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

JOURNAL OF GEOMETRY AND PHYSICS

卷:61

Article

Escher, J.¹  Kohlmann, M.¹  Lenells, J.² 

[1] Leibniz Univ Hannover, Inst Appl Math, D-30167 Hannover, Germany

[2] Baylor Univ, Dept Math, Waco, TX 76798 USA

关键词: Camassa-Holm equation;    Degasperis-Procesi equation;    Semidirect product;    Geodesic flow;    Sectional curvature;   

DOI  :  10.1016/j.geomphys.2010.10.011

来源: Elsevier

【 摘 要 】

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. (C) 2010 Elsevier B.V. All rights reserved.

【 授权许可】

Free