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
Date:2024
Main Authors: Calini, Annalisa, Ivey, Thomas, Musso, Emilio
Format: Article
Language:English
Published: Інститут математики НАН України 2024
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
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Summary: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.
ISSN:1815-0659