THE CAMASSA-HOLM EQUATION AS A GEODESIC FLOW FOR THE H¹ RIGHT-INVARIANT METRIC

Abstract

The fundamental role played by the Lie groups in mechanics, and especially by the dual space of the Lie algebra of the group and the coadjoint action are illustrated through the Camassa-Holm equation (CH). In 1996 Misio lek observed that CH is a geodesic flow equation on the group of diffeomorphisms, preserving the H 1 metric. This example is analogous to the Euler equations in hydrodynamics, which describe geodesic flow for a right-invariant metric on the infinite-dimensional group of diffeomorphisms preserving the volume element of the domain of fluid flow and to the Euler equations of rigid body whith a fixed point, describing geodesics for a left-invariant metric on SO(3).The momentum map and an explicit parametrization of the Virasoro group, related to recently obtained solutions for the CH equation are presented.

Conformal and Geometric Properties of the Camassa-Holm Hierarchy

“…∞ 0 e −|g(ξ,t)−g(ξ,t)| p(ξ, t)dξ − ω, ġ(ξ, t) = u(g(ξ, t), t), therefore g(x, t) in (68) is the diffeomorphism (Virasoro group element) in the purely solitonic case [12]. The situation when the condition q(x, 0) ≡ m(x, 0) + ω > 0 on the initial data does not hold is more complicated and requires separate analysis [36] (if m(x, 0)+ω changes sign there are infinitely many positive eigenvalues accumulating at infinity and singularities might appear in finite time [8,7,6]).…”

Equations of the Camassa-Holm hierarchy

“…The CH solitary waves are stable solitons if ω > 0 [6,12,13,21] or peakons if ω = 0 [1,22,23]. The KdV and CH equations can also be interpreted as geodesic flow equations for the respective L 2 and H 1 metrics on the Bott-Virasoro group [24,25,26,27,28,29].…”