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

SIGMA 9 (2013), 022, 21 pages      arXiv:1303.1259      https://doi.org/10.3842/SIGMA.2013.022
Contribution to the Special Issue “Symmetries of Differential Equations: Frames, Invariants and Applications”

Integrable Flows for Starlike Curves in Centroaffine Space

Annalisa Calini a, b, Thomas Ivey a and Gloria Marí Beffa c
a) College of Charleston, Charleston SC, USA
b) National Science Foundation, Arlington VA, USA
c) University of Wisconsin, Madison WI, USA

Received September 07, 2012, in final form February 27, 2013; Published online March 06, 2013

Abstract
We construct integrable hierarchies of flows for curves in centroaffine R3 through a natural pre-symplectic structure on the space of closed unparametrized starlike curves. We show that the induced evolution equations for the differential invariants are closely connected with the Boussinesq hierarchy, and prove that the restricted hierarchy of flows on curves that project to conics in RP2 induces the Kaup-Kuperschmidt hierarchy at the curvature level.

Key words: integrable curve evolutions; centroaffine geometry; Boussinesq hierarchy; bi-Hamiltonian systems.

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