Bivariate orthogonal polynomials, 2D Toda lattices and Lax-type pairs

Schur Flows and Orthogonal Polynomials on the Unit Circle

Abstract. The relation between the Toda lattices and similar nonlinear chains and orthogonal polynomials on the real line has been elaborated immensely for the last decades. We examine another system of differential-difference equations known as the Schur flow, within the framework of the theory of orthogonal polynomials on the unit circle. This system can be displayed in equivalent form as the...

Multidimensional Toda Lattices: Continuous and Discrete Time

In this paper we present multidimensional analogues of both the continuousand discrete-time Toda lattices. The integrable systems that we consider here have two or more space coordinates. To construct the systems, we generalize the orthogonal polynomial approach for the continuous and discrete Toda lattices to the case of multiple orthogonal polynomials.

Biorthogonal Laurent polynomials, Töplitz determinants, minimal Toda orbits and isomonodromic tau functions

We consider the class of biorthogonal polynomials that are used to solve the inverse spectral problem associated to elementary co-adjoint orbits of the Borel group of upper triangular matrices; these orbits are the phase space of generalized integrable lattices of Toda type. Such polynomials naturally interpolate between the theory of orthogonal polynomials on the line and orthogonal polynomial...

Equations of motion for zeros of orthogonal polynomials related to the Toda lattices

The Pfaff lattice on symplectic matrices

Abstract. The Pfaff lattice is an integrable system arising from the SR-group factorization in an analogous way to how the Toda lattice arises from the QR-group factorization. In our recent paper [Intern. Math. Res. Notices, (2007) rnm120], we studied the Pfaff lattice hierarchy for the case where the Lax matrix is defined to be a lower Hessenberg matrix. In this paper we deal with the case of ...