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 TMF, 2004, Volume 138, Number 2, Pages 283–296 (Mi tmf25)

Lax Pairs for Equations Describing Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Reductions of the Lamé Equations

O. I. Mokhov

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

Abstract: We solve the problem of describing compatible nonlocal Poisson brackets of hydrodynamic type. We prove that for nonsingular pairs of compatible nonlocal Poisson brackets of hydrodynamic type, there exist special local coordinates such that the metrics and the Weingarten operators of both brackets are diagonal. The nonlinear evolution equations describing all nonsingular pairs of compatible nonlocal Poisson brackets of hydrodynamic type are derived in these special coordinates, and the integrability of these equations is proved using the inverse scattering transform. The Lax pairs with a spectral parameter for these equations are found. We construct various classes of integrable reductions of the derived equations. These classes of reductions are of an independent differential-geometric and applied interest. In particular, if one of the compatible Poisson brackets is local, we obtain integrable reductions of the classical Lamé equations describing all orthogonal curvilinear coordinate systems in a flat space; if one of the compatible brackets is generated by a constant-curvature metric, the corresponding equations describe integrable reductions of the equations for orthogonal curvilinear coordinate systems in a space of constant curvature.

Keywords: nonlocal Poisson brackets of hydrodynamic type, compatible metrics, compatible Poisson brackets, inverse scattering transform, orthogonal curvilinear coordinate systems, integrable systems

DOI: https://doi.org/10.4213/tmf25

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English version:
Theoretical and Mathematical Physics, 2004, 138:2, 238–249

Bibliographic databases:

Citation: O. I. Mokhov, “Lax Pairs for Equations Describing Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Reductions of the Lamé Equations”, TMF, 138:2 (2004), 283–296; Theoret. and Math. Phys., 138:2 (2004), 238–249

Citation in format AMSBIB
\Bibitem{Mok04} \by O.~I.~Mokhov \paper Lax Pairs for Equations Describing Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Reductions of the Lamé Equations \jour TMF \yr 2004 \vol 138 \issue 2 \pages 283--296 \mathnet{http://mi.mathnet.ru/tmf25} \crossref{https://doi.org/10.4213/tmf25} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2061741} \zmath{https://zbmath.org/?q=an:1178.37084} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2004TMP...138..238M} \transl \jour Theoret. and Math. Phys. \yr 2004 \vol 138 \issue 2 \pages 238--249 \crossref{https://doi.org/10.1023/B:TAMP.0000015071.25148.90} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000220283500007} 

• http://mi.mathnet.ru/eng/tmf25
• https://doi.org/10.4213/tmf25
• http://mi.mathnet.ru/eng/tmf/v138/i2/p283

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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, “Riemann invariants of semisimple non-locally bi-Hamiltonian systems of hydrodynamic type and compatible metrics”, Russian Math. Surveys, 65:6 (2010), 1183–1185
2. 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
3. D. A. Berdinskii, I. P. Rybnikov, “On orthogonal curvilinear coordinate systems in constant curvature spaces”, Siberian Math. J., 52:3 (2011), 394–401
4. Cieslinski J.L. Kobus A., “Lax Triples for Integrable Surfaces in Three-Dimensional Space”, Adv. Math. Phys., 2016, 8386420
5. O. I. Mokhov, “O metrikakh diagonalnoi krivizny”, Fundament. i prikl. matem., 21:6 (2016), 171–182
6. O. I. Mokhov, “Pencils of compatible metrics and integrable systems”, Russian Math. Surveys, 72:5 (2017), 889–937
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