Abstract:
We reduce an arbitrary pair of compatible nonlocal Poisson brackets of hydrodynamic type generated by metrics of constant Riemannian curvature (compatible Mokhov–Ferapontov brackets) to a canonical form, find an integrable system describing all such pairs, and, for an arbitrary solution of this integrable system, i.e., for any pair of compatible Poisson brackets in question, construct (in closed form) integrable bi-Hamiltonian systems of
hydrodynamic type possessing this pair of compatible Poisson brackets of hydrodynamic type. The corresponding special canonical forms of metrics of constant Riemannian curvature are considered. A theory of special Liouville
coordinates for Poisson brackets is developed. We prove that the classification of these compatible Poisson brackets is equivalent to the classification of special Liouville coordinates for Mokhov–Ferapontov brackets.
Keywords:
metric of constant curvature, integrable hierarchy, system of hydrodynamic type, bi-Hamiltonian system, compatible Poisson brackets, Poisson bracket of hydrodynamic type, compatible metrics, flat pencil of metrics, Liouville bracket, Liouville coordinates.
Citation:
O. I. Mokhov, “The Liouville Canonical Form for Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Hierarchies”, Funktsional. Anal. i Prilozhen., 37:2 (2003), 28–40; Funct. Anal. Appl., 37:2 (2003), 103–113
\Bibitem{Mok03}
\by O.~I.~Mokhov
\paper The Liouville Canonical Form for Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Hierarchies
\jour Funktsional. Anal. i Prilozhen.
\yr 2003
\vol 37
\issue 2
\pages 28--40
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\jour Funct. Anal. Appl.
\yr 2003
\vol 37
\issue 2
\pages 103--113
\crossref{https://doi.org/10.1023/A:1024469316049}
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Linking options:
https://www.mathnet.ru/eng/faa146
https://doi.org/10.4213/faa146
https://www.mathnet.ru/eng/faa/v37/i2/p28
This publication is cited in the following 8 articles:
Garton J., “Botanical Symbolism in Vicino Orsini'S Sacro Bosco”, Stud. Hist. Gard. Des. Landsc., 41:2, SI (2021), 141–154
O. I. Mokhov, “Pencils of compatible metrics and integrable systems”, Russian Math. Surveys, 72:5 (2017), 889–937
Cirilo-Lombardo D.J., “Integrable Hydrodynamic Equations For Initial Chiral Currents and Infinite Hydrodynamic Chains From WZNW Model and String Model of WZNW Type With Su(2), So(3), Sp(2), Su(Infinity), So(Infinity), Sp(Infinity) Constant Torsions”, Int. J. Mod. Phys. A, 29:24 (2014), 1450134
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
Victor D. Gershun, “Integrable String Models in Terms of Chiral Invariants of SU(n)SU(n), SO(n)SO(n),
SP(n)SP(n) Groups”, SIGMA, 4 (2008), 041, 16 pp.
Pavlov, MV, “Algebro-geometric approach in the theory of integrable hydrodynamic type systems”, Communications in Mathematical Physics, 272:2 (2007), 469
Pavlov, MV, “Hydrodynamic chains and the classification of their Poisson brackets”, Journal of Mathematical Physics, 47:12 (2006), 123514
O. I. Mokhov, “Quasi-Frobenius Algebras and Their Integrable NN-Parameter Deformations Generated by Compatible (N×N)(N×N) Metrics of Constant Riemannian Curvature”, Theoret. and Math. Phys., 136:1 (2003), 908–916
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