On the Question of the Bäcklund Transformations and Jordan Generalizations of the Second Painlevé Equation
We demonstrate the way to derive the second Painlevé equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mn>2</mn></msub></semantics></math></in...
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| Auteurs principaux: | , |
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| Format: | article |
| Langue: | EN |
| Publié: |
MDPI AG
2021
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| Sujets: | |
| Accès en ligne: | https://doaj.org/article/11557ea5b95c4311a6ab51a3236fd972 |
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| Résumé: | We demonstrate the way to derive the second Painlevé equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mn>2</mn></msub></semantics></math></inline-formula> and its Bäcklund transformations from the deformations of the Nonlinear Schrödinger equation (NLS), all the while preserving the strict invariance with respect to the Schlesinger transformations. The proposed algorithm allows for a construction of Jordan algebra-based completely integrable multiple-field generalizations of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mn>2</mn></msub></semantics></math></inline-formula> while also producing the corresponding Bäcklund transformations. We suggest calling such models the JP-systems. For example, a Jordan algebra <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>J</mi><msub><mrow></mrow><mrow><mi>Mat</mi><mo>(</mo><mi>N</mi><mo>,</mo><mi>N</mi><mo>)</mo></mrow></msub></msub></semantics></math></inline-formula> with the Jordan product in the form of a semi-anticommutator is shown to generate an integrable matrix generalization of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mn>2</mn></msub></semantics></math></inline-formula>, whereas the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>V</mi><msub><mrow></mrow><mi>N</mi></msub></msub></semantics></math></inline-formula> algebra produces a different JP-system that serves as a generalization of the Sokolov’s form of a vectorial NLS. |
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