Abstract
We solve the problem of describing all nonsingular pairs of compatible flat metrics (or, in other words, nonsingular flat pencils of metrics) in the general N-component case. This problem is equivalent to the problem of describing all compatible Dubrovin–Novikov brackets (compatible nondegenerate local Poisson brackets of hydrodynamic type) playing an important role in the theory of integrable systems of hydrodynamic type and also in modern differential geometry and field theory. We prove that all nonsingular pairs of compatible flat metrics are described by a system of nonlinear differential equations that is a special nonlinear differential reduction of the classical Lamé equations, and we present a scheme for integrating this system by the method of the inverse scattering problem. The integration procedure is based on using the Zakharov method for integrating the Lamé equations (a version of the inverse scattering method).
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REFERENCES
O. I. Mokhov, “Compatible and almost compatible pseudo-Riemannian metrics,” math.DG/0005051 (2000).
V. E. Zakharov, Duke Math. J., 94, 103–139 (1998).
B. Dubrovin, “Geometry of 2D topological field theories,” in: Integrable Systems and Quantum Groups. (Lect. Notes Math., Vol. 1620, M. Francaviglia and S. Greco, eds.), Springer, Berlin (1996), pp. 120–348; hepth/ 9407018 (1994).
B. Dubrovin, “Differential geometry of the space of orbits of a Coxeter group,” Preprint SISSA-29/93/FM, SISSA, Trieste (1993); hep-th/9303152 (1993).
B. Dubrovin, “Flat pencils of metrics and Frobenius manifolds,” Preprint SISSA 25/98/FM, SISSA, Trieste (1998); math.DG/9803106 (1998).
E. V. Ferapontov, “Nonlocal Hamiltonian operators of hydrodynamic type: differential geometry and applications,” in: Topics in Topology and Mathematical Physics (S. P. Novikov, ed.), Am. Math. Soc., Providence, R.I. (1995), pp. 33–58.
O. I. Mokhov, Russ. Math. Surv., 52, 1310-1311 (1997).
O. I. Mokhov, Russ. Math. Surv., 53, 396–397 (1998).
O. I. Mokhov, Proc. Steklov Math. Inst., 225, 269–284 (1999).
O. I. Mokhov, Rep. Math. Phys., 43, No. 1/2, 247–256 (1999).
O. I. Mokhov and E. V. Ferapontov, Russ. Math. Surv., 45, 218–219 (1990).
E. V. Ferapontov, Funct. Anal. Appl., 25, 195–204 (1991).
E. V. Ferapontov, J. Sov. Math., 55, 1970–1995 (1991).
O. I. Mokhov, Phys. Lett. A, 166, 215–216 (1992).
O. I. Mokhov and E. V. Ferapontov, Funct. Anal. Appl., 28, 123–125 (1994).
O. I. Mokhov, Russ. Math. Surv., 53, 515–622 (1998).
B. A. Dubrovin and S. P. Novikov, Sov. Math. Dokl., 27, 665–669 (1983).
B. A. Dubrovin and S. P. Novikov, Russ. Math. Surv., 44, 35–124 (1989).
F. Magri, J. Math. Phys., 19, No. 5, 1156–1162 (1978).
I. M. Gelfand and I. Ya. Dorfman, Funct. Anal. Appl., 13, 246-262 (1979).
B. Fuchssteiner, Nonlinear Anal. Theor. Meth. Appl., 3, 849–862 (1979).
A. S. Fokas and B. Fuchssteiner, Lett. Nuovo Cimento, 28, No. 8, 299–303 (1980).
P. Olver, Applications of Lie Groups to Di.erential Equations, Springer, New York (1986).
I. Dorfman, Dirac Structures and Integrability of Nonlinear Evolution Equations, Wiley, Chichester, England (1993).
O. I. Mokhov, Russ. Math. Surv., 40, 233–234 (1985).
O. I. Mokhov, Funct. Anal. Appl., 21, 217–223 (1987).
D. B. Cooke, J. Math. Phys., 32, 109–119 (1991).
D. B. Cooke, J. Math. Phys., 32, 3071–3076 (1991).
C. S. Gardner, J. Math. Phys., 12, 1548–1551 (1971).
V. E. Zakharov and L. D. Faddeev, Funct. Anal. Appl., 5, 280–287 (1971).
Y. Nutku, J. Math. Phys., 28, 2579–2585 (1987).
P. Olver and Y. Nutku, J. Math. Phys., 29, 1610–1619 (1988).
M. Arik, F. Neyzi, Y. Nutku, P. J. Olver, and J. M. Verosky, J. Math. Phys., 30, 1338–1344 (1989).
F. Neyzi, J. Math. Phys., 30, 1695–1698 (1989).
H. Gümral and Y. Nutku, J. Math. Phys., 31, 2606–2611 (1990).
E. V. Ferapontov and M. V. Pavlov, Physica D, 52, 211–219 (1991).
A. Nijenhuis, Indagationes Math., 13, No. 2, 200–212 (1951).
A. Haantjes, Indagationes Math., 17, No. 2, 158–162 (1955).
G. Darboux, Leçons sur les systèmes orthogonaux et les coordonnèes curvilignes (2nd ed.), Gauthier-Villars, Paris (1910).
E. Cartan, Les systems differentiels exterieurs et leurs applications geometriques, Hermann, Paris (1945).
L. Bianchi, Opere, Vol. 3, Sistemi tripli orthogonali, Edizioni Cremonese, Roma (1955).
I. M. Krichever, Funct. Anal. Appl., 31, 25–39 (1997).
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Mokhov, O.I. Integrability of the Equations for Nonsingular Pairs of Compatible Flat Metrics. Theoretical and Mathematical Physics 130, 198–212 (2002). https://doi.org/10.1023/A:1014235331479
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DOI: https://doi.org/10.1023/A:1014235331479