A shallow water equation as a geodesic flow on the Bott-Virasoro group

1998/02/01 by Gerard Misio łek · 20 citations
Physics and Astronomy · Mathematics · #Nonlinear Waves and Solitons #Geometry and complex manifolds #Black Holes and Theoretical Physics #Mathematics #Geodesic flow #Geodesic #Invariant (physics) #Curvature #Metric (unit) #Group (periodic table) #Yamabe flow #Geodesic map #Pure mathematics #Sectional curvature #Mathematical analysis #Virasoro algebra #Flow (mathematics) #Conformal map #Mathematical physics #Scalar curvature #Algebra over a field #Geometry #Physics

paper · pdf · doi:10.1016/s0393-0440(97)00010-7

openalex publication_date 1998/02/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/08

Abstract

The Camassa-Holm equation is shown to give rise to a geodesic flow of a certain right invariant metric on the Bott-Virasoro group. The sectional curvature of this metric is computed and shown to assume positive and negative signs.

Citations

• Comparison Theorems in Riemannian Geometry
• A New Integrable Shallow Water Equation

Cited by

• Traveling wave solutions of the Camassa–Holm equation
• Euler equations on homogeneous spaces and Virasoro orbits
• Orbital stability of solitary waves for a shallow water equation
• The Camassa–Holm equation for water waves moving over a shear flow
• Finite propagation speed for the Camassa–Holm equation
• Conservation laws of the Camassa–Holm equation
• Low-Regularity Solutions of the Periodic Camassa–Holm Equation
• On the scattering problem for the Camassa-Holm equation
• Evolution of the Scattering Coefficients of the Camassa–Holm Equation, for General Initial Data
• A Convergent Finite Difference Scheme for the Camassa–Holm Equation with General H¹ Initial Data
• Global existence of weak solutions to the Camassa-Holm equation
• Existence time for the Camassa-Holm equation and the critical Sobolev index
• Existence of permanent and breaking waves for a shallow water equation: a geometric approach
• A Variational Approach to the Stability of Periodic Peakons
• Periodic conservative solutions of the Camassa–Holm equation
• Stability for the periodic Camassa-Holm equation
• Variational derivation of the Camassa-Holm shallow water equation with non-zero vorticity
• About the Explicit Characterization of Hamiltonians of the Camassa-Holm Hierarchy
• The Hunter–Saxton equation describes the geodesic flow on a sphere
• Camassa–Holm equation [wikipedia]