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arXiv · 1009.0188

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

Joachim Escher·Martin Kohlmann·Jonatan Lenells
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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.

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Joachim Escher, Martin Kohlmann, Jonatan Lenells. 2010-09-01. The geometry of the two-component Camassa-Holm and Degasperis-Procesi equations. https://doi.org/10.1016/j.geomphys.2010.10.011

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