Description of the n-orthogonal curvilinear coordinate systems and hamiltonian integrable systems of hydrodynamic type, I: Integration of the lamé equations

Duke Mathematical Journal

Volumn 94, Issue 1, 1998, Pages 103-139

  Zakharov, Vladimir E ^a,b  

^a LANDAU INSTITUTE FOR THEORETICAL PHYSICS   (Russian Federation)

^b University of Arizona   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0003022531     PISSN: 00127094     EISSN: None     Source Type: Journal    
DOI: 10.1215/S0012-7094-98-09406-6     Document Type: Article

References (16)

1. F. Backes, Sur les transformations de Ribaucour des systèmes triples orthogonaux, Acad. Royal Belg. Bull. Cl. Sci. 57 (1971), 768-788.
2. L. Bianchi, Opere, Vol. 3: Sistemi Tripli Orthogonali, Edizioni Cremonese, Roma, 1955.
3. G. Bouligand, Une forme donnée à la recherche des systèmes triples orthogonaux, C. R. Acad. Sci. Paris Sèr I Math. 236 (1953), 2462-2463.
4. E. Cartan, Leçons sur la Géométrie des espaces de Riemann, Gauthier-Villars, Paris, 1928.
5. G. Darboux, Leçons sur le Systèmes Orthogonaux et les Coordonnées Curvilignes, Gauthier-Villars, Paris, 1910.
6. B. A. Dubrovin, Integrable systems in topological field theory, Nuclear Phys. B 379 (1992), 627-689.
7. B. A. Dubrovin, On the differential geometry of strongly integrable systems of hydrodynamic type, Funktsional. Anal. Appl. 24 (1990), 280-285.
8. B. A. Dubrovin and S. Novikov, Hamiltonian formalism of one-dimensional systems of hydrodynamic type and the Bogolyubov-Whitham averaging method, Dokl. Akad. Nauk. SSSR 270 (1983), 781-785.
9. A. Fokas and V. Zakharov, The dressing method and nonlocal Riemann-Hilbert problems, J. Nonlinear Sci. 2 (1992), 109-134.
10. A. R. Forsyth, Lectures on the Differential Geometry of Curves and Surfaces, Cambridge Univ. Press, Cambridge, 1920.
11. S. Tsarev, Poisson’s brackets and one-dimensional Hamiltonian systems of hydrodynamic type, Dokl. Nauk. SSSR 282 (1985), 534-537.
12. S. Tsarev, “Classical differential geometry and integrability of systems of hydrodynamic type” in Applications of Analytic and Geometric Methods to Nonlinear Differential Equations (Exeter, 1992), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 413, Kluwer, Dordrecht, 1993, 241-249.
13. S. Tsarev, The geometry of Hamiltonian systems of hydrodynamic type: The generalized hodograph method, Math. USSR-Izv. 37 (1991), 397-419.
14. V. Zakharov, “On the dressing method” in Inverse Methods in Action (Montpellier, 1989), Inverse Probl. Theoret. Imaging, Springer-Verlag, Berlin, 1990, 602-623.
15. V. Zakharov and S. Manakov, Construction of multidimensional nonlinear integrable systems and their solitons, Funktsional. Anal, i Prilozhen 19 (1985), 11-25.
16. V. Zakharov and A. Shabat, An integration scheme for the nonlinear equations of mathematical physics by the method of the inverse scattering problem, Funktsional. Anal, i Prilozhen 8 (1974), 43-54 and 226-235.