Euler equations on homogeneous spaces and Virasoro orbits

Advances in Mathematics

Volumn 176, Issue 1, 2003, Pages 116-144

  Khesin, Boris ^a   Misiołek, Gerard ^b  

^a UNIVERSITY OF TORONTO   (Canada)

^b UNIVERSITY OF NOTRE DAME   (United States)

Author keywords

Bi Hamiltonian structures; Euler equations; Geodesic flows; Virasoro orbits

Indexed keywords

EID: 0037612018     PISSN: 00018708     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0001-8708(02)00063-4     Document Type: Article

References (28)

1. V. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble) 16 (1966) 319-361;
2. Mathematical Methods of Classical Mechanics, Springer, Berlin, New York, 1989, xvi + 508pp.
3. V. Arnold, B. Khesin, Topological Methods in Hydrodynamics, Springer, New York, 1998, xv + 376pp.
4. R. Beals, D.H. Sattinger, J. Szrmigielski, Acoustic scattering and the extended Korteweg-de Vries hierarchy, Adv. in Math. 140 (1998) 190-206;
5. Inverse scattering solutions of the Hunter-Saxton equations, preprint, 2001.
6. Y. Brenier, Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations, Comm. Pure Appl. Math. 52 (4) (1999) 411-452.
7. R. Camassa, D. Holm, An integrable shallow water equation with peaked solutions, Phys. Lett. Rev. 71 (1993) 1661-1664.
8. A. Fokas, B. Fuchssteiner, Symplectic structures, their Bäklund transformations and hereditary symmetries, Physica D 4 (1981/82) 47-66.
9. 1, Funct. Anal. Appl. 23 (1989) 85-93.
10. L. Guieu, C. Roger, Algebre et groupe de Virasoro; aspects geometriques et algebriques, in: Cours Specialises, Societe Mathematique de France, to appear.
11. V. Guillemin, S. Sternberg, On collective complete integrability according to the method of Thimm, Ergodic Theory Dyn. Systems 3 (2) (1983) 219-230.
12. J.K. Hunter, R. Saxton, Dynamics of director fields, SIAM J. Appl. Math. 51 (6) (1991) 1498-1521.
13. J. Hunter, Y. Zheng, On a completely integrable nonlinear hyperbolic variational equation, Physica D 79 (1994) 361-386.
14. A. Kirillov, The orbits of the group of diffeomorphisms of the circle, and local Lie superalgebras, Funct. Anal. Appl. 15 (1981) 135-136;
15. The orbit method, II: infinite-dimensional Lie groups and Lie algebras, Contemp. Math, 145 (1993) 33-63.
16. A. Kirillov, Kahler structure on the K-orbits of a group of diffeomorphisms of the circle, Funct. Anal. Appl. 21 (2) (1987) 42-45.
17. M. Kruskal, Nonlinear wave equations, in: Dynamical Systems, Theory and Applications, Lecture Notes in Phys., Vol. 38, Springer, Berlin, 1975, pp. 310-354.
18. S. Manakov, A note on the integration of Euler's equations of the dynamics of an n-dimensional rigid body, Funct. Anal. Appl. 10 (1976) 93-94.
19. F. Magri, A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19 (1978) 1156-1162.
20. G. Misiołek, Classical solutions of the periodic Camassa-Holm equation, Geom. Funct. Anal. 12 (2002) 1080-1104.
21. V. Ovsienko, On the Denogardus great number and Hooke's law, Kvant 8 (1989) 8-16; Kvant selecta: Algebra and Analysis, II, Math. World, vol. 15, Amer. Math. Soc., Providence, RI, 1999, pp. 153-159.
22. V. Ovsienko, B. Khesin, The (super) KdV equation as an Euler equation, Funct. Anal. Appl. 21 (4) (1987) 81-82.
23. V. Ovsienko, B. Khesin, Symplectic leaves of the Gelfand-Dikii brackets and homotopy classes of non-degenerate curves, Funct. Anal. Appl. 24 (1) (1990) 33-40.
24. G. Segal, Unitary representations of some infinite-dimensional groups, Comm. Math. Phys. 80 (1981) 301-342;
25. The geometry of the KdV equation, in: Topological Methods in Quantum Field Theory (Trieste, 1990), or Internat. J. Modern Phys. A 6(16) (1991), 2859-2869.
26. A. Thimm, Integrable geodesic flows on homogeneous spaces, Ergodic Theory Dyn. Systems 1 (1981) 491-517.
27. E. Witten, Coadjoint orbits of the Virasoro group, Comm. Math. Phys. 114 (1988) 1-53.
28. I. Zakharevich, Manuscript, in preparation.