Regular Polygonal Vortex Filament Evolution and Exponential Sums

When taking a regular planar polygon of M sides and length 2π as the initial datum of the vortex filament equation, Xt = Xs ∧Xss , the solution becomes polygonal at times of the form tpq = (p/q)(2π/M2), with gcd(p, q) = 1, and the corresponding polygon has Mq sides, if q is odd, and Mq/2 sides, if q...

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Autores: Chamizo, Fernando, De la Hoz Méndez, Francisco
Tipo de recurso: artículo
Fecha de publicación:2024
País:España
Institución:Universidad del País Vasco
Repositorio:Addi. Archivo Digital para la Docencia y la Investigación
OAI Identifier:oai:addi.ehu.eus:10810/70804
Acceso en línea:http://hdl.handle.net/10810/70804
Access Level:acceso abierto
Palabra clave:vortex filament equation
nonlinear Schrödinger equation
rotation matrices
trigonometric sums
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spelling Regular Polygonal Vortex Filament Evolution and Exponential SumsChamizo, FernandoDe la Hoz Méndez, Franciscovortex filament equationnonlinear Schrödinger equationrotation matricestrigonometric sumsWhen taking a regular planar polygon of M sides and length 2π as the initial datum of the vortex filament equation, Xt = Xs ∧Xss , the solution becomes polygonal at times of the form tpq = (p/q)(2π/M2), with gcd(p, q) = 1, and the corresponding polygon has Mq sides, if q is odd, and Mq/2 sides, if q is even. Moreover, that polygon is skew (except when q = 1 or q = 2, where the initial shape is recovered), and the angle ρ between two adjacent sides is a constant. In this paper, we give a rigorous proof of the conjecture that states that, at a time tpq , cosq (ρ/2) = cos(π/M), if q is odd, and cosq (ρ/2) = cos 2(π/M), if q is even. Since the transition of one side of the polygon to the next one is given by a rotation in R3 determined by a generalized Gauss sum, the idea of the proof consists in showing that a certain product of those rotations is a rotation of angle 2π/M, which is equivalent to proving that some exponential sums with arithmetic content are purely imaginary.Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. Fer- nando Chamizo was partially supported by the PID2020-113350GB-I00 grant of the MICIU (Spain) and by “Severo Ochoa Programme for Centres of Excellence in R&D” (CEX2019-000904-S). Francisco de la Hoz was partially supported by the research group grant IT1615-22 funded by the Basque Government, and by the project PID2021-126813NB-I00 funded by MICIU/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”.Springer Nature202420242024info:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/10810/70804reponame:Addi. Archivo Digital para la Docencia y la Investigacióninstname:Universidad del País VascoInglésinfo:eu-repo/grantAgreement/MICINN/PID2020-113350GB-I00/info:eu-repo/grantAgreement/MICINN/CEX2019-000904-S/info:eu-repo/grantAgreement/MICINN/PID2021-126813NB-I00/https://link.springer.com/article/10.1007/s10440-024-00697-4info:eu-repo/semantics/openAccesshttp://creativecommons.org/licenses/by/3.0/es/© The Author(s) 2024. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.Atribución 3.0 Españaoai:addi.ehu.eus:10810/708042026-06-18T09:23:17Z
dc.title.none.fl_str_mv Regular Polygonal Vortex Filament Evolution and Exponential Sums
title Regular Polygonal Vortex Filament Evolution and Exponential Sums
spellingShingle Regular Polygonal Vortex Filament Evolution and Exponential Sums
Chamizo, Fernando
vortex filament equation
nonlinear Schrödinger equation
rotation matrices
trigonometric sums
title_short Regular Polygonal Vortex Filament Evolution and Exponential Sums
title_full Regular Polygonal Vortex Filament Evolution and Exponential Sums
title_fullStr Regular Polygonal Vortex Filament Evolution and Exponential Sums
title_full_unstemmed Regular Polygonal Vortex Filament Evolution and Exponential Sums
title_sort Regular Polygonal Vortex Filament Evolution and Exponential Sums
dc.creator.none.fl_str_mv Chamizo, Fernando
De la Hoz Méndez, Francisco
author Chamizo, Fernando
author_facet Chamizo, Fernando
De la Hoz Méndez, Francisco
author_role author
author2 De la Hoz Méndez, Francisco
author2_role author
dc.subject.none.fl_str_mv vortex filament equation
nonlinear Schrödinger equation
rotation matrices
trigonometric sums
topic vortex filament equation
nonlinear Schrödinger equation
rotation matrices
trigonometric sums
description When taking a regular planar polygon of M sides and length 2π as the initial datum of the vortex filament equation, Xt = Xs ∧Xss , the solution becomes polygonal at times of the form tpq = (p/q)(2π/M2), with gcd(p, q) = 1, and the corresponding polygon has Mq sides, if q is odd, and Mq/2 sides, if q is even. Moreover, that polygon is skew (except when q = 1 or q = 2, where the initial shape is recovered), and the angle ρ between two adjacent sides is a constant. In this paper, we give a rigorous proof of the conjecture that states that, at a time tpq , cosq (ρ/2) = cos(π/M), if q is odd, and cosq (ρ/2) = cos 2(π/M), if q is even. Since the transition of one side of the polygon to the next one is given by a rotation in R3 determined by a generalized Gauss sum, the idea of the proof consists in showing that a certain product of those rotations is a rotation of angle 2π/M, which is equivalent to proving that some exponential sums with arithmetic content are purely imaginary.
publishDate 2024
dc.date.none.fl_str_mv 2024
2024
2024
dc.type.none.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv http://hdl.handle.net/10810/70804
url http://hdl.handle.net/10810/70804
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv info:eu-repo/grantAgreement/MICINN/PID2020-113350GB-I00/
info:eu-repo/grantAgreement/MICINN/CEX2019-000904-S/
info:eu-repo/grantAgreement/MICINN/PID2021-126813NB-I00/
https://link.springer.com/article/10.1007/s10440-024-00697-4
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by/3.0/es/
Atribución 3.0 España
eu_rights_str_mv openAccess
rights_invalid_str_mv http://creativecommons.org/licenses/by/3.0/es/
Atribución 3.0 España
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Springer Nature
publisher.none.fl_str_mv Springer Nature
dc.source.none.fl_str_mv reponame:Addi. Archivo Digital para la Docencia y la Investigación
instname:Universidad del País Vasco
instname_str Universidad del País Vasco
reponame_str Addi. Archivo Digital para la Docencia y la Investigación
collection Addi. Archivo Digital para la Docencia y la Investigación
repository.name.fl_str_mv
repository.mail.fl_str_mv
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