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

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

The structure of the extended affine Weyl symmetry group of higher Painlevé equations of <i>N</i> periodicity depends on whether <i>N</i> is even or odd. We find that for even <i>N</i>, the symmetry group <inline-formula><math xmlns="http://www.w3.or...

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Bibliographic Details
Main Authors: Henrik Aratyn, José Francisco Gomes, Gabriel Vieira Lobo, Abraham Hirsz Zimerman
Format: Article
Language:English
Published: MDPI AG 2024-11-01
Series:Mathematics
Subjects:
Painlevé equations
affine Weyl symmetries
Bäcklund transformations
dressing chain equations
Kummer polynomials
Online Access:https://www.mdpi.com/2227-7390/12/23/3701
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Summary:The structure of the extended affine Weyl symmetry group of higher Painlevé equations of <i>N</i> periodicity depends on whether <i>N</i> is even or odd. We find that for even <i>N</i>, the symmetry group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mrow><mover accent="true"><mi>A</mi><mo>^</mo></mover></mrow><mrow><mi>N</mi><mo>−</mo><mn>1</mn></mrow><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msubsup></semantics></math></inline-formula> contains the conventional Bäcklund transformations <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>s</mi><mi>j</mi></msub><mo>,</mo><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>N</mi></mrow></semantics></math></inline-formula>, the group of automorphisms consisting of cycling permutations but also reflections on a periodic circle of <i>N</i> 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>=</mo><mn>4</mn></mrow></semantics></math></inline-formula>, we explicitly show how even reflection automorphisms cause degeneracy of a class of rational solutions obtained on the orbit of the translation operators of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mrow><mover accent="true"><mi>A</mi><mo>^</mo></mover></mrow><mn>3</mn><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msubsup></semantics></math></inline-formula>. We obtain the closed expressions for the solutions and their degenerated counterparts in terms of the determinants of the Kummer polynomials.
ISSN:2227-7390

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