#### Document Type

Conference Paper

#### Rights

This item is available under a Creative Commons License for non-commercial use only

#### Disciplines

Pure mathematics, Applied mathematics

#### Abstract

A class of equations describing the geodesic flow for a right-invariant metric on the group of diffeomorphisms of Rn is reviewed from the viewpoint of their Lie-Poisson structures. A subclass of these equations is analogous to the Euler equations in hydrodynamics (for n = 3), preserving the volume element of the domain of fluid flow. An example in n = 1 dimension is the Camassa-Holm equation, which is a geodesic flow equation on the group of diffeomorphisms, preserving the H1 metric.

#### Recommended Citation

R. Ivanov, Poisson Structures of Equations associated with groups of diffeomorphisms, in: Trends in Differential geometry, Complex analysis and Mathematical Physics (ed. K. Sekigawa et al.), World Scientific, 2009, pp 99 – 108.

#### Included in

Geometry and Topology Commons, Non-linear Dynamics Commons, Partial Differential Equations Commons

## Publication Details

Trends in Differential geometry, Complex analysis and Mathematical Physics (ed. K. Sekigawa et al.), World Scientific, 2009