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...

Descripción completa

Guardado en:

Detalles Bibliográficos
Autores principales:	Artyom V. Yurov, Valerian A. Yurov
Formato:	article
Lenguaje:	EN
Publicado:	MDPI AG 2021
Materias:	Painlevé equations Bäcklund transformations Schlesinger transformations JS-systems JP-systems Mathematics QA1-939
Acceso en línea:	https://doaj.org/article/11557ea5b95c4311a6ab51a3236fd972
Etiquetas:	Agregar Etiqueta Sin Etiquetas, Sea el primero en etiquetar este registro!

id	oai:doaj.org-article:11557ea5b95c4311a6ab51a3236fd972
record_format	dspace
spelling	oai:doaj.org-article:11557ea5b95c4311a6ab51a3236fd9722021-11-25T19:06:42ZOn the Question of the Bäcklund Transformations and Jordan Generalizations of the Second Painlevé Equation10.3390/sym131120952073-8994https://doaj.org/article/11557ea5b95c4311a6ab51a3236fd9722021-11-01T00:00:00Zhttps://www.mdpi.com/2073-8994/13/11/2095https://doaj.org/toc/2073-8994We 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.Artyom V. YurovValerian A. YurovMDPI AGarticlePainlevé equationsBäcklund transformationsSchlesinger transformationsJS-systemsJP-systemsMathematicsQA1-939ENSymmetry, Vol 13, Iss 2095, p 2095 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	Painlevé equations Bäcklund transformations Schlesinger transformations JS-systems JP-systems Mathematics QA1-939
spellingShingle	Painlevé equations Bäcklund transformations Schlesinger transformations JS-systems JP-systems Mathematics QA1-939 Artyom V. Yurov Valerian A. Yurov On the Question of the Bäcklund Transformations and Jordan Generalizations of the Second Painlevé Equation
description	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.
format	article
author	Artyom V. Yurov Valerian A. Yurov
author_facet	Artyom V. Yurov Valerian A. Yurov
author_sort	Artyom V. Yurov
title	On the Question of the Bäcklund Transformations and Jordan Generalizations of the Second Painlevé Equation
title_short	On the Question of the Bäcklund Transformations and Jordan Generalizations of the Second Painlevé Equation
title_full	On the Question of the Bäcklund Transformations and Jordan Generalizations of the Second Painlevé Equation
title_fullStr	On the Question of the Bäcklund Transformations and Jordan Generalizations of the Second Painlevé Equation
title_full_unstemmed	On the Question of the Bäcklund Transformations and Jordan Generalizations of the Second Painlevé Equation
title_sort	on the question of the bäcklund transformations and jordan generalizations of the second painlevé equation
publisher	MDPI AG
publishDate	2021
url	https://doaj.org/article/11557ea5b95c4311a6ab51a3236fd972
work_keys_str_mv	AT artyomvyurov onthequestionofthebacklundtransformationsandjordangeneralizationsofthesecondpainleveequation AT valerianayurov onthequestionofthebacklundtransformationsandjordangeneralizationsofthesecondpainleveequation
_version_	1718410261493186560

On the Question of the Bäcklund Transformations and Jordan Generalizations of the Second Painlevé Equation

Ejemplares similares