VIRTUAL SEMINAR SERIES -- INTEGRABLE SYSTEMS: Complete classification of rational solutions of A_2n-Painlevé systems
David Gómez-Ullate Oteiza (Universidad de Cádiz)
Wednesday 10th March, 2021 13:30-14:30 Zoom seminar hosted by ICMS
Abstract
We provide a complete classification and an explicit representation of rational solutions to the fourth Painlevé equation 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 A_2n-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. The special polynomials associated with higher order A_2n-Painlevé systems generalize the Okamoto and generalized Hermite polynomials which appear in the rational solutions of PIV (n=1).
[1] D Gómezâ€Ullate, Y Grandati, R Milson, Complete classification of rational solutions of A_2n-Painlevé systems, arXiv preprint arXiv:2010.00076
[2] PA Clarkson, D Gómezâ€Ullate, Y Grandati, R Milson, Cyclic Maya diagrams and rational solutions of higher order Painlevé systems
Studies in Applied Mathematics 144 (3), 357-385 (2020)
[3] D Gómezâ€Ullate, Y Grandati, R Milson, Durfee rectangles and pseudoâ€Wronskian equivalences for Hermite polynomials
Studies in Applied Mathematics 141 (4), 596-625 (2018)
For more information, and how to access the seminar through Zoom, see the webpage of the events here.
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