Geometric Aspects of the Periodic μ-Degasperis-Procesi Equation

Abstract

We consider the periodic μDP equation (a modified version of the Degasperis-Procesi equation) as the geodesic flow of a right-invariant affine connection ∇ on the Fréchet Lie group Diff^∞($\mathbb{S}^1$) of all smooth and orientation-preserving diffeomorphisms of the circle $\mathbb{S}^1\,=\,\mathbb{R}/\mathbb{Z}$. On the Lie algebra C^∞($\mathbb{S}^1$) of Diff^∞($\mathbb{S}^1$), this connection is canonically given by the sum of the Lie bracket and a bilinear operator. For smooth initial data, we show the short time existence of a smooth solution of μDP which depends smoothly on time and on the initial data. Furthermore, we prove that the exponential map defined by ∇ is a smooth local diffeomorphism of a neighbourhood of zero in C^∞($\mathbb{S}^1$) onto a neighbourhood of the unit element in Diff^∞($\mathbb{S}^1$). Our results follow from a general approach on non-metric Euler equations on Lie groups, a Banach space approximation of the Fréchet space C^∞($\mathbb{S}^1$), and a sharp spatial regularity result for the geodesic flow.

Mathematics Subject Classification (2000). Primary 53D25; Secondary 37K65.