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 Funktsional. Anal. i Prilozhen.: Year: Volume: Issue: Page: Find

 Funktsional. Anal. i Prilozhen., 2003, Volume 37, Issue 2, Pages 28–40 (Mi faa146)

The Liouville Canonical Form for Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Hierarchies

O. I. Mokhov

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

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

DOI: https://doi.org/10.4213/faa146

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English version:
Functional Analysis and Its Applications, 2003, 37:2, 103–113

Bibliographic databases:

UDC: 514.7+517.956.35

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

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/faa146
• https://doi.org/10.4213/faa146
• http://mi.mathnet.ru/eng/faa/v37/i2/p28

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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, “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
2. Pavlov, MV, “Hydrodynamic chains and the classification of their Poisson brackets”, Journal of Mathematical Physics, 47:12 (2006), 123514
3. Pavlov, MV, “Algebro-geometric approach in the theory of integrable hydrodynamic type systems”, Communications in Mathematical Physics, 272:2 (2007), 469
4. Victor D. Gershun, “Integrable String Models in Terms of Chiral Invariants of $\mathrm{SU}(n)$, $\mathrm{SO}(n)$, $\mathrm{SP}(n)$ Groups”, SIGMA, 4 (2008), 041, 16 pp.
5. 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
6. 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
7. O. I. Mokhov, “Pencils of compatible metrics and integrable systems”, Russian Math. Surveys, 72:5 (2017), 889–937
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