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The geometry of the two-component Camassa-Holm and Degasperis-Procesi equations. (English) Zbl 1210.58007

Summary: 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.

MSC:

58D05 Groups of diffeomorphisms and homeomorphisms as manifolds
70E45 Higher-dimensional generalizations in rigid body dynamics
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