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

Joachim Escher (1) , Boris Kolev (2)

1 IFAM - Institute for Applied Mathematics
2 LATP - Laboratoire d'Analyse, Topologie, Probabilités

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.

Mots clés: Euler equation; diffeomorphisms group of the circle; Degasperis-Procesi equation

Domaines: Physique mathématique [math-ph]; Physique mathématique [math-ph]

Joachim Escher, Boris Kolev. The Degasperis-Procesi equation as a non-metric Euler equation. Mathematische Zeitschrift, 2011, 269 (3), pp 1137-1153. ⟨10.1007/s00209-010-0778-2⟩. ⟨hal-00409022⟩