Integrable Flows for Starlike Curves in Centroaffine Space

TL;DR

This paper develops integrable flow hierarchies for starlike curves in centroaffine space, revealing connections to the Boussinesq and Kaup-Kuperschmidt hierarchies through differential invariants.

Contribution

It introduces a natural pre-symplectic structure on the space of curves and links the induced evolution equations to well-known integrable hierarchies.

Findings

Connection between curve flows and Boussinesq hierarchy

Restricted flows induce Kaup-Kuperschmidt hierarchy

Establishment of a geometric framework for integrable flows

Abstract

We construct integrable hierarchies of flows for curves in centroaffine ${\mathbb R}^3$ 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 ${\mathbb{RP}}^2$ induces the Kaup-Kuperschmidt hierarchy at the curvature level.