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.

arXiv comment31 pages

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• Variational derivation of two-component Camassa-Holm shallow water system
• On a two-component $\pi$-Camassa--Holm system
• Real-valued algebro-geometric solutions of the two-component Camassa-Holm hierarchy
• A note on multi-dimensional Camassa-Holm type systems on the torus
• Lipschitz metric for the two-component Camassa--Holm system
• A semi-discrete scheme derived from variational principles for global conservative solutions of a Camassa-Holm system
• Neumann Type Integrable Reduction to the Negative-Order Coupled Harry--Dym Hierarchy
• The two-dimensional periodic $b$-equation on the diffeomorphism group of the torus
• A two-component geodesic equation on a space of constant positive curvature
• Blow-up phenomena for the rotation-two-component Camassa-Holm system
• Cauchy problem for a generalized cross-coupled Camassa-Holm system with waltzing peakons and higher-order nonlinearities
• Liouville correspondences between multi-component integrable hierarchies