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

This article is cited in 10 scientific papers (total in 10 papers)

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
b University of Paderborn

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

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Received: 04.07.2001

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
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    Citing articles on Google Scholar: Russian citations, English citations
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    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  mathnet  crossref  crossref  mathscinet  isi  elib
    2. O. I. Mokhov, “Integrable bi-Hamiltonian hierarchies generated by compatible metrics of constant Riemannian curvature”, Russian Math. Surveys, 57:5 (2002), 999–1001  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    5. O. I. Mokhov, “Lax Pairs for Equations Describing Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Reductions of the Lamй Equations”, Theoret. and Math. Phys., 138:2 (2004), 238–249  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    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  crossref  mathscinet  zmath  isi  scopus  scopus
    7. O. I. Mokhov, “The Classification of Nonsingular Multidimensional Dubrovin–Novikov Brackets”, Funct. Anal. Appl., 42:1 (2008), 33–44  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    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  mathnet  crossref  crossref  mathscinet  adsnasa  isi
    9. O. I. Mokhov, “O metrikakh diagonalnoi krivizny”, Fundament. i prikl. matem., 21:6 (2016), 171–182  mathnet
    10. O. I. Mokhov, “Pencils of compatible metrics and integrable systems”, Russian Math. Surveys, 72:5 (2017), 889–937  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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