Central Affine Curve Flow on the Plane

TL;DR

This paper provides a comprehensive analysis of Pinkall's central affine curve flow on the plane, including new methods for constructing integrable structures, transformations, explicit solutions, and solving the Cauchy problem for periodic data.

Contribution

It introduces a systematic approach to construct higher flows, conservation laws, and bi-Hamiltonian structures, along with Baecklund transformations, explicit solutions, and solutions to the Cauchy problem.

Findings

Constructed higher commuting flows and conservation laws.

Developed Baecklund transformations and permutability formulas.

Provided explicit solutions and solved the Cauchy problem for periodic data.

Abstract

We give the following results for Pinkall's central affine curve flow on the plane: (i) a systematic and simple way to construct the known higher commuting curve flows, conservation laws, and a bi-Hamiltonian structure, (ii) Baecklund transformations and a permutability formula, (iii) infinitely many families of explicit solutions. We also solve the Cauchy problem for periodic initial data.