KdV hierarchies and their relation to Kac-Moody algebras

The Korteweg-de Vries (KdV) equation equation was one of the first integrable systems which caught the attention of many researchers during the 19th century due to its several distinguishing characteristics. On the one hand, it admitted a special kind of solution called a soliton: a solitary wave ca...

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Autores:
Aragón Rodríguez, Manuel Alejandro
Tipo de recurso:
Trabajo de grado de pregrado
Fecha de publicación:
2024
Institución:
Universidad de los Andes
Repositorio:
Séneca: repositorio Uniandes
Idioma:
eng
OAI Identifier:
oai:repositorio.uniandes.edu.co:1992/76149
Acceso en línea:
https://hdl.handle.net/1992/76149
Palabra clave:
Integrable systems
Lie algebras
KdV equation
Lax equation
Física
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openAccess
License
Attribution-NonCommercial-NoDerivatives 4.0 International
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oai_identifier_str oai:repositorio.uniandes.edu.co:1992/76149
network_acronym_str UNIANDES2
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dc.title.none.fl_str_mv KdV hierarchies and their relation to Kac-Moody algebras
title KdV hierarchies and their relation to Kac-Moody algebras
spellingShingle KdV hierarchies and their relation to Kac-Moody algebras
Integrable systems
Lie algebras
KdV equation
Lax equation
Física
title_short KdV hierarchies and their relation to Kac-Moody algebras
title_full KdV hierarchies and their relation to Kac-Moody algebras
title_fullStr KdV hierarchies and their relation to Kac-Moody algebras
title_full_unstemmed KdV hierarchies and their relation to Kac-Moody algebras
title_sort KdV hierarchies and their relation to Kac-Moody algebras
dc.creator.fl_str_mv Aragón Rodríguez, Manuel Alejandro
dc.contributor.advisor.none.fl_str_mv Téllez Acosta, Gabriel
dc.contributor.author.none.fl_str_mv Aragón Rodríguez, Manuel Alejandro
dc.contributor.jury.none.fl_str_mv Reyes Lega, Andrés Fernando
dc.contributor.researchgroup.none.fl_str_mv Facultad de Ciencias::Física estadística
dc.subject.keyword.eng.fl_str_mv Integrable systems
topic Integrable systems
Lie algebras
KdV equation
Lax equation
Física
dc.subject.keyword.none.fl_str_mv Lie algebras
KdV equation
Lax equation
dc.subject.themes.none.fl_str_mv Física
description The Korteweg-de Vries (KdV) equation equation was one of the first integrable systems which caught the attention of many researchers during the 19th century due to its several distinguishing characteristics. On the one hand, it admitted a special kind of solution called a soliton: a solitary wave capable of traveling without losing its form and that whose interactions are elastical; on the other hand, the KdV equation exhibited a large quantity of independent conserved quantities regardless of its high nonlinearity. One of the major breakthroughs in the study of the KdV equation came from Drinfeld and Sokolov in 1985, who showed there existed an algebraic structure underlying the KdV system associated to the recently famous Kac-Moody algebras. Moreover, beginning from an arbitrary affine Kac-Moody algebra, Drinfeld and Sokolov gave a method of constructing a hierarchy of evolution equations that could be solved under the same scheme under which the KdV equation could. The purpose of this work is to provide a fundamental background in which Drinfeld and Sokolov’s construction can be understood.
publishDate 2024
dc.date.issued.none.fl_str_mv 2024-05-31
dc.date.accessioned.none.fl_str_mv 2025-04-21T15:52:22Z
dc.date.available.none.fl_str_mv 2025-04-21T15:52:22Z
dc.type.none.fl_str_mv Trabajo de grado - Pregrado
dc.type.driver.none.fl_str_mv info:eu-repo/semantics/bachelorThesis
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dc.language.iso.none.fl_str_mv eng
language eng
dc.relation.references.none.fl_str_mv Gu Chaohao (Ed.) “Soliton theory and its applications”. In: Springer-Verlag, 1995
M. A. Aragon. “Opers as a generalisation of complex projective structures”. Universidad de los Andes, 2023
[G. K. Balanis. “The plasma inverse problem”. In: J. Math. Phys. 13.7 (1972)
[J. Boussinesq. “Essai sur la theorie des eaux courantes”. In: Memoires presentes par divers savants ‘ l’Acad. des Sci. Inst. Nat. France XXIII (1877)
M. D. Kruskal & R. M. Miura C. S. Gardner C. S. Greene. “Method for solving the Korteweg-de Vries equation”. In: Phys. Rev. Lett. 19 (1967)
G. de Vries D. Korteweg. “On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves”. In: Phil. Mag. 39 (1985)
Boris Dubrovin, Di Yang, and Don Zagier. On tau-functions for the KdV hierarchy. 2021. arXiv: 1812.08488 [math-ph]
M. Dunajski. Soliton, instantons and twistors. Oxford gradute texts in mathematics, 2010
L. D. Faddeev. “Properties of the S-matrix of the one-dimensional Schrodinger equation”. In: Amer. Math. Soc. Transl. 2nd ser. (1967), pp. 139–166
H. Flaschka. “The Toda lattice. II”. In: Phys. Rev. B 9 (1974)
C. S. Gardener. “The Korteweg-de Vries equation and generalizations IV”. In: J. Math. Phys. 12 (1971)
M. Henon. “Integrals of the Toda lattice”. In: Phys. Rev. B 9 (1974)
J. E. Humphreys. Introduction to Lie Algebras and Representation Theory. Vol. 9. Graduate Texts in Mathematics. New York: Springer, 1972
[H. E. Moses I. Kay. “Reflectionless transmission through dielectrics and scattering potentials”. In: J. Appl. Phys. 27 (1956)
B. M. Levitan I. M. Gelfand. “On the determination of a differential equation from its spectral function”. In: Amer. Math. soc. Transl. 2nd ser. 1 (1955)
Nathan Jacobson. Lie Algebras. New York: Dover Publications, 1979
E. M. de Jager. On the origin of the Korteweg-de Vries equation. 2011. arXiv: math/0602661 [math.HO]
V. G. Kac. Infinite Dimensional Lie Algebras. 3rd. Cambridge University Press, 1990
B. Kostant. “The Solution to a Generalized Toda Lattice and Representation Theory”. In: Advances in Mathematics 34 (1979)
G. Lamb. “Elements of soliton theory”. In: (1980)
P. Lax. “Integrals of nonlinear equations of evolution and solitary waves”. In: Comm. on pure and pplied Math. 21 (1968)
P. A. Clarkson M. J. Ablowitz. Solitons, nonlinear evolution equations and inverse scattering. Cambridge University Press, 1991.
M. Toda M. Wadati. “The exact N-soliton solution of the Korteweg-de Vries equation”. In: J. Phys. Soc. Japan 32 (1972)
V. A. Marchenko. “Sturm-Liouville Operators and Applications”. In: 1986. url: https://api.semanticscholar.org/CorpusID:118317624
R. M. Miura. “Korteweg-de Vries equation and generalization”. In: J. Math. Physics 9.9 (1968)
A. C. Newell. Solitons in mathematics and physics. Ed. by Society for Industrial and Applied Mathematics. 1985
Russell. “Report on waves”. In: Rept. Fourteenth Meeting of the British Association for the Advancement of Science. Ed. by J. Murray. 1844
M. Toda. “Vibration of a chain with a non-linear interaction”. In: J. Phys. Soc. Jpn. 22 (1967)
L. D. Faddeev V. E. Zhakarov. “The Korteweg-de Vries equation: a completely integrable Hamiltonian system”. In: Funct. Anal. Appl. 5 (1968)
V. V. Sokolov V. G. Drinfeld. “Lie algebras and equations of Korteweg-de Vries type”. In: Journal of Soviet Mathematics 30 (1985)
M. Verde. “Asymptotic expansion of phase shifts at high energies”. In: Nuovo cimento 2 (1955)
F. W. Warner. Foundations of Differentiable Manifolds and Lie Groups. Vol. 94. Graduate Texts in Mathematics. New York: Springer, 1983
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dc.publisher.department.none.fl_str_mv Departamento de Física
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spelling Téllez Acosta, Gabrielvirtual::24039-1Aragón Rodríguez, Manuel AlejandroReyes Lega, Andrés Fernandovirtual::24040-1Facultad de Ciencias::Física estadística2025-04-21T15:52:22Z2025-04-21T15:52:22Z2024-05-31https://hdl.handle.net/1992/76149instname:Universidad de los Andesreponame:Repositorio Institucional Sénecarepourl:https://repositorio.uniandes.edu.co/The Korteweg-de Vries (KdV) equation equation was one of the first integrable systems which caught the attention of many researchers during the 19th century due to its several distinguishing characteristics. On the one hand, it admitted a special kind of solution called a soliton: a solitary wave capable of traveling without losing its form and that whose interactions are elastical; on the other hand, the KdV equation exhibited a large quantity of independent conserved quantities regardless of its high nonlinearity. One of the major breakthroughs in the study of the KdV equation came from Drinfeld and Sokolov in 1985, who showed there existed an algebraic structure underlying the KdV system associated to the recently famous Kac-Moody algebras. Moreover, beginning from an arbitrary affine Kac-Moody algebra, Drinfeld and Sokolov gave a method of constructing a hierarchy of evolution equations that could be solved under the same scheme under which the KdV equation could. The purpose of this work is to provide a fundamental background in which Drinfeld and Sokolov’s construction can be understood.Pregrado47 páginasapplication/pdfengUniversidad de los AndesFísicaFacultad de CienciasDepartamento de FísicaAttribution-NonCommercial-NoDerivatives 4.0 Internationalhttp://creativecommons.org/licenses/by-nc-nd/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2KdV hierarchies and their relation to Kac-Moody algebrasTrabajo de grado - Pregradoinfo:eu-repo/semantics/bachelorThesisinfo:eu-repo/semantics/acceptedVersionhttp://purl.org/coar/resource_type/c_7a1fTexthttp://purl.org/redcol/resource_type/TPIntegrable systemsLie algebrasKdV equationLax equationFísicaGu Chaohao (Ed.) “Soliton theory and its applications”. In: Springer-Verlag, 1995M. A. Aragon. “Opers as a generalisation of complex projective structures”. Universidad de los Andes, 2023[G. K. Balanis. “The plasma inverse problem”. In: J. Math. Phys. 13.7 (1972)[J. Boussinesq. “Essai sur la theorie des eaux courantes”. In: Memoires presentes par divers savants ‘ l’Acad. des Sci. Inst. Nat. France XXIII (1877)M. D. Kruskal & R. M. Miura C. S. Gardner C. S. Greene. “Method for solving the Korteweg-de Vries equation”. In: Phys. Rev. Lett. 19 (1967)G. de Vries D. Korteweg. “On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves”. In: Phil. Mag. 39 (1985)Boris Dubrovin, Di Yang, and Don Zagier. On tau-functions for the KdV hierarchy. 2021. arXiv: 1812.08488 [math-ph]M. Dunajski. Soliton, instantons and twistors. Oxford gradute texts in mathematics, 2010L. D. Faddeev. “Properties of the S-matrix of the one-dimensional Schrodinger equation”. In: Amer. Math. Soc. Transl. 2nd ser. (1967), pp. 139–166H. Flaschka. “The Toda lattice. II”. In: Phys. Rev. B 9 (1974)C. S. Gardener. “The Korteweg-de Vries equation and generalizations IV”. In: J. Math. Phys. 12 (1971)M. Henon. “Integrals of the Toda lattice”. In: Phys. Rev. B 9 (1974)J. E. Humphreys. Introduction to Lie Algebras and Representation Theory. Vol. 9. Graduate Texts in Mathematics. New York: Springer, 1972[H. E. Moses I. Kay. “Reflectionless transmission through dielectrics and scattering potentials”. In: J. Appl. Phys. 27 (1956)B. M. Levitan I. M. Gelfand. “On the determination of a differential equation from its spectral function”. In: Amer. Math. soc. Transl. 2nd ser. 1 (1955)Nathan Jacobson. Lie Algebras. New York: Dover Publications, 1979E. M. de Jager. On the origin of the Korteweg-de Vries equation. 2011. arXiv: math/0602661 [math.HO]V. G. Kac. Infinite Dimensional Lie Algebras. 3rd. Cambridge University Press, 1990B. Kostant. “The Solution to a Generalized Toda Lattice and Representation Theory”. In: Advances in Mathematics 34 (1979)G. Lamb. “Elements of soliton theory”. In: (1980)P. Lax. “Integrals of nonlinear equations of evolution and solitary waves”. In: Comm. on pure and pplied Math. 21 (1968)P. A. Clarkson M. J. Ablowitz. Solitons, nonlinear evolution equations and inverse scattering. Cambridge University Press, 1991.M. Toda M. Wadati. “The exact N-soliton solution of the Korteweg-de Vries equation”. In: J. Phys. Soc. Japan 32 (1972)V. A. Marchenko. “Sturm-Liouville Operators and Applications”. In: 1986. url: https://api.semanticscholar.org/CorpusID:118317624R. M. Miura. “Korteweg-de Vries equation and generalization”. In: J. Math. Physics 9.9 (1968)A. C. Newell. Solitons in mathematics and physics. Ed. by Society for Industrial and Applied Mathematics. 1985Russell. “Report on waves”. In: Rept. Fourteenth Meeting of the British Association for the Advancement of Science. Ed. by J. Murray. 1844M. Toda. “Vibration of a chain with a non-linear interaction”. In: J. Phys. Soc. Jpn. 22 (1967)L. D. Faddeev V. E. Zhakarov. “The Korteweg-de Vries equation: a completely integrable Hamiltonian system”. In: Funct. Anal. Appl. 5 (1968)V. V. Sokolov V. G. Drinfeld. “Lie algebras and equations of Korteweg-de Vries type”. In: Journal of Soviet Mathematics 30 (1985)M. Verde. “Asymptotic expansion of phase shifts at high energies”. In: Nuovo cimento 2 (1955)F. W. Warner. Foundations of Differentiable Manifolds and Lie Groups. Vol. 94. Graduate Texts in Mathematics. 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