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

Abstract

We explore the connection between an infinite system of particles in R 2 described by a bi-dimensional version of the Toda equations with the theory of orthogonal polynomials in two variables. We define a 2D Toda lattice in the sense that we consider only one time variable and two space variables describing a mesh of interacting particles over the plane. We show that this 2D Toda lattice is related with the matrix coefficients of the three term relations of bivariate orthogonal polynomials associated with an exponential modification of a positive measure. Moreover, block Lax pairs for 2D Toda lattices are deduced.

Non-Classical Symmetry Analysis of a Class of Nonlinear Lattice Equations

“…In 1993, Levi [8] proposed the classical Lie symmetry method for differential-difference equations. The Toda equation is a classical model of differential-difference equation [9][10][11][12], whose symmetries have been studied [13] and the differential-difference Lie symmetry method was applied to solve a class of Toda-like lattice equations [14]. In 2022, a survey of the connection between orthogonal polynomials, Toda lattices and related lattices, and Painlevé equations (discrete and continuous) was given [15].…”