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Inverse problem of the spectral analysis and non-Abelian chains of nonlinear equations

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Abstract

Using the method of the inverse spectral problem we construct solutions of the Cauchy problem for some systems of nonlinear difference-differential equations with operator unknowns in the semi-infinite case.

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Literature cited

  1. Yu. M. Berezanskii, “Integration of nonlinear difference equations by means of the method of the inverse spectral problem,” Dokl. Akad. Nauk SSSR,281, No. 1, 16–19 (1985).

    Google Scholar 

  2. Yu. T. Berezanskii, “The integration of semi-infinite Toda chain by means of the inverse spectral problem,” Rep. Math. Phys.,24, No. 1, 21–47 (1986).

    Google Scholar 

  3. Yu. M. Berezanskii, M. I. Gekhtman, and M. E. Shmoish, “Integration of some chains of nonlinear difference equations by the method of the inverse spectral problem,” Ukr. Mat. Zh.,38, No. 1, 84–89 (1986).

    Google Scholar 

  4. M. G. Krein, “Infinite J-matrices and a matrix-moment problem,” Dokl. Akad. Nauk SSSR,69, No. 2, 125–128 (1949).

    Google Scholar 

  5. Yu. M. Berezanskii, “Expansions in eigenfunctions of second-order partial difference equations,” Tr. Mosk. Mat. Obshch.,5, 203–268 (1956).

    Google Scholar 

  6. V. G. Tarnopol'skii, “On the self-adjointness of difference operators with operator coefficients,” Dop. Akad. Nauk Ukr. RSR, Ser. A,11, 1189–1192 (1959).

    Google Scholar 

  7. Yu. M. Berezanskii, Expansions in Eigenfunctions of Self-Adjoint Operators [in Russian], Naukova Dumka, Kiev (1965).

    Google Scholar 

  8. M. Brushi, S. V. Manakov, O. Ragnisco, and D. Levi, “The non-Abelian Toda lattice (discrete analogue of the matrix Schroedinger spectral problem),” J. Math. Phys.,21, 2749–2759 (1980).

    Google Scholar 

  9. I. M. Krichever, “A periodic non-Abelian Toda lattice and its generalization to two dimensions,” Usp. Mat. Nauk,36, No. 2, 72–80 (1981).

    Google Scholar 

  10. P. Deift, L. C. Li, and C. Tomei, “Toda flows with infinitely many variables,” J. Funct. Anal.,64, No. 3, 358–402 (1985).

    Google Scholar 

  11. Yu. L. Daletskii and M. G. Krein, Stability of Solutions of Differential Equations in a Banach Space [in Russian], Nauka, Moscow (1970).

    Google Scholar 

  12. I. M. Krichever, “Methods of algebraic geometry of the theory of non-linear equations,” Usp. Mat. Nauk,32, No. 6, 183–208 (1977).

    Google Scholar 

  13. S. V. Manakov, “On complete integrability and stochastization of discrete dynamic systems,” Zh. Éksp. Teor. Fiz.,67, No. 2, 543–545 (1974).

    Google Scholar 

  14. N. V. Zhernakov, “Integration of Toda lattices in the class of Hilbert-Schmidt operators,” Ukr. Mat. Zh.,39, No. 5, 645–648 (1987).

    Google Scholar 

  15. A. Yu. Daletskii and G. B. Podkolzin, “Group approach to the integration of infinite Toda chains,” Ukr. Mat. Zh.,40, No. 4, 518–521 (1988).

    Google Scholar 

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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 6, pp. 730–747, June, 1990.

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Berezanskii, Y.M., Gekhtman, M.I. Inverse problem of the spectral analysis and non-Abelian chains of nonlinear equations. Ukr Math J 42, 645–658 (1990). https://doi.org/10.1007/BF01058907

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  • DOI: https://doi.org/10.1007/BF01058907

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