Ukrainian Mathematical Journal

, Volume 60, Issue 4, pp 521–539 | Cite as

Integration of a modified double-infinite Toda lattice by using the inverse spectral problem

  • Yu. M. Berezans’kyi


An approach to finding a solution of the Cauchy problem for a modified double-infinite Toda lattice by using the inverse spectral problem is given.


Cauchy Problem Spectral Measure Jacobi Matrice Toda Lattice Inverse Spectral Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    B. M. Levitan, Inverse Sturm-Liouville Problems [in Russian], Nauka, Moscow (1984).zbMATHGoogle Scholar
  2. 2.
    M. Kac and P. van Moerbeke, “On an explicitly soluble system of nonlinear differential equations related to certain Toda lattices,” Adv. Math., 16, No. 2, 160–169 (1975).zbMATHCrossRefGoogle Scholar
  3. 3.
    J. Moser, “Three integrable Hamilton systems connected with isospectral deformations,” Adv. Math., 16, No. 2, 197–220 (1975).zbMATHCrossRefGoogle Scholar
  4. 4.
    Yu. M. Berezans’kyi, “Integration of nonlinear difference equations by the inverse scattering method,” Dokl. Akad. Nauk SSSR, 281, No. 1, 16–19 (1985).MathSciNetGoogle Scholar
  5. 5.
    Yu. M. Berezansky, “The integration of semi-infinite Toda chain by means of inverse spectral problem,” Rep. Math. Phys., 24, No. 1, 21–47 (1986).CrossRefMathSciNetGoogle Scholar
  6. 6.
    I. M. Krichever, “Periodic non-Abelian Toda chain and its two-dimensional generalization,” Usp. Mat. Nauk, 36, No. 2, 72–80 (1981).MathSciNetGoogle Scholar
  7. 7.
    M. Kac and P. van Moerbeke, “On some periodic Toda lattice,” Proc. Nat. Acad. Sci. USA, 72, No. 7, 2879–2880 (1975).zbMATHCrossRefGoogle Scholar
  8. 8.
    S. V. Manakov, “On complete integrability and stochastization in discrete dynamical media,” Zh. Éksp. Teor. Fiz., 67, No. 2, 543–555 (1974).MathSciNetGoogle Scholar
  9. 9.
    H. Flaschka, “On the Toda lattice. I,” Phys. Rev. B, 9, 1924–1925 (1974).CrossRefMathSciNetGoogle Scholar
  10. 10.
    H. Flaschka, “On the Toda lattice. II,” Progr. Theor. Phys., 51, No. 3, 703–716 (1974).CrossRefMathSciNetGoogle Scholar
  11. 11.
    N. V. Zhernakov, “Direct and inverse problems for a periodic Jacobi matrix,” Ukr. Math. J., 38, No. 6, 665–668 (1986).zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    N. V. Zhernakov, “Integration of Toda chains in the class of Hilbert-Schmidt operators,” Ukr. Math. J., 39, No. 5, 527–530 (1987).zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    N. V. Zhernakov, Spectral Theory of Some Classes of Jacobi Matrices and Its Application to the Integration of the Toda Chain [in Russian], Doctoral-Degree Thesis (Physics and Mathematics), Kiev (1987).Google Scholar
  14. 14.
    F. Gesztesy and W. Renger, “New classes of Toda soliton solutions,” Commun. Math. Phys., 184, 27–50 (1997).zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    M. Bruschi, S. V. Manakov, O. Ragnisco, and D. Levi, “The nonabelian Toda lattice-discrete analogue of the matrix Schrödinger spectral problem,” J. Math. Phys., 21, 2749–2753 (1980).zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    M. Bruschi, O. Ragnisco, and D. Levi, “Evolution equations associated with the discrete analog of the matrix Schrödinger spectral problem solvable by the inverse spectral transform,” J. Math. Phys., 22, 2463–2471 (1981).zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    A. Yu. Daletskii and G. B. Podkolzin, “Group approach to the integration of the infinite Toda chain,” Ukr. Mat. Zh., 40, No. 4, 518–521 (1988).MathSciNetGoogle Scholar
  18. 18.
    Yu. M. Berezanskii, Expansions in Eigenfunctions of Self-Adjoint Operators, American Mathematical Society, Providence, RI (1968).Google Scholar
  19. 19.
    M. G. Krein, “Infinite J-matrices and matrix moment problem,” Dokl. Akad. Nauk SSSR, 69, No. 2, 125–128 (1949).zbMATHMathSciNetGoogle Scholar
  20. 20.
    Yu. M. Berezanskii and M. I. Gekhtman, “Inverse problem of the spectral analysis and non-Abelian chains,” Ukr. Math. J., 42, No. 6, 645–658 (1990).CrossRefMathSciNetGoogle Scholar
  21. 21.
    Yu. M. Berezansky and M. E. Dudkin, “The direct and inverse spectral problem for the block Jacobi-type unitary matrices,” Meth. Funct. Anal. Top., 11, No. 4, 327–345 (2005).zbMATHMathSciNetGoogle Scholar
  22. 22.
    Yu. M. Berezansky and M. E. Dudkin, “The complex moment problem and direct and inverse spectral problems for the block Jacobi-type bounded normal matrices,” Meth. Funct. Anal. Top., 12, No. 1, 1–31 (2006).zbMATHMathSciNetGoogle Scholar
  23. 23.
    Yu. M. Berezans’kyi and A. A. Mokhon’ko, “Integration of some nonlinear differential-difference equations using the spectral theory of block Jacobi matrices,” Funkts. Anal. Prilozhen., 42, No. 1, 1–21 (2008).CrossRefGoogle Scholar
  24. 24.
    Yu. M. Berezansky, Z. G. Sheftel, and G. F. Us, Functional Analysis, Birkhäuser, Basel (1996).zbMATHGoogle Scholar
  25. 25.
    V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, Theory of Solitons. The Inverse Scattering Method [in Russian], Nauka, Moscow (1980).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  • Yu. M. Berezans’kyi
    • 1
  1. 1.Institute of MathematicsNational Academy of Sciences of UkraineKyivUkraine

Personalised recommendations