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.

31 pages

References in corpus (4)

• On an integrable two-component Camassa-Holm shallow water system
• A New Integrable Equation with Peakon Solutions
• Geodesic flows on semidirect-product Lie groups: geometry of singular measure-valued solutions
• Geodesic Flow and Two (Super) Component Analog of the Camassa-Holm Equation

Cited by in corpus (4)

• The curvature of semidirect product groups associated with two-component Hunter-Saxton systems
• Variational derivation of two-component Camassa-Holm shallow water system
• A semi-discrete scheme derived from variational principles for global conservative solutions of a Camassa-Holm system
• Blow-up phenomena for the rotation-two-component Camassa-Holm system