RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Archive
Impact factor
Subscription
License agreement
Submit a manuscript

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



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






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


Funktsional. Anal. i Prilozhen., 2008, Volume 42, Issue 1, Pages 1–21 (Mi faa2886)  

This article is cited in 7 scientific papers (total in 7 papers)

Integration of Some Differential-Difference Nonlinear Equations Using the Spectral Theory of Normal Block Jacobi Matrices

Yu. M. Berezanskiia, A. A. Mokhon'kob

a Institute of Mathematics, Ukrainian National Academy of Sciences
b National Taras Shevchenko University of Kyiv

Abstract: The following method for integrating the Cauchy problem for a Toda lattice on the half-line is well known: to a solution $u(t)$, $t\in[0,\infty)$, of the problem, one assigns a self-adjoint semi-infinite Jacobi matrix $J(t)$ whose spectral measure $d\rho(\lambda;t)$ undergoes simple evolution in time $t$. The solution of the Cauchy problem goes as follows. One writes out the spectral measure $d\rho(\lambda;0)$ for the initial value $u(0)$ of the solution and the corresponding Jacobi matrix $J(0)$ and then computes the time evolution $d\rho(\lambda;t)$ of this measure. Using the solution of the inverse spectral problem, one reconstructs the Jacobi matrix $J(t)$ from $d\rho(\lambda;t)$ and hence finds the desired solution $u(t)$.
In the present paper, this approach is generalized to the case in which the role of $J(t)$ is played by a block Jacobi matrix generating a normal operator in the orthogonal sum of finite-dimensional spaces with spectral measure $d\rho(\zeta;t)$ defined on the complex plane. Some recent results on the spectral theory of these normal operators permit one to use the integration method described above for a rather wide class of differential-difference nonlinear equations replacing the Toda lattice.

Keywords: block Jacobi matrix, generalized eigenvector, spectral representation, Toda lattice

DOI: https://doi.org/10.4213/faa2886

Full text: PDF file (294 kB)
References: PDF file   HTML file

English version:
Functional Analysis and Its Applications, 2008, 42:1, 1–18

Bibliographic databases:

UDC: 517.53+517.91
Received: 29.05.2007

Citation: Yu. M. Berezanskii, A. A. Mokhon'ko, “Integration of Some Differential-Difference Nonlinear Equations Using the Spectral Theory of Normal Block Jacobi Matrices”, Funktsional. Anal. i Prilozhen., 42:1 (2008), 1–21; Funct. Anal. Appl., 42:1 (2008), 1–18

Citation in format AMSBIB
\Bibitem{BerMok08}
\by Yu.~M.~Berezanskii, A.~A.~Mokhon'ko
\paper Integration of Some Differential-Difference Nonlinear Equations Using the Spectral Theory of Normal Block Jacobi Matrices
\jour Funktsional. Anal. i Prilozhen.
\yr 2008
\vol 42
\issue 1
\pages 1--21
\mathnet{http://mi.mathnet.ru/faa2886}
\crossref{https://doi.org/10.4213/faa2886}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2423974}
\zmath{https://zbmath.org/?q=an:1168.34038}
\elib{http://elibrary.ru/item.asp?id=10441215}
\transl
\jour Funct. Anal. Appl.
\yr 2008
\vol 42
\issue 1
\pages 1--18
\crossref{https://doi.org/10.1007/s10688-008-0001-y}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000255229000001}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-41549124040}


Linking options:
  • http://mi.mathnet.ru/eng/faa2886
  • https://doi.org/10.4213/faa2886
  • http://mi.mathnet.ru/eng/faa/v42/i1/p1

    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. Berezans'kyi Yu.M., “Integration of the modified double-infinite Toda lattice with the help of inverse spectral problem”, Ukrainian Math. J., 60:4 (2008), 521–539  crossref  mathscinet  isi
    2. Berezansky Yu.M., “Spectral Theory of the Infinite Block Jacobi Type Normal Matrices, Orthogonal Polynomials on a Complex Domain, and the Complex Moment Problem”, Modern Analysis and Applications: Mark Krein Centenary Conference, Operator Theory Advances and Applications, 191, 2009, 37–50  mathscinet  zmath  isi
    3. Mokhonko O., “Nonisospectral Flows on Self-adjoint, Unitary and Normal Semi-infinite Block Jacobi Matrices”, Modern Analysis and Applications: Mark Krein Centenary Conference, Operator Theory Advances and Applications, 190, 2009, 387–395  mathscinet  zmath  isi
    4. Guseinov G.Sh., “A Class of Complex Solutions to the Finite Toda Lattice”, Math. Comput. Model., 57:5-6 (2013), 1190–1202  crossref  mathscinet  isi
    5. Huseynov A., Guseinov G.Sh., “Solution of the Finite Complex Toda Lattice by the Method of Inverse Spectral Problem”, Appl. Math. Comput., 219:10 (2013), 5550–5563  crossref  mathscinet  zmath  isi  elib
    6. Dudkin M.E. Kozak V.I., “Jacobi-Type Block Matrices Corresponding to the Two-Dimensional Moment Problem: Polynomials of the Second Kind and Weyl Function”, Ukr. Math. J., 68:4 (2016), 557–569  crossref  mathscinet  isi  scopus
    7. Gekhtman M., “Inverse Moment Problem For Non-Abelian Coxeter Double Bruhat Cells”, Methods Funct. Anal. Topol., 22:2 (2016), 117–136  mathscinet  zmath  isi
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
    Number of views:
    This page:607
    Full text:202
    References:54
    First page:16

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2020