Extended Symmetry of Higher Painlevé Equations of Even Periodicity and Their Rational Solutions

Henrik Aratyn (), José Francisco Gomes, Gabriel Vieira Lobo and Abraham Hirsz Zimerman
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Henrik Aratyn: Department of Physics, University of Illinois at Chicago, 845 W. Taylor Str., Chicago, IL 60607-7059, USA
José Francisco Gomes: Instituto de Física Teórica-UNESP, Rua Dr Bento Teobaldo Ferraz 271, Bloco II, São Paulo 01140-070, Brazil
Gabriel Vieira Lobo: Instituto de Física Teórica-UNESP, Rua Dr Bento Teobaldo Ferraz 271, Bloco II, São Paulo 01140-070, Brazil
Abraham Hirsz Zimerman: Instituto de Física Teórica-UNESP, Rua Dr Bento Teobaldo Ferraz 271, Bloco II, São Paulo 01140-070, Brazil

Mathematics, 2024, vol. 12, issue 23, 1-25

Abstract: The structure of the extended affine Weyl symmetry group of higher Painlevé equations of N periodicity depends on whether N is even or odd. We find that for even N , the symmetry group A ^ N − 1 ( 1 ) contains the conventional Bäcklund transformations s j , j = 1 , … , N , the group of automorphisms consisting of cycling permutations but also reflections on a periodic circle of N points, which is a novel feature uncovered in this paper. The presence of reflection automorphisms is connected to the existence of degenerated solutions, and for N = 4 , we explicitly show how even reflection automorphisms cause degeneracy of a class of rational solutions obtained on the orbit of the translation operators of A ^ 3 ( 1 ) . We obtain the closed expressions for the solutions and their degenerated counterparts in terms of the determinants of the Kummer polynomials.

Keywords: Painlevé equations; affine Weyl symmetries; Bäcklund transformations; dressing chain equations; Kummer polynomials (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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