Schur flows and orthogonal polynomials on the unit circle

Abstract

Connections between Toda lattices (Toda chains) and similar non-linear chains and the theory of orthogonal polynomials on the real axis have been studied in detail during the last decades. Another system of difference differential equations, known as the Schur flow, is considered in this paper in the framework of the theory of orthogonal polynomials on the unit circle. A Lax pair for this system is found and the dynamics of the corresponding spectral measure is described. The general result is illustrated by an example of Bessel modified measures on the unit circle: the large-time asymptotic behaviour of their reflection coefficients is determined.