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.
nonlinear dynamical system Langmuir chain moments problem multidimensional continuous fraction
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J. K. Moser, “Three integrable Hamiltonian systems connected with isospectral deformations,”Adv. Math.,16, 197–220 (1975).CrossRefMATHGoogle Scholar
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
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
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
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
E. M. Nikishin and V. N. Sorokin,Rational Approximations and Orthogonality [in Russian] Nauka, Moscow (1988).Google Scholar
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
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
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
N. I. Akhiezer,Classical Moments Problem [in Russian], Fizmatgiz, Moscow (1961).Google Scholar
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
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