Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 027, 30 pages      arXiv:2308.10125
Contribution to the Special Issue on Symmetry, Invariants, and their Applications in honor of Peter J. Olver

mKdV-Related Flows for Legendrian Curves in the Pseudohermitian 3-Sphere

Annalisa Calini a, Thomas Ivey a and Emilio Musso b
a) Department of Mathematics, College of Charleston, Charleston, SC 29424, USA
b) Department of Mathematical Sciences, Politecnico di Torino, Italy

Received September 26, 2023, in final form March 13, 2024; Published online April 02, 2024

We investigate geometric evolution equations for Legendrian curves in the 3-sphere which are invariant under the action of the unitary group ${\rm U}(2)$. 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, are 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 which evolve by rigid motions in ${\rm U}(2)$. Generalizations of our results to higher-order evolutions and curves in similar geometries are also discussed.

Key words: mKdV; Legendrian curves; geometric flows; pseudohermitian CR geometry.

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