mKdV-Related Flows for Legendrian Curves in the Pseudohermitian 3-Sphere
We investigate geometric evolution equations for Legendrian curves in the 3-sphere, which are invariant under the action of the unitary group U(2). We define a natural symplectic structure on the space of Legendrian loops and show that the modified Korteweg-de Vries equation, along with its associat...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2024 |
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2024
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/212168 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | mKdV-Related Flows for Legendrian Curves in the Pseudohermitian 3-Sphere. Annalisa Calini, Thomas Ivey and Emilio Musso. SIGMA 20 (2024), 027, 30 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862543428952260608 |
|---|---|
| author | Calini, Annalisa Ivey, Thomas Musso, Emilio |
| author_facet | Calini, Annalisa Ivey, Thomas Musso, Emilio |
| citation_txt | mKdV-Related Flows for Legendrian Curves in the Pseudohermitian 3-Sphere. Annalisa Calini, Thomas Ivey and Emilio Musso. SIGMA 20 (2024), 027, 30 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We investigate geometric evolution equations for Legendrian curves in the 3-sphere, which are invariant under the action of the unitary group U(2). We define a natural symplectic structure on the space of Legendrian loops and show that the modified Korteweg-de Vries equation, along with its associated hierarchy, is realized as curvature evolutions induced by a sequence of Hamiltonian flows. For the flow among these that induces the mKdV equation, we investigate the geometry of solutions that evolve by rigid motions in U(2). Generalizations of our results to higher-order evolutions and curves in similar geometries are also discussed.
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| first_indexed | 2026-03-12T21:57:31Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-212168 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-12T21:57:31Z |
| publishDate | 2024 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Calini, Annalisa Ivey, Thomas Musso, Emilio 2026-01-30T08:17:04Z 2024 mKdV-Related Flows for Legendrian Curves in the Pseudohermitian 3-Sphere. Annalisa Calini, Thomas Ivey and Emilio Musso. SIGMA 20 (2024), 027, 30 pages 1815-0659 2020 Mathematics Subject Classification: 37K10; 37K25; 53E99; 57K33 arXiv:2308.10125 https://nasplib.isofts.kiev.ua/handle/123456789/212168 https://doi.org/10.3842/SIGMA.2024.027 We investigate geometric evolution equations for Legendrian curves in the 3-sphere, which are invariant under the action of the unitary group U(2). We define a natural symplectic structure on the space of Legendrian loops and show that the modified Korteweg-de Vries equation, along with its associated hierarchy, is realized as curvature evolutions induced by a sequence of Hamiltonian flows. For the flow among these that induces the mKdV equation, we investigate the geometry of solutions that evolve by rigid motions in U(2). Generalizations of our results to higher-order evolutions and curves in similar geometries are also discussed. We thank Cornelia Vizman for very helpful discussions of the hat calculus and the reviewers for their useful comments and suggestions. The authors gratefully acknowledge the support and warm hospitality of the Department of Mathematics at the Politecnico di Torino. A. Calini would also like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme Dispersive hydrodynamics: mathematics, simulation and experiments, with applications in nonlinear waves, when part of this work was undertaken. This work was partially supported by PRIN 2017 Real and Complex Manifolds: Topology, Geometry and holomorphic dynamics (protocollo 2017JZ2SW5-004); by the Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni within the Istituto Nazionale di Alta Matematica “Francesco Severi” (INdAM); and by EPSRC grant number EP/R014604/1. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications mKdV-Related Flows for Legendrian Curves in the Pseudohermitian 3-Sphere Article published earlier |
| spellingShingle | mKdV-Related Flows for Legendrian Curves in the Pseudohermitian 3-Sphere Calini, Annalisa Ivey, Thomas Musso, Emilio |
| title | mKdV-Related Flows for Legendrian Curves in the Pseudohermitian 3-Sphere |
| title_full | mKdV-Related Flows for Legendrian Curves in the Pseudohermitian 3-Sphere |
| title_fullStr | mKdV-Related Flows for Legendrian Curves in the Pseudohermitian 3-Sphere |
| title_full_unstemmed | mKdV-Related Flows for Legendrian Curves in the Pseudohermitian 3-Sphere |
| title_short | mKdV-Related Flows for Legendrian Curves in the Pseudohermitian 3-Sphere |
| title_sort | mkdv-related flows for legendrian curves in the pseudohermitian 3-sphere |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212168 |
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