Reduction of bi-Hamiltonian systems and separation of variables: An example from the Boussinesq hierarchy

Theoretical and Mathematical Physics

Volumn 122, Issue 2, 2000, Pages 176-192

  Falqui, G ^a   Magri, F ^b   Tondo, G ^c  

^a SISSA   (Italy)

^b UNIVERSITY OF MILAN   (Italy)

^c UNIVERSITY OF TRIESTE   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034348926     PISSN: 00405779     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF02551195     Document Type: Article

References (29)

1. 1. F. Magri and T. Marsico, "Some developments of the concepts of Poisson manifolds in the sense of A. Lichnerowicz," in: Gravitation, Electromagnetism, and Geometric Structures (G. Ferrarese, ed.), Pitagora Editrice, Bologna (1996), pp. 207-222; T. Marsico, "Una caratterizzazione geometrica dei sistemi che ammettono rappresentazione alla Lax estesa," Doctoral diss., Università di Milano, Milano (1996).
2. 1. F. Magri and T. Marsico, "Some developments of the concepts of Poisson manifolds in the sense of A. Lichnerowicz," in: Gravitation, Electromagnetism, and Geometric Structures (G. Ferrarese, ed.), Pitagora Editrice, Bologna (1996), pp. 207-222; T. Marsico, "Una caratterizzazione geometrica dei sistemi che ammettono rappresentazione alla Lax estesa," Doctoral diss., Università di Milano, Milano (1996).
3. 2. C. Morosi and G. Tondo, J. Phys. A, 30, 2799-2806 (1997).
4. 3. S. I. Alber, Commun. Pure Appl. Math., 34, 259-272 (1981).
5. 4. M. Antonowicz, A. P. Fordy, and S. Wojciechowski, Phys. Lett. A, 124, 455-462 (1987).
6. 5. O. I. Bogoyavlensky and S. P. Novikov, Funct. Anal. Appl., 10, 8-11 (1976).
7. 6. B. A. Dubrovin, I. M. Krichever, and S. P. Novikov, "Integrable Systems I [in Russian]," in: Dynamical Systems IV (Encycl. Math. Sci., Vol. 4) (V. I. Arnol'd and S. P. Novikov, eds.), VINITI, Moscow (1985), pp. 179-285; English transl., Springer, Berlin (1990), pp. 173-280.
8. 7. I. M. Gel'fand and I. Zakharevich, "On the local geometry of a bi-Hamiltonian structure," in: The Gelfand Mathematical Seminars 1990-1992 (L. Corwin et al., eds.), Birkhäuser, Boston (1993), pp. 51-112.
9. 8. P. Casati, G. Falqui, F. Magri, and M. Pedroni, J. Math. Phys., 38, 4605-4628 (1997).
10. 9. G. Falqui, F. Magri, and M. Pedroni, Commun. Math. Phys., 197, 303-324 (1998).
11. 10. M. Sato and Y. Sato, "Soliton equations as dynamical systems on infinite-dimensional Grassmann manifold," in: Nonlinear PDEs in Applied Sciences (US-Japan Seminar, Tokyo) (P. Lax and H. Fujita, eds.), North-Holland, Amsterdam (1982), pp. 259-271.
12. 11. L. A. Dickey, Soliton Equations and Hamiltonian Systems (Adv. Ser. in Math. Phys., Vol. 12), World Scientific, Singapore (1991).
13. 12. F. Magri and C. Morosi, "A geometric characterization of integrable Hamiltonian systems through the theory of Poisson-Nijenhuis manifolds," Preprint Quaderno S/19, Univ. Milano, Milano (1984).
14. 13. A. G. Reyman and M. A. Semenov-Tian-Shansky, "Group-theoretical methods in the theory of finite-dimensional integrable systems [in Russian]," in: Dynamical Systems VII (Encycl. Math. Sci., Vol. 16) (V. I. Arnol'd and S. P. Novikov, eds.), VINITI, Moscow (1987), pp. 119-194; English transl., Springer, Berlin (1994), pp. 116-225.
15. 14. G. Magnano and F. Magri, Rev. Math. Phys., 3, 403-466 (1991).
16. 15. M. Pedroni and P. Vanhaecke, Regular and Chaotic Dynamics, 3, 132-160 (1998).
17. 16. J. E. Marsdena and T. Ratiu, Lett. Math. Phys., 11, 161-169 (1986).
18. 17. P. Casati, F. Magri, and M. Pedroni, "Bi-Hamiltonian manifolds and τ-function," in: Mathematical Aspects of Classical Field Theory 1991 (Cont. Math., Vol. 132) (M. J. Gotay et al., eds.), Am. Math. Soc., Providence, RI (1992), pp. 213-234.
19. 18. A. P. Fordy and S. D. Harris, "Hamiltonian structures in stationary manifold coordinates," in: Algebraic Aspects of Integrable Systems (Progr. Nonlinear Differ. Equat. Appl., Vol. 26) (A. S. Fokas and I. M. Gelfand, eds.), Birkhäuser, Boston (1996), pp. 103-130.
20. 19. A. Fokas and B. Fuchssteiner, Physica D, 4, 47-66 (1981).
21. 20. R. Brouzet, R. Caboz, J. Rabenivo, and V. Ravoson, J. Phys. A, 29, 2069-2076 (1996).
22. 21. E. Sklyanin, Progr. Theor. Phys. Suppl., 118, 35-60 (1995).
23. 22. S. De Filippo, G. Marmo, M. Salerno, and G. Vilasi, Nuovo Cimento B, 83, 97-112 (1984).
24. 23. S. Benenti, "Separability structures on Riemannian manifolds," in: Differential Geometrical Methods in Mathematical Physics (Proc. Conf. Aix-en-Provence, Sept. 3-7, 1979, and Salamanca, Sept. 10-14, 1979) (Lect. Notes Math., Vol. 836) (P. L. García, A. Pérez-Rendón, and J. M. Souriau, eds.), Springer, Berlin (1980), pp. 512-538.
25. 24. G. Tondo, J. Phys. A, 28, 5097-5115 (1995).
26. 25. T. Levi-Civita, Math. Annal., 59, 383-397 (1904).
27. 26. G. Tondo, "On the integrability of Hénon-Heiles-type systems," in: Non-Linear Physics: Theory and Experiment (E. Alfinito et al., eds.), World Scientific, Singapore (1996), pp. 313-320.
28. 27. M. Błaszak, Multi-Hamiltonian Theory of Dynamical Systems, Springer, Berlin (1998).
29. 28. C. Morosi and G. Tondo, "Bi-Hamiltonian structures, quasi-bi-Hamiltonian systems, and separation variables," Rep. Math. Phys. (forthcoming).