Completely integrable nonlinear dynamical systems of the Langmuir chains type

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.