The Degasperis–Procesi equation as a non-metric Euler equation

2009/08/04 by Joachim Escher, Boris Kolev
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Advanced Differential Geometry Research #Connection (principal bundle) #Diffeomorphism #Euler equations #Flow (mathematics) #Functional equation #Geodesic #Interpretation (philosophy) #Nonlinear Waves and Solitons #Ode #Riccati equation #math-ph #math.MP #msc:35Q53 #msc:58D05

paper · pdf · doi:10.1007/s00209-010-0778-2

published as Mathematische Zeitschrift 269, 3 (2011) pp 1137-1153 · 17 pages

arxiv created 2009/08/04 · openalex publication_date 2010/09/27 · arxiv updated 2012/01/06 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

In this paper we present a geometric interpretation of the periodic Degasperis-Procesi equation as the geodesic flow of a right invariant symmetric linear connection on the diffeomorphism group of the circle. We also show that for any evolution in the family of b-equations there is neither gain nor loss of the spatial regularity of solutions. This in turn allows us to view the Degasperis-Procesi and the Camassa-Holm equation as an ODE on the Fréchet space of all smooth functions on the circle.

Citations

• Inverse Scattering Transform for the Camassa-Holm equation
• The inverse function theorem of Nash and Moser
• Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation
• The Camassa–Holm equation as a geodesic flow on the diffeomorphism group
• Wave breaking for nonlinear nonlocal shallow water equations
• The Hydrodynamical Relevance of the Camassa–Holm and Degasperis–Procesi Equations
• Water waves and integrability