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Mathematical Notes

, Volume 62, Issue 4, pp 488–500 | Cite as

Completely integrable nonlinear dynamical systems of the Langmuir chains type

  • V. N. Sorokin
Article

Abstract

The solution of the Cauchy problem for semi-infinite chains of ordinary differential equations, studied first by O. I. Bogoyavlenskii in 1987, is obtained in terms of the decomposition in a multidimensional continuous fraction of Markov vector functions (the resolvent functions) related to the chain of a nonsymmetric operator; the decomposition is performed by the Euler-Jacobi-Perron algorithm. The inverse spectral problem method, based on Lax pairs, on the theory of joint Hermite-Padé approximations, and on the Sturm-Liouville method for finite difference equations is used.

Key words

nonlinear dynamical system Langmuir chain moments problem multidimensional continuous fraction 

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References

  1. 1.
    J. K. Moser, “Three integrable Hamiltonian systems connected with isospectral deformations,”Adv. Math.,16, 197–220 (1975).CrossRefMATHGoogle Scholar
  2. 2.
    O. I. Bogoyavlenskii, “Some construction of integrable dynamical systems,”Izv. Akad. Nauk SSSR Ser. Mat. [Math. USSR-Izv.],51, No. 4, 737–766 (1987).MATHMathSciNetGoogle Scholar
  3. 3.
    O. I. Bogoyavlenskii, “Integrable dynamical systems related with the KdF equation,”Izv. Akad. Nauk SSSR Ser. Mat. [Math. USSR-Izv.],51, No. 6, 1123–1141 (1987).MATHMathSciNetGoogle Scholar
  4. 4.
    A. S. Osipov, “Discrete analog of the Korteweg-de Vries equation (KdF): integration by the inverse problem method,”Mat. Zametki [Math. Notes],56, No. 6, 141–144 (1994).MATHMathSciNetGoogle Scholar
  5. 5.
    V. A. Yurko, “On integration of nonlinear dynamical systems by the inverse spectral problem method,”Mat. Zametki [Math. Notes],57, No. 6, 945–949 (1995).MATHMathSciNetGoogle Scholar
  6. 6.
    E. M. Nikishin and V. N. Sorokin,Rational Approximations and Orthogonality [in Russian] Nauka, Moscow (1988).Google Scholar
  7. 7.
    V. I. Parusnikov, “The Jacobi-Perron algorithm and joint approximation of functions,”Mat. Sb. [Math. USSR-Sb.],114, No. 2, 322–333 (1981).MATHMathSciNetGoogle Scholar
  8. 8.
    V. A. Kalyagin, “Of the Hermite-Padé approximation and spectral theory of nonsymmetric operators,”Mat. Sb. [Math. USSR-Sb.],185, No. 6, 79–100 (1994).MATHGoogle Scholar
  9. 9.
    V. N. Sorokin,Connection Between Matrix Hermite-Padé Approximation and Matrix Continued Fractions in the Example of a Particular Toeplitz Matrix, Reprint No. ANO-346, Université des Sciences et Technologies de Lille, Lille (1995).Google Scholar
  10. 10.
    N. I. Akhiezer,Classical Moments Problem [in Russian], Fizmatgiz, Moscow (1961).Google Scholar
  11. 11.
    E. M. Nikishin, “On Joint Padé Approximations,”Mat. Sb. [Math. USSR-Sb.],113, No. 4, 499–519 (1980).MATHMathSciNetGoogle Scholar
  12. 12.
    V. N. Sorokin, “Convergence of joint Padé approximations to functions of Stieltjes type,”Izv. Vyssh. Uchebn. Zaved. Mat. [Soviet Math. J. (Iz. VUZ)], No. 7, 48–56 (1987).MATHMathSciNetGoogle Scholar
  13. 13.
    A. S. Vshivtsev and V. N. Sorokin, “Perturbation theory for the Schrödinger equation with the polynomial potential,”Izv. Vyssh. Uchebn. Zaved. Fiz. [Russian Phys. J.], No. 1, 95–101 (1994).MathSciNetGoogle Scholar

Copyright information

© Plenum Publishing Corporation 1998

Authors and Affiliations

  • V. N. Sorokin
    • 1
  1. 1.M. V. Lomonosov Moscow State UniversityMoscowRussia

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