Riemannian geometry on the diffeomorphism group of the circle

Arkiv for Matematik

Volumn 45, Issue 2, 2007, Pages 297-325

  Lenells, Jonatan ^a  

^a LUND UNIVERSITY   (Sweden)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 38149115470     PISSN: 00042080     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11512-007-0047-8     Document Type: Article

References (28)

1. Arnold, V., 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. Burgers, J., The Nonlinear Diffusion Equation, Reidel, Dordrecht, 1974.
3. Camassa, R. and Holm, D. D., An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (1993), 1661-1664.
4. Constantin, A., On the Cauchy problem for the periodic Camassa-Holm equation, J. Differential Equations 141 (1997), 218-235.
5. Constantin, A., On the inverse spectral problem for the Camassa-Holm equation, J. Funct. Anal. 155 (1998), 352-363.
6. Constantin, A., Existence of permanent and breaking waves for a shallow water equation: a geometric approach, Ann. Inst. Fourier (Grenoble) 50 (2000), 321-362.
7. Constantin, A. and Escher, J., Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math. 51 (1998), 475-504.
8. Constantin, A. and Escher, J., On the blow-up rate and the blow-up set of breaking waves for a shallow water equation, Math. Z. 233 (2000), 75-91.
9. Constantin, A., Kappeler, T., Kolev, B. and Topalov, P., On geodesic exponential maps of the Virasoro group, Ann. Global Anal. Geom. 31 (2007), 155-180.
10. Constantin, A. and Kolev, B., On the geometric approach to the motion of inertial mechanical systems, J. Phys. A 35 (2002), R51-R79.
11. Constantin, A. and Kolev, B., Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv. 78 (2003), 787-804.
12. Constantin, A. and Kolev, B., Integrability of invariant metrics on the diffeomorphism group of the circle, J. Nonlinear Sci. 16 (2006), 109-122.
13. Constantin, A. and McKean, H. P., A shallow water equation on the circle, Comm. Pure Appl. Math. 52 (1999), 949-982.
14. Ebin, D. G. and Marsden, J., Groups of diffeomorphisms and the notion of an incompressible fluid, Ann. of Math. 92 (1970), 102-163.
15. Hamilton, R. S., The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. 7 (1982), 65-222.
16. Holm, D. D., Kouranbaeva, S., Marsden, J. E., Ratiu, T. and Shkoller, S., A nonlinear analysis of the averaged Euler equations, Arnold Festschrift Vol. 2, Preprint, 1999. arXiv:chao-dyn/9903036.
17. Hunter, J. K. and Saxton, R., Dynamics of director fields, SIAM J. Appl. Math. 51 (1991), 1498-1521.
18. Iorio, R. J. J. and Iorio, V. d. M., Fourier Analysis and Partial Differential Equations, Cambridge Studies in Advanced Mathematics 70, Cambridge University Press, Cambridge, 2001.
19. Keller, H. H., Differential Calculus in Locally Convex Spaces, Lecture Notes in Mathematics 417, Springer, Berlin-Heidelberg, 1974.
20. Kouranbaeva, S., The Camassa-Holm equation as a geodesic flow on the diffeomorphism group, J. Math. Phys. 40 (1999), 857-868.
21. Lang, S., Differential and Riemannian Manifolds, Graduate Texts in Mathematics 160, Springer, Berlin-Heidelberg, 1995.
22. Lenells, J., Traveling wave solutions of the Camassa-Holm equation, J. Differential Equations 217 (2005), 393-430.
23. Marsden, J. E. and Ratiu, T. S., Introduction to Mechanics and Symmetry, Texts in Applied Mathematics 17, Springer, Berlin-Heidelberg, 1999.
24. McKean, H. P., Breakdown of the Camassa-Holm equation, Comm. Pure Appl. Math. 57 (2004), 416-418.
25. Misiołek, G., A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys. 24 (1998), 203-208.
26. Omori, H., Infinite-Dimensional Lie Groups, Translations of Mathematical Monographs 158, American Mathematical Society, Providence, RI, 1997.
27. Ovsienko, V. Y. and Khesin, B. A., The super Korteweg-de Vries equation as an Euler equation, Funktsional. Anal. i Prilozhen. 21:4 (1987), 81-82 (Russian). English transl.: Functional Anal. Appl. 21 (1988), 329-331.
28. 1 metric and mean hydrodynamics, J. Funct. Anal. 160 (1998), 337-365.