RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
General information
Latest issue
Archive
Impact factor
Subscription
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


TMF, 2010, Volume 165, Number 3, Pages 440–471 (Mi tmf6587)  

This article is cited in 3 scientific papers (total in 3 papers)

The relative frame bundle of an infinite-dimensional flag variety and solutions of integrable hierarchies

G. F. Helmincka, A. G. Helminckb, A. V. Opimakhc

a Korteweg–de Vries Institute for Mathematics, University of Amsterdam, Amsterdam, The Netherlands
b North Carolina State University, Raleigh, USA
c Orenburg State Pedagogical University, Orenburg, Russia

Abstract: We develop a group theory approach for constructing solutions of integrable hierarchies corresponding to the deformation of a collection of commuting directions inside the Lie algebra of upper-triangular $\mathbb Z{\times}\mathbb Z$ matrices. Depending on the choice of the set of commuting directions, the homogeneous space from which these solutions are constructed is the relative frame bundle of an infinite-dimensional flag variety or the infinite-dimensional flag variety itself. We give the evolution equations for the perturbations of the basic directions in the Lax form, and they reduce to a tower of differential and difference equations for the coefficients of these perturbed matrices. The Lax equations follow from the linearization of the hierarchy and require introducing a proper analogue of the Baker–Akhiezer function.

Keywords: upper-triangular $\mathbb Z{\times}\mathbb Z$ matrices, Lax equations, zero curvature form

DOI: https://doi.org/10.4213/tmf6587

Full text: PDF file (537 kB)
References: PDF file   HTML file

English version:
Theoretical and Mathematical Physics, 2010, 165:3, 1610–1636

Bibliographic databases:

Document Type: Article

Citation: G. F. Helminck, A. G. Helminck, A. V. Opimakh, “The relative frame bundle of an infinite-dimensional flag variety and solutions of integrable hierarchies”, TMF, 165:3 (2010), 440–471; Theoret. and Math. Phys., 165:3 (2010), 1610–1636

Citation in format AMSBIB
\Bibitem{HelHelOpi10}
\by G.~F.~Helminck, A.~G.~Helminck, A.~V.~Opimakh
\paper The~relative frame bundle of an~infinite-dimensional flag variety and solutions of integrable hierarchies
\jour TMF
\yr 2010
\vol 165
\issue 3
\pages 440--471
\mathnet{http://mi.mathnet.ru/tmf6587}
\crossref{https://doi.org/10.4213/tmf6587}
\transl
\jour Theoret. and Math. Phys.
\yr 2010
\vol 165
\issue 3
\pages 1610--1636
\crossref{https://doi.org/10.1007/s11232-010-0133-0}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000288427000003}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-79151472227}


Linking options:
  • http://mi.mathnet.ru/eng/tmf6587
  • https://doi.org/10.4213/tmf6587
  • http://mi.mathnet.ru/eng/tmf/v165/i3/p440

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    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. Helminck G.F., Helminck A.G., Opimakh A.V., “Equivalent forms of multi component Toda hierarchies”, J. Geom. Phys., 61:4 (2011), 847–873  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    2. Helminck G.F., Opimakh A.V., “The zero curvature form of integrable hierarchies in the $\mathbb Z\times\mathbb Z$-matrices”, Algebr. Colloq., 19:2 (2012), 237–262  crossref  mathscinet  zmath  isi  elib
    3. Helminck G.F. Helminck A.G., “Infinite Dimensional Symmetric Spaces and Lax Equations Compatible With the Infinite Toda Chain”, J. Geom. Phys., 85 (2014), 60–74  crossref  mathscinet  zmath  adsnasa  isi  scopus
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
    Number of views:
    This page:318
    Full text:41
    References:21
    First page:19

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019