Integrable systems in three-dimensional Riemannian geometry

Journal of Nonlinear Science

Volumn 12, Issue 2, 2002, Pages 143-167

  Mari Beffa G ^a   Sanders, J A ^b   Wang, Jing Ping ^b  

^a UNIVERSITY OF WISCONSIN MADISON   (United States)

^b VU UNIVERSITY AMSTERDAM   (Netherlands)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0013097341     PISSN: 09388974     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00332-001-0472-y     Document Type: Article

References (28)

1. M. Adler. On a trace functional for formal pseudo-differential operators and the symplectic structure of the Korteweg-de Vries type equations. Invent. Math., 50(3):219-248, 1978/79.
2. L. Bianchi. Applicazioni geometriche del metodo delle approssimazione succesive di Picard. In [Bia53], page 137.
3. L. Bianchi. Lezioni di geometria differenziale, volume 1. Spoerri, 1892.
4. L. Bianchi. Opere. Vol. I, Parte seconda. Edizioni Cremonese della Casa Editrice Perrella, Roma, 1953. A cura dell'Unione Matematica Italiana e col contributo del Consiglio Nazionale delle Ricerche.
5. A. Calini. Recent developments in integrable curve dynamics. In Geometric approaches to differential equations (Canberra, 1995), pages 56-99. Cambridge Univ. Press, Cambridge, 2000.
6. G. Darboux. Leçons sur les systèmes orthogonaux et les coordonnées curvilignes. Gauthier-Villars, Paris, 1910. Reprinted in [Dar93].
7. G. Darboux. Leçons sur les systèmes orthogonaux et les coordonnées curvilignes. Principes de géométrie analytique. Éditions Jacques Gabay, Sceaux, 1993. The first title is a reprint of the second (1910) edition; the second title is a reprint of the 1917 original. Cours de Géométrie de la Faculté des Sciences. [Course on Geometry of the Faculty of Science].
8. I. Dorfman. Dirac structures and integrability of nonlinear evolution equations. John Wiley & Sons Ltd., Chichester, 1993.
9. V. G. Drinfel'd and V. V. Sokolov. Lie algebras and equations of Korteweg-de Vries type. In Current problems in mathematics. Vol. 24, Itogi Nauki i Tekhniki, pages 81-180. Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984.
10. A. Doliwa and P. M. Santini. An elementary geometric characterization of the integrable motions of a curve. Phys. Lett. A, 185(4):373-384, 1994.
11. I. M. Gel'fand and L. A. Dikiǐ. A family of Hamiltonian structures connected with integrable nonlinear differential equations. Akad. Nauk SSSR Inst. Prikl. Mat. Preprint, 136:41, 1978.
12. R. Hasimoto. A soliton on a vortex filament. J. Fluid Mech., 51:477-485, 1972.
13. Noel J. Hicks. Notes on differential geometry. D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965.
14. Thomas A. Ivey. Integrable geometric evolution equations for curves. Technical report, 2001. Contemp. Math., 285:71-84, 2001.
15. 2 = 0. J. Math. Pure Appl., 18:71-72, 1853.
16. J. Langer and R. Perline. Poisson geometry of the filament equation. J. Nonlinear Sci., 1(1):71-93, 1991.
17. J. Langer and R. Perline. The planar filament equation. In Mechanics day (Waterloo, ON, 1992), pages 171-180. Am. Math. Soc., Providence, RI, 1996.
18. F. Magri. A simple model of the integrable Hamiltonian equation. J. Math. Phys., 19(5):1156-1162, 1978.
19. G. Marí Beffa. The theory of differential invariants and KdV Hamiltonian evolutions. Bull. Soc. Math. France, 127(3):363-391, 1999.
20. G. Marí Beffa. Conformal analogue of the Adler-Gel'fand-Dikiǐ bracket in two dimensions. J. Phys. A, 33(25):4689-4707, 2000.
21. G. Marí Beffa. The theory of differential invariance and infinite dimensional Hamiltonian evolutions. In Poisson geometry (Warsaw, 1998), pages 187-196. Polish Acad. Sci., Warsaw, 2000.
22. J. Marsden and A. Weinstein. Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids. Physica D, 7(1-3):305-323, 1983. Order in chaos (Los Alamos, NM, 1982).
23. P. J. Olver. Applications of Lie groups to differential equations, volume 107 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1993.
24. P. Petersen. Riemannian geometry. Springer-Verlag, New York, 1998.
25. J. A. Sanders and J. P. Wang. Integrable systems and their recursion operators. Nonlinear Anal., 47(8):5213-5240, 2001.
26. J. A. Sanders and J. P. Wang. On recursion operators. Physica D, 149(1-2):1-10, 2001.
27. Y. Yasui and N. Sasaki. Differential geometry of the vortex filament equation. J. Geom. Phys., 28(1-2): 195-207, 1998.
28. V. E. Zakharov. Description of the n-orthogonal curvilinear coordinate systems and Hamiltonian integrable systems of hydrodynamic type. I. Integration of the Lamé equations. Duke Math. J., 94(1):103-139, 1998.