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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| author | Henrik Aratyn José Francisco Gomes Gabriel Vieira Lobo Abraham Hirsz Zimerman |
| author_facet | Henrik Aratyn José Francisco Gomes Gabriel Vieira Lobo Abraham Hirsz Zimerman |
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| description | 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. |
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| spelling | doaj-art-e05ddae2e46441c5a8e6cd6b16b2da9d2024-12-13T16:27:28ZengMDPI AGMathematics2227-73902024-11-011223370110.3390/math12233701Extended Symmetry of Higher Painlevé Equations of Even Periodicity and Their Rational SolutionsHenrik Aratyn0José Francisco Gomes1Gabriel Vieira Lobo2Abraham Hirsz Zimerman3Department of Physics, University of Illinois at Chicago, 845 W. Taylor Str., Chicago, IL 60607-7059, USAInstituto de Física Teórica-UNESP, Rua Dr Bento Teobaldo Ferraz 271, Bloco II, São Paulo 01140-070, BrazilInstituto de Física Teórica-UNESP, Rua Dr Bento Teobaldo Ferraz 271, Bloco II, São Paulo 01140-070, BrazilInstituto de Física Teórica-UNESP, Rua Dr Bento Teobaldo Ferraz 271, Bloco II, São Paulo 01140-070, BrazilThe 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.https://www.mdpi.com/2227-7390/12/23/3701Painlevé equationsaffine Weyl symmetriesBäcklund transformationsdressing chain equationsKummer polynomials |
| spellingShingle | Henrik Aratyn José Francisco Gomes Gabriel Vieira Lobo Abraham Hirsz Zimerman Extended Symmetry of Higher Painlevé Equations of Even Periodicity and Their Rational Solutions Mathematics Painlevé equations affine Weyl symmetries Bäcklund transformations dressing chain equations Kummer polynomials |
| title | Extended Symmetry of Higher Painlevé Equations of Even Periodicity and Their Rational Solutions |
| title_full | Extended Symmetry of Higher Painlevé Equations of Even Periodicity and Their Rational Solutions |
| title_fullStr | Extended Symmetry of Higher Painlevé Equations of Even Periodicity and Their Rational Solutions |
| title_full_unstemmed | Extended Symmetry of Higher Painlevé Equations of Even Periodicity and Their Rational Solutions |
| title_short | Extended Symmetry of Higher Painlevé Equations of Even Periodicity and Their Rational Solutions |
| title_sort | extended symmetry of higher painleve equations of even periodicity and their rational solutions |
| topic | Painlevé equations affine Weyl symmetries Bäcklund transformations dressing chain equations Kummer polynomials |
| url | https://www.mdpi.com/2227-7390/12/23/3701 |
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