Integrable vs. nonintegrable geodesic soliton behavior

Physica D: Nonlinear Phenomena

Volumn 150, Issue 3-4, 2001, Pages 237-263

  Fringer, Oliver B ^a,b   Holm, Darryl D ^b  

^a Stanford University   (United States)

^b CENTER FOR NONLINEAR STUDIES   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ALGEBRA; ASYMPTOTIC STABILITY; COMPUTER SIMULATION; DEGREES OF FREEDOM (MECHANICS); GEODESY; HAMILTONIANS; MATHEMATICAL MODELS; PARTIAL DIFFERENTIAL EQUATIONS;

EULER-POINCARE EQUATIONS; GEODESIC MOTIONS;

SOLITONS;

EID: 0035311815     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0167-2789(00)00215-3     Document Type: Article

References (25)

1. Poincaré H. Sue une forme nouvelle des équations de la méhanique. C.R. Acad. Sci., Paris. 132:1901;369-371.
2. Arnold V.I. Sur la géométrie differentielle des groupes de Lie de dimenson infinie et ses applications à l'hydrodynamique des fluids parfaits. Ann. Inst. Fourier, Grenoble. 16:1966;319-361.
3. J.E. Marsden, T.S. Ratiu, Introduction to Mechanics and Symmetry, Texts in Applied Mathematics, Vol. 17, Springer, New York, 1994.
4. Holm D.D., Marsden J.E., Ratiu T.S. Euler-Poincaré equations and semidirect products with applications to continuum theories. Adv. Math. 137:1998;1-81.
5. Ovsienko V.Y., Khesin B.A. Korteweg-de Vries superequations as an Euler equation. Funct. Anal. Appl. 21:1987;329-331.
6. Segal G. The geometry of the KdV equation. Int. J. Mod. Phys. A. 132:1991;2859-2869.
7. Misiolek G. A shallow water equation as a geodesic flow on the Bott-Virosoro group. J. Geom. Phys. 24:1998;203-208.
8. Camassa R., Holm D.D. An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71:1993;1661-1664.
9. R. Camassa, D.D. Holm, J.M. Hyman, A new integrable shallow water equation, Advances in Applied Mechanics, Vol. 31, Academic Press, Boston, 1994, pp. 1-33.
10. Holm D.D., Marsden J.E., Ratiu T.S. Euler-Poincaré models of ideal fluids with nonlinear dispersion. Phys. Rev. Lett. 80:1998;4173-4177.
11. 1 metric and mean hydrodynamics, J. Funct. Anal. 160 (1998) 337-365.
12. D.D. Holm, S. Kouranbaeva, J.E. Marsden, T.S. Ratiu, S. Shkoller, A nonlinear analysis of the averaged Euler equations, Fields Inst. Comm., Arnold, Vol. 2, American Mathematical Society, Providence, RI, 1998, in press.
13. Brenier Y. The least action principle and the related concept of generalized flows for incompressible perfect fluids. J. Am. Math. Soc. 2:1989;225-255.
14. Shnirelman A.I. Generalized fluid flows, their approximation and applications. Geom. Funct. Anal. 4:1994;586-620.
15. D.D. Holm, J.E. Marsden, S. Shkoller, Generalized flows of shallow water waves, in preparation.
16. Hunter J., Zheng Y. On a completely integrable nonlinear hyperbolic variational equation. Physica D. 79:1994;361-386.
17. Alber M., Camassa R., Holm D.D., Marsden J.E. The geometry of peaked solitons and billiard solutions of a class of integrable PDEs. Lett. Math. Phys. 32:1994;137-151.
18. Alber M., Camassa R., Holm D.D., Marsden J.E. On the link between umbilic geodesics and soliton solutions of nonlinear PDEs. Proc. R. Soc. 450:1995;677-692.
19. A. Constantin, H. McKean, A shallow water equation on the circle, Commun. Pure Appl. Math. 52 (1999) 949-982.
20. Velan M.S., Lakshmanan M. Lie symmetries and infinite-dimensional Lie-algebras of certain. (1+1) -dimensional nonlinear evolution-equations J. Phys. A. 30:1997;3261-3271.
21. Calogero F. An integrable Hamiltonian system. Phys. Lett. A. 201:1995;306-310.
22. Calogero F., Francoise J.P. A completely integrable Hamiltonian system. J. Math. Phys. 37:1996;2863-2871.
23. J.M. Hyman, Personal communication.
24. R. Beales, D.H. Sattinger, J. Szmigielski, Multipeakons and the Classical Moment Problem, Adv. Math. 154 (2000) 229-257.
25. Constantin A. On the inverse spectral problem for the Camassa-Holm equation. J. Funct. Anal. 155:1998;352-363.