The continuum limit of the Toda lattice and discrete orthogonal polynomials

A method for integration of the Cauchy problem for the hyperbolic system (the so-called continuum limit of the Toda lattice, see above) is proposed. ∂ α / ∂ t = - ($\beta$-α)/4 ∂ α/ ∂ x, ∂ $\beta$ / ∂ t = - ($\beta$-α)/4 ∂ $\beta$/ ∂ x, α(x,0)=α(x), $\beta$(x,0)=$\beta$(x), α(0,t)=$\beta$(0,t)=α(0), α(1,t)=$\beta$(1,t)=α(1). The method is based on some extremal problems of the theory of logarithmic potentials. The method is justified by means of the known results of the asymptotic theory of the polynomials orthogonal with respect to a discrete measure.