The Degasperis-Procesi equation as a non-metric Euler equation

TL;DR

This paper provides a geometric interpretation of the Degasperis-Procesi equation as a non-metric Euler equation, revealing new insights into its structure and regularity properties.

Contribution

It introduces a geometric framework for the Degasperis-Procesi equation as a geodesic flow on the diffeomorphism group, and analyzes solution regularity for a family of b-equations.

Findings

Degasperis-Procesi equation as a geodesic flow on diffeomorphism group

No gain or loss of spatial regularity in solutions of b-equations

Degasperis-Procesi and Camassa-Holm equations as ODEs on smooth function space

Abstract

In this paper we present a geometric interpretation of the periodic Degasperis-Procesi equation as the geodesic flow of a right invariant symmetric linear connection on the diffeomorphism group of the circle. We also show that for any evolution in the family of $b$-equations there is neither gain nor loss of the spatial regularity of solutions. This in turn allows us to view the Degasperis-Procesi and the Camassa-Holm equation as an ODE on the Fr\'echet space of all smooth functions on the circle.