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 TMF, 2002, Volume 130, Number 2, Pages 233–250 (Mi tmf299)

Integrability of the Equations for Nonsingular Pairs of Compatible Flat Metrics

O. I. Mokhovab

a Landau Institute for Theoretical Physics, Centre for Non-linear Studies

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).

DOI: https://doi.org/10.4213/tmf299

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English version:
Theoretical and Mathematical Physics, 2002, 130:2, 198–212

Bibliographic databases:

Citation: O. I. Mokhov, “Integrability of the Equations for Nonsingular Pairs of Compatible Flat Metrics”, TMF, 130:2 (2002), 233–250; Theoret. and Math. Phys., 130:2 (2002), 198–212

Citation in format AMSBIB
\Bibitem{Mok02} \by O.~I.~Mokhov \paper Integrability of the Equations for Nonsingular Pairs of Compatible Flat Metrics \jour TMF \yr 2002 \vol 130 \issue 2 \pages 233--250 \mathnet{http://mi.mathnet.ru/tmf299} \crossref{https://doi.org/10.4213/tmf299} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1922009} \zmath{https://zbmath.org/?q=an:1029.37046} \elib{http://elibrary.ru/item.asp?id=13403260} \transl \jour Theoret. and Math. Phys. \yr 2002 \vol 130 \issue 2 \pages 198--212 \crossref{https://doi.org/10.1023/A:1014235331479} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000174582900002} 

• http://mi.mathnet.ru/eng/tmf299
• https://doi.org/10.4213/tmf299
• http://mi.mathnet.ru/eng/tmf/v130/i2/p233

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. O. I. Mokhov, “Compatible Dubrovin–Novikov Hamiltonian Operators, Lie Derivative, and Integrable Systems of Hydrodynamic Type”, Theoret. and Math. Phys., 133:2 (2002), 1557–1564
2. O. I. Mokhov, “Integrable bi-Hamiltonian hierarchies generated by compatible metrics of constant Riemannian curvature”, Russian Math. Surveys, 57:5 (2002), 999–1001
3. O. I. Mokhov, “Quasi-Frobenius Algebras and Their Integrable $N$-Parameter Deformations Generated by Compatible $(N\times N)$ Metrics of Constant Riemannian Curvature”, Theoret. and Math. Phys., 136:1 (2003), 908–916
4. O. I. Mokhov, “The Liouville Canonical Form for Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Hierarchies”, Funct. Anal. Appl., 37:2 (2003), 103–113
5. O. I. Mokhov, “Lax Pairs for Equations Describing Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Reductions of the Lamé Equations”, Theoret. and Math. Phys., 138:2 (2004), 238–249
6. Sergyeyev A, “A simple way of making a Hamiltonian system into a bi-Hamiltonian one”, Acta Applicandae Mathematicae, 83:1–2 (2004), 183–197
7. O. I. Mokhov, “The Classification of Nonsingular Multidimensional Dubrovin–Novikov Brackets”, Funct. Anal. Appl., 42:1 (2008), 33–44
8. O. I. Mokhov, “Compatible metrics and the diagonalizability of nonlocally bi-Hamiltonian systems of hydrodynamic type”, Theoret. and Math. Phys., 167:1 (2011), 403–420
9. O. I. Mokhov, “O metrikakh diagonalnoi krivizny”, Fundament. i prikl. matem., 21:6 (2016), 171–182
10. O. I. Mokhov, “Pencils of compatible metrics and integrable systems”, Russian Math. Surveys, 72:5 (2017), 889–937
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