Teoreticheskaya i Matematicheskaya Fizika
RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB  
General information
Latest issue
Archive
Impact factor
Guidelines for authors
License agreement
Submit a manuscript

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



TMF:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Teoreticheskaya i Matematicheskaya Fizika, 2004, Volume 138, Number 2, Pages 283–296
DOI: https://doi.org/10.4213/tmf25
(Mi tmf25)
 

This article is cited in 8 scientific papers (total in 8 papers)

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

O. I. Mokhov

Landau Institute for Theoretical Physics, Centre for Non-linear Studies
Full-text PDF (247 kB) Citations (8)
References:
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 Lamй 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.
Received: 04.01.2003
English version:
Theoretical and Mathematical Physics, 2004, Volume 138, Issue 2, Pages 238–249
DOI: https://doi.org/10.1023/B:TAMP.0000015071.25148.90
Bibliographic databases:
Language: Russian
Citation: O. I. Mokhov, “Lax Pairs for Equations Describing Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Reductions of the Lamй 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 Lamй 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://mathscinet.ams.org/mathscinet-getitem?mr=2061741}
\zmath{https://zbmath.org/?q=an:1178.37084}
\adsnasa{https://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{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000220283500007}
Linking options:
  • https://www.mathnet.ru/eng/tmf25
  • https://doi.org/10.4213/tmf25
  • https://www.mathnet.ru/eng/tmf/v138/i2/p283
  • This publication is cited in the following 8 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Теоретическая и математическая физика Theoretical and Mathematical Physics
     
      Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2024