The geometry of the two-component Camassa–Holm and Degasperis–Procesi equations

Abstract

We use geometric methods to study two natural two-component generalizations of the periodic Camassa–Holm and Degasperis–Procesi equations. We show that these generalizations can be regarded as geodesic equations on the semidirect product of the diffeomorphism group of the circle Diff ( S 1 ) with some space of sufficiently smooth functions on the circle. Our goals are to understand the geometric properties of these two-component systems and to prove local well-posedness in various function spaces. Furthermore, we perform some explicit curvature calculations for the two-component Camassa–Holm equation, giving explicit examples of large subspaces of positive curvature.