Geometric aspects of the periodic $\mu$-Degasperis-Procesi equation

Abstract

We consider the periodic $\muDP$ equation (a modified version of the Degasperis-Procesi equation) as the geodesic flow of a right-invariant affine connection $\nabla$ on the Fr\'echet Lie group $\Diff^{\infty}(\S^1)$ of all smooth and orientation-preserving diffeomorphisms of the circle $\S^1=\R/\Z$. On the Lie algebra $\C^{\infty}(\S^1)$ of $\Diff^{\infty}(\S^1)$, 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 $\muDP$ which depends smoothly on time and on the initial data. Furthermore, we prove that the exponential map defined by $\nabla$ is a smooth local diffeomorphism of a neighbourhood of zero in $\C^{\infty}(\S^1)$ onto a neighbourhood of the unit element in $\Diff^{\infty}(\S^1)$. Our results follow from a general approach on non-metric Euler equations on Lie groups, a Banach space approximation of the Fr\'echet space $\C^{\infty}(\S^1)$, and a sharp spatial regularity result for the geodesic flow.