdf · doi:10.48550/arxiv.2010.00076
43 pages, 7 figures, typed in AMS-LaTeX
openalex publication_date 2020/09/30 · arxiv created 2021/04/12 · arxiv updated 2021/04/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We provide a complete classification and an explicit representation of rational solutions to the fourth Painlevé equation PIV and its higher order generalizations known as the A2n-Painlevé or Noumi-Yamada systems. The construction of solutions makes use of the theory of cyclic dressing chains of Schrödinger operators. Studying the local expansions of the solutions around their singularities we find that some coefficients in their Laurent expansion must vanish, which express precisely the conditions of trivial monodromy of the associated potentials. The characterization of trivial monodromy potentials with quadratic growth implies that all rational solutions can be expressed as Wronskian determinants of suitably chosen sequences of Hermite polynomials. The main classification result states that every rational solution to the A2n-Painlevé system corresponds to a cycle of Maya diagrams, which can be indexed by an oddly coloured integer sequence. Finally, we establish the link with the standard approach to building rational solutions, based on applying Bäcklund transformations on seed solutions, by providing a representation for the symmetry group action on coloured sequences and Maya cycles.