Extended Relativistic Toda Lattice, L-Orthogonal Polynomials and Associated Lax Pair

Detalhes bibliográficos
Autor(a) principal: Bracciali, Cleonice F. [UNESP]
Data de Publicação: 2018
Outros Autores: Silva, Jairo S., Sri Ranga, A. [UNESP]
Tipo de documento: Artigo
Idioma: eng
Título da fonte: Repositório Institucional da UNESP
Texto Completo: http://dx.doi.org/10.1007/s10440-018-00229-x
http://hdl.handle.net/11449/187153
Resumo: When a measure Ψ(x) on the real line is subjected to the modification dΨ( t )(x) = e− t xdΨ(x) , then the coefficients of the recurrence relation of the orthogonal polynomials in x with respect to the measure Ψ( t )(x) are known to satisfy the so-called Toda lattice formulas as functions of t. In this paper we consider a modification of the form e− t ( p x + q / x ) of measures or, more generally, of moment functionals, associated with orthogonal L-polynomials and show that the coefficients of the recurrence relation of these L-orthogonal polynomials satisfy what we call an extended relativistic Toda lattice. Most importantly, we also establish the so called Lax pair representation associated with this extended relativistic Toda lattice. These results also cover the (ordinary) relativistic Toda lattice formulations considered in the literature by assuming either p= 0 or q= 0. However, as far as Lax pair representation is concern, no complete Lax pair representations were established before for the respective relativistic Toda lattice formulations. Some explicit examples of extended relativistic Toda lattice and Langmuir lattice are also presented. As further results, the lattice formulas that follow from the three term recurrence relations associated with kernel polynomials on the unit circle are also established.
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spelling Extended Relativistic Toda Lattice, L-Orthogonal Polynomials and Associated Lax PairKernel polynomials on the unit circleL-orthogonal polynomialsLax pairsRelativistic Toda latticeWhen a measure Ψ(x) on the real line is subjected to the modification dΨ( t )(x) = e− t xdΨ(x) , then the coefficients of the recurrence relation of the orthogonal polynomials in x with respect to the measure Ψ( t )(x) are known to satisfy the so-called Toda lattice formulas as functions of t. In this paper we consider a modification of the form e− t ( p x + q / x ) of measures or, more generally, of moment functionals, associated with orthogonal L-polynomials and show that the coefficients of the recurrence relation of these L-orthogonal polynomials satisfy what we call an extended relativistic Toda lattice. Most importantly, we also establish the so called Lax pair representation associated with this extended relativistic Toda lattice. These results also cover the (ordinary) relativistic Toda lattice formulations considered in the literature by assuming either p= 0 or q= 0. However, as far as Lax pair representation is concern, no complete Lax pair representations were established before for the respective relativistic Toda lattice formulations. Some explicit examples of extended relativistic Toda lattice and Langmuir lattice are also presented. As further results, the lattice formulas that follow from the three term recurrence relations associated with kernel polynomials on the unit circle are also established.Departamento de Matemática Aplicada UNESP–Univ Estadual PaulistaDepartamento de Matemática Universidade Federal do MaranhãoDepartamento de Matemática Aplicada UNESP–Univ Estadual PaulistaUniversidade Estadual Paulista (Unesp)Universidade Federal do MaranhãoBracciali, Cleonice F. [UNESP]Silva, Jairo S.Sri Ranga, A. [UNESP]2019-10-06T15:27:05Z2019-10-06T15:27:05Z2018-01-01info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articlehttp://dx.doi.org/10.1007/s10440-018-00229-xActa Applicandae Mathematicae.1572-90360167-8019http://hdl.handle.net/11449/18715310.1007/s10440-018-00229-x2-s2.0-8505812087183003224526224670000-0002-6823-4204Scopusreponame:Repositório Institucional da UNESPinstname:Universidade Estadual Paulista (UNESP)instacron:UNESPengActa Applicandae Mathematicaeinfo:eu-repo/semantics/openAccess2022-02-09T11:13:33Zoai:repositorio.unesp.br:11449/187153Repositório InstitucionalPUBhttp://repositorio.unesp.br/oai/requestopendoar:29462024-08-05T23:00:56.417256Repositório Institucional da UNESP - Universidade Estadual Paulista (UNESP)false
dc.title.none.fl_str_mv Extended Relativistic Toda Lattice, L-Orthogonal Polynomials and Associated Lax Pair
title Extended Relativistic Toda Lattice, L-Orthogonal Polynomials and Associated Lax Pair
spellingShingle Extended Relativistic Toda Lattice, L-Orthogonal Polynomials and Associated Lax Pair
Bracciali, Cleonice F. [UNESP]
Kernel polynomials on the unit circle
L-orthogonal polynomials
Lax pairs
Relativistic Toda lattice
title_short Extended Relativistic Toda Lattice, L-Orthogonal Polynomials and Associated Lax Pair
title_full Extended Relativistic Toda Lattice, L-Orthogonal Polynomials and Associated Lax Pair
title_fullStr Extended Relativistic Toda Lattice, L-Orthogonal Polynomials and Associated Lax Pair
title_full_unstemmed Extended Relativistic Toda Lattice, L-Orthogonal Polynomials and Associated Lax Pair
title_sort Extended Relativistic Toda Lattice, L-Orthogonal Polynomials and Associated Lax Pair
author Bracciali, Cleonice F. [UNESP]
author_facet Bracciali, Cleonice F. [UNESP]
Silva, Jairo S.
Sri Ranga, A. [UNESP]
author_role author
author2 Silva, Jairo S.
Sri Ranga, A. [UNESP]
author2_role author
author
dc.contributor.none.fl_str_mv Universidade Estadual Paulista (Unesp)
Universidade Federal do Maranhão
dc.contributor.author.fl_str_mv Bracciali, Cleonice F. [UNESP]
Silva, Jairo S.
Sri Ranga, A. [UNESP]
dc.subject.por.fl_str_mv Kernel polynomials on the unit circle
L-orthogonal polynomials
Lax pairs
Relativistic Toda lattice
topic Kernel polynomials on the unit circle
L-orthogonal polynomials
Lax pairs
Relativistic Toda lattice
description When a measure Ψ(x) on the real line is subjected to the modification dΨ( t )(x) = e− t xdΨ(x) , then the coefficients of the recurrence relation of the orthogonal polynomials in x with respect to the measure Ψ( t )(x) are known to satisfy the so-called Toda lattice formulas as functions of t. In this paper we consider a modification of the form e− t ( p x + q / x ) of measures or, more generally, of moment functionals, associated with orthogonal L-polynomials and show that the coefficients of the recurrence relation of these L-orthogonal polynomials satisfy what we call an extended relativistic Toda lattice. Most importantly, we also establish the so called Lax pair representation associated with this extended relativistic Toda lattice. These results also cover the (ordinary) relativistic Toda lattice formulations considered in the literature by assuming either p= 0 or q= 0. However, as far as Lax pair representation is concern, no complete Lax pair representations were established before for the respective relativistic Toda lattice formulations. Some explicit examples of extended relativistic Toda lattice and Langmuir lattice are also presented. As further results, the lattice formulas that follow from the three term recurrence relations associated with kernel polynomials on the unit circle are also established.
publishDate 2018
dc.date.none.fl_str_mv 2018-01-01
2019-10-06T15:27:05Z
2019-10-06T15:27:05Z
dc.type.status.fl_str_mv info:eu-repo/semantics/publishedVersion
dc.type.driver.fl_str_mv info:eu-repo/semantics/article
format article
status_str publishedVersion
dc.identifier.uri.fl_str_mv http://dx.doi.org/10.1007/s10440-018-00229-x
Acta Applicandae Mathematicae.
1572-9036
0167-8019
http://hdl.handle.net/11449/187153
10.1007/s10440-018-00229-x
2-s2.0-85058120871
8300322452622467
0000-0002-6823-4204
url http://dx.doi.org/10.1007/s10440-018-00229-x
http://hdl.handle.net/11449/187153
identifier_str_mv Acta Applicandae Mathematicae.
1572-9036
0167-8019
10.1007/s10440-018-00229-x
2-s2.0-85058120871
8300322452622467
0000-0002-6823-4204
dc.language.iso.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv Acta Applicandae Mathematicae
dc.rights.driver.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.source.none.fl_str_mv Scopus
reponame:Repositório Institucional da UNESP
instname:Universidade Estadual Paulista (UNESP)
instacron:UNESP
instname_str Universidade Estadual Paulista (UNESP)
instacron_str UNESP
institution UNESP
reponame_str Repositório Institucional da UNESP
collection Repositório Institucional da UNESP
repository.name.fl_str_mv Repositório Institucional da UNESP - Universidade Estadual Paulista (UNESP)
repository.mail.fl_str_mv
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