New Integrable Curve Flows in the Pseudoconformal 3-Sphere

Abstract: The pseudoconformal 3-sphere $S^3$ is the projectivization of the null cone in $\mathbb C^3$ with the standard pseudo-Hermitian inner product. The Lie group $SU(2,1)$ fixing this metric naturally acts on the sphere, preserving a contact structure, and can be identified with the pseudoconformal frame bundle of $S^3$. By normalizing lifts to the frame bundle, we define scalar geometric invariants for Legendrian curves (L-curves) in $S^3$, and for curves transverse to the contact planes (T-curves). We seek invariant geometric flows for these parametrized curves that induce integrable evolution systems for the invariants. While there is an infinite sequence of geometric flows for L-curves inducing the Boussinesq hierarchy, for T-curves there is another infinite sequence of flows that induces a sequence of 3-component evolution systems for the invariants, evidently a novel integrable bi-Hamiltonian hierarchy. This closely resembles the NLS hierarchy, itself realized by a sequence of curve flows in Euclidean 3-space, including the vortex filament equation. We discuss some common features of these hierarchies, describe the geometry and dynamics of travelling wave solutions (also arising as critical curves for Lagrangians derived from the conserved densities) and conclude with some open questions.

differential geometry

Audience: researchers in the topic

Differential Geometry Seminar Torino

Series comments: This is a hybrid seminar organized by the Differential Geometry groups of Università and Politecnico di Torino.

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