Euler equations on homogeneous spaces and Virasoro orbits

Abstract

We show that the following three systems related to various hydrodynamical approximations: the Korteweg-de Vries equation, the Camassa-Holm equation, and the Hunter-Saxton equation, have the same symmetry group and similar bihamiltonian structures. It turns out that their configuration space is the Virasoro group and all three dynamical systems can be regarded as equations of the geodesic flow associated to different right-invariant metrics on this group or on appropriate homogeneous spaces. In particular, we describe how Arnold's approach to the Euler equations as geodesic flows of one-sided invariant metrics extends from Lie groups to homogeneous spaces.We also show that the above three cases describe all generic bihamiltonian systems which are related to the Virasoro group and can be integrated by the translation argument principle: they correspond precisely to the three different types of generic Virasoro orbits. Finally, we discuss interrelation between the above metrics and Kahler structures on Virasoro orbits as well as open questions regarding integrable systems corresponding to a finer classification of the orbits.

Overview of the Geometries of Shape Spaces and Diffeomorphism Groups

“…Its geodesic nature was discovered in [72]. It is also a completely integrable, bihamiltonian equation with an infinite number of conservation laws [61].…”

Global Existence for the Generalized Two-Component Hunter–Saxton System

“…It was noted by Constantin & Ivanov [10] that the Hunter-Saxton system allows for peakon solutions; moreover, Lenells & Lechtenfeld [28] showed that it can be interpreted as the Euler equation on the superconformal algebra of contact vector fields on the 1 2-dimensional supercircle, which is in accordance with the by now well-known geometric interpretation of the Hunter-Saxton equation as the geodesic flow of the right-invariantḢ 1 (S) metric on the space of orientationpreserving circle diffeomorphisms modulo rigid rotations [25,[27][28][29] (see also [11-13, 30, 31] for related geodesic flow equations).…”

Geometric Differences between the Burgers and the Camassa-Holm Equations

“…A similar approach can be pursued for the general Camassa-Holm equation u t − u txx + 3uu x − 2u x u xx − uu xxx + cu xxx = 0, c ∈ R which can be obtained as the Euler equation for the H 1 right-invariant metric on the Virasoro group [21]. For the Virasoro group, short-time existence of the geodesic flow for the H k metric was established in [9] using the approach given in Section 4.…”

“…Other equations from mathematical physics were found to have an interpretation as geodesic flows on diffeomorphism groups (see for example [21,22,31,32]). …”