Tri-Hamiltonian Structures of Egorov Systems of Hydrodynamic Type

Functional Analysis and its Applications

Volumn 37, Issue 1, 2003, Pages 32-45

  Pavlov, M V ^a   Tsarev, S P ^a  

^a Astaf'ev Krasnoyarsk State Pedagogical University   (Russian Federation)

Author keywords

Benney chain; Egorov metric; Hamiltonian structure; Hydrodynamic type systems; Riemann invariant; Whitham equations

Indexed keywords

EID: 0037245698     PISSN: 00162663     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1022971910438     Document Type: Article

References (26)

1. V. L. Alekseev, "On nonlocal Hamiltonian operators of hydrodynamic type associated with the Whitham equation," Usp. Mat. Nauk, 50, No. 6, 165-166 (1995).
2. B. A. Dubrovin and S. P. Novikov, "Hamiltonian formalism of one-dimensional systems of the hydrodynamic type and the Bogolyubov-Whitham averaging method," Dokl. Akad. Nauk SSSR, 270, No. 4, 781-785 (1993).
3. D. F. Egorov, Works in Differential Geometry [in Russian], Nauka, Moscow, 1970.
4. V. E. Zakharov, "Benney equations and the semiclassical approximation in the inverse problem method," Funkts. Anal. Prilozhen., 14, No. 2, 15-24 (1980).
5. O. I. Mokhov and E. V. Ferapontov, "Nonlocal Hamiltonian operators of hydrodynamic type that are connected with metrics of constant curvature," Usp. Mat. Nauk, 45, No. 3, 191-192 (1990).
6. M. V. Pavlov, "The nonlinear Schrödinger equation and the Bogolyubov-Whitham averaging method," Teor. Mat. Fiz., 71, No. 3, 351-356 (1987).
7. M. V. Pavlov, "Elliptic coordinates and multi-Hamiltonian structures of hydrodynamic type systems," Dokl. Ross. Akad. Nauk, 339, No. 1, 21-23 (1994).
8. M. V. Pavlov, "Multi-Hamiltonian structures of the Witham equations," Dokl. Ross. Akad. Nauk, 338, No. 2, 165-167 (1994).
9. M. V. Pavlov, "Local Hamiltonian structures of Benney's system," Dokl. Ross. Akad. Nauk, 338, No. 1, 33-34 (1994).
10. M. V. Pavlov and S. P. Tsarev, "Conservation laws for the Benney equations," Usp. Mat. Nauk, 46, No. 4, 169-170 (1991).
11. E. V. Ferapontov, "Differential geometry of nonlocal Hamiltonian operators of hydrodynamic type," Funkts. Anal. Prilozhen., 25, No. 3, 37-49 (1991).
12. S. P. Tsarev, "Poisson brackets and one-dimensional Hamiltonian systems of hydrodynamic type," Dokl. Akad. Nauk SSSR, 282, No. 3, 534-537 (1985).
13. S. P. Tsarev, "The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method," Izv. Akad. Nauk SSSR, Ser. Mat., 54, No. 5, 1048-1068 (1990).
14. S. P. Tsarev, Differential-Geometric Integration Methods for Systems of Hydrodynamic Type, Ph.D. Thesis, Moscow, 1993.
15. V. L. Alekseev and M. V. Pavlov, "Hamiltonian structures of the Whitham equations," Proceedings of the Conference on NLS, Chernogolovka, 1994.
16. D. J. Benney, "Some properties of long nonlinear waves," Stud. Appl. Math., 52, 45-50 (1973).
17. G. Darboux, Leçons sur les systèmes orthogonaux et les coordonnées curvilignes, Gautier-Villars, Paris, 1910.
18. B. A. Dubrovin, "Geometry of 2D topological field theories," In: Integrable Systems and Quantum Groups (Montecatini Terme, 1993), M. Francaviglia, S. Greco, eds., Lecture Notes in Math., Vol. 1620, 1996, pp. 120-348.
19. B. A. Dubrovin, "Integrable systems and classification of 2D topological field theories," In: Integrable Systems, The J.-L. Verdier Memorial Conference, Actes du Colloque International de Luminy (O. Babelon, P. Cartier, and Y. Kosmann-Schwarzbach, eds.), Birkhäuser, 1993, pp. 313-359.
20. E. V. Ferapontov and M. V. Pavlov, "Quasiclassical limit of Coupled KdV equations. Riemann invariants and multi-Hamiltonian structure," Phys. D, 52, 211-219 (1991).
21. M. G. Forest and J. E. Lee, "Geometry and modulation theory for the periodic nonlinear Schrödinger equation," In: Oscillation Theory, Computation, and Methods of Compensated Compactness (C. Dafermos et al., eds.), IMA, Vol. 2, Springer-Verlag, New York, 1986, pp. 35-69.
22. H. Flaschka, M. G. Forest, and D. W. McLaughlin, "Multiphase averaging and the inverse spectral solution of the Korteweg-de Vries equation," Comm. Pure Appl. Math., 33, 739-784 (1980).
23. J. Gibbons, "Collisionless Boltzmann equations and integrable moment equations," Phys. D, 3, No. 3, 503-511 (1981).
24. H. Gumral and Y. Nutku, "Multi-Hamiltonian structure of equations of hydrodynamic type," J. Math. Phys., 31, No. 11, 2606-2611 (1990).
25. M. V. Pavlov, "Integrable systems and metrics of constant curvature," J. Nonlinear Math. Phys., 9, Suppl. 1, 173-191 (2002).
26. W. Oevel, J. Sidorenko, and W. Strampp, "Hamiltonian structures of Melnikov system and its reductions," Inverse Problems, 9, 737-747 (1993).