1998/07/22 by Shinar Kouranbaeva · 10 citations
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Geometry and complex manifolds #Nonlinear Waves and Solitons #math-ph #math.GR #math.MP #msc:57R57 #msc:58B99
paper · pdf · doi:10.1063/1.532690
10 single-spaced pages, Geometric Methods in Fluid Equations: Submitted to the Journal of Mathematical Physics
arxiv created 1998/07/22 · openalex publication_date 1999/02/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Misiolek [J. Geom. Phys. 24, 203–208 (1998)] has shown that the Camassa–Holm equation is a geodesic flow on the Bott–Virasoro group. In this paper it is shown that the Camassa–Holm equation for the case κ=0 is the geodesic spray of the weak Riemannian metric on the diffeomorphism group of the line or the circle obtained by right translating the H1 inner product over the entire group. This paper uses the right-trivialization technique to rigorously verify that the Euler–Poincaré theory for Lie groups can be applied to diffeomorphism groups. The observation made in this paper has led to physically meaningful generalizations of the CH-equation to higher dimensional manifolds.