A two-component geodesic equation on a space of constant positive curvature. (English) Zbl 1242.53056
Summary: We propose a new two-component geodesic equation with the unusual property that the underlying space has constant positive curvature. In the special case of one space dimension, the equation reduces to the two-component Hunter-Saxton equation.
MSC:
| 53C30 | Differential geometry of homogeneous manifolds |
| 58D05 | Groups of diffeomorphisms and homeomorphisms as manifolds |
| 53C44 | Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) |
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
References:
| [1] | Arnold, V. I., Sur la géometrie différentielle des groupes de Lie de dimension infinie et ses application à l’hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble), 16, 319-361 (1966) · Zbl 0148.45301 |
| [2] | Ebin, D.; Marsden, J. E., Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. Math., 92, 102-163 (1970) · Zbl 0211.57401 |
| [3] | Vizman, C., Geodesic equations on diffeomorphism groups, SIGMA Symmetry Integrability Geom. Methods Appl., 4, 22 (2008), Paper 030 · Zbl 1153.58006 |
| [4] | B. Khesin, J. Lenells, G. Misiołek, S.C. Preston, Geometry of diffeomorphism groups, complete integrability and geometric statistics, Preprint, http://arxiv.org/abs/1105.0643; B. Khesin, J. Lenells, G. Misiołek, S.C. Preston, Geometry of diffeomorphism groups, complete integrability and geometric statistics, Preprint, http://arxiv.org/abs/1105.0643 · Zbl 1275.58006 |
| [5] | Hunter, J. K.; Saxton, R., Dynamics of director fields, SIAM J. Appl. Math., 51, 1498-1521 (1991) · Zbl 0761.35063 |
| [6] | Constantin, A.; Ivanov, R., On an integrable two-component Camassa-Holm shallow water system, Phys. Lett. A, 372, 7129-7132 (2008) · Zbl 1227.76016 |
| [7] | Wunsch, M., On the Hunter-Saxton system, Discrete Contin. Dyn. Syst. Ser. B, 12, 647-656 (2009) · Zbl 1176.35028 |
| [8] | Khesin, B.; Misiołek, G., Euler equations on homogeneous spaces and Virasoro orbits, Adv. Math., 176, 116-144 (2003) · Zbl 1017.37039 |
| [9] | Lenells, J., The Hunter-Saxton equation describes the geodesic flow on a sphere, J. Geom. Phys., 57, 2049-2064 (2007) · Zbl 1125.35085 |
| [10] | Kohlmann, M., The curvature of semidirect product groups associated with two-component Hunter-Saxton systems, J. Phys. A, 44, 225203, 17 (2011) · Zbl 1217.37069 |
| [11] | Freed, D. S., The geometry of loop groups, J. Differential Geom., 28, 223-276 (1988) · Zbl 0619.58003 |
| [12] | McKean, H. P., Curvature of an \(\infty \)-dimensional manifold related to Hill’s equation, J. Differential Geom., 17, 523-529 (1982) · Zbl 0481.58013 |
| [13] | Lenells, J., Weak geodesic flow and global solutions of the Hunter-Saxton equation, Discrete Contin. Dyn. Syst., 18, 643-656 (2007) · Zbl 1140.35388 |
| [14] | Wunsch, M., Weak geodesic flow on a semidirect product and global solutions to the periodic Hunter-Saxton system, Nonlinear Anal., 74, 4951-4960 (2011) · Zbl 1252.35118 |
| [15] | Escher, J.; Kohlmann, M.; Lenells, J., The geometry of the two-component Camassa-Holm and Degasperis-Procesi equations, J. Geom. Phys., 61, 436-452 (2011) · Zbl 1210.58007 |
| [16] | Kobayashi, S.; Nomizu, K., Foundations of Differential Geometry II (1969), John Wiley & Sons: John Wiley & Sons New York · Zbl 0175.48504 |
| [17] | Rosenberg, S., (The Laplacian on a Riemannian Manifold. The Laplacian on a Riemannian Manifold, London Mathematical Society Student Texts, vol. 31 (1997), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 0868.58074 |
| [18] | Arnold, V. I., (Mathematical Methods of Classical Mechanics. Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, vol. 60 (1989), Springer-Verlag: Springer-Verlag New York) · Zbl 0692.70003 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
