Abstract
We prove that two Dubrovin–Novikov Hamiltonian operators are compatible if and only if one of these operators is the Lie derivative of the other operator along a certain vector field. We consider the class of flat manifolds, which correspond to arbitrary pairs of compatible Dubrovin–Novikov Hamiltonian operators. Locally, these manifolds are defined by solutions of a system of nonlinear equations, which is integrable by the method of the inverse scattering problem. We construct the integrable hierarchies generated by arbitrary pairs of compatible Dubrovin–Novikov Hamiltonian operators.
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REFERENCES
B. A. Dubrovin and S. P. Novikov, Sov. Math. Dokl., 27, 665–669 (1983).
I. Dorfman, Dirac Structures and Integrability of Nonlinear Evolution Equations, Wiley, Chichester, UK (1993).
F. Magri, J. Math. Phys., 19, 1156–1162 (1978).
B. Dubrovin, “Geometry of 2D topological field theories,” in: Integrable Systems and Quantum Groups (Lect. Notes Math., Vol. 1620, M. Francaviglia et al., eds.), Springer, Berlin (1996), pp. 120–348; hep-th/9407018 (1994).
B. Dubrovin, “Di.erential geometry of the space of orbits of a Coxeter group,” Preprint SISSA-29/93/FM, SISSA, Trieste; hep-th/9303152 (1993).
B. Dubrovin, “Flat pencils of metrics and Frobenius manifolds,” Preprint SISSA 25/98/FM, SISSA, Trieste; math.DG/9803106 (1998).
A. P. Fordy and O. I. Mokhov, Phys. D, 152–153, 475–490 (2001).
O. I. Mokhov, Funct. Anal. Appl., 35, 100–110 (2001); math.DG/0005051 (2000).
O. I. Mokhov, Russ. Math. Surveys, 52, 1310–1311 (1997).
O. I. Mokhov, Theor. Math. Phys., 130, 198–212 (2002); math.DG/0005081 (2000); Russ. Math. Surveys, 56, 416–418 (2001).
E. V. Ferapontov, “Compatible Poisson brackets of hydrodynamic type,” math.DG/0005221 (2000).
O. I. Mokhov, Proc. Steklov Math. Inst., 225, 269–284 (1999).
O. I. Mokhov, Rep. Math. Phys., 43, 247–256 (1999).
O. I. Mokhov, Russ. Math. Surveys, 53, 396–397 (1998).
I. M. Gelfand and I.Ya. Dorfman, Funct. Anal. Appl., 13, No. 4, 248–262 (1979); B. Fuchssteiner, Nonlinear Anal. Theor. Meth. Appl., 3, 849–862 (1979); A. S. Fokas and B. Fuchssteiner, Lett. Nuovo Cimento, 28, 299–303 (1980); P. Olver, Applications of Lie Groups to Differential Equations, Springer, New York (1986); O. I. Mokhov, Russ. Math. Surveys, 53, 515–622 (1998).
O. I. Mokhov and E. V. Ferapontov, Russ. Math. Surveys, 45, 218–219 (1990); E. V. Ferapontov, Funct. Anal. Appl., 25, 195–204 (1991); J. Soviet Math., 55, 1970–1995 (1991); O. I. Mokhov and E. V. Ferapontov, Funct. Anal. Appl., 28, 123–125 (1994); O. I. Mokhov, Phys. Lett. A, 166, 215–216 (1992); E. V. Ferapontov, “Nonlocal Hamiltonian operators of hydrodynamic type: Differential geometry and applications,” in: Topics in Topology and Mathematical Physics (S. P. Novikov, ed.), Amer. Math. Soc., Providence, R. I. (1995), pp. 33–58.
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Mokhov, O.I. Compatible Dubrovin–Novikov Hamiltonian Operators, Lie Derivative, and Integrable Systems of Hydrodynamic Type. Theoretical and Mathematical Physics 133, 1557–1564 (2002). https://doi.org/10.1023/A:1021155028895
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DOI: https://doi.org/10.1023/A:1021155028895