Solvable dynamical systems and isospectral matrices defined in terms of the zeros of orthogonal or otherwise special polynomials

TL;DR

This paper reviews properties of special polynomials and introduces a general method to construct isospectral matrices based on their zeros, leading to solvable nonlinear differential systems.

Contribution

It presents a novel approach to derive solvable nonlinear differential systems from the zeros of special polynomials, expanding the understanding of isospectral matrices.

Findings

Construction of isospectral matrices from polynomial zeros

Reduction of complex equations to solvable ODE systems

Development of algebraically solvable nonlinear differential equations

Abstract

Several recently discovered properties of multiple families of special polynomials (some orthogonal and some not) that satisfy certain differential, difference or q-difference equations are reviewed. A general method of construction of isospectral matrices defined in terms of the zeros of such polynomials is discussed. The method involves reduction of certain partial differential, differential difference and differential q-difference equations to systems of ordinary differential equations and their subsequent linearization about the zeros of polynomials in question. Via this process, solvable (in terms of algebraic operations) nonlinear first order systems of ordinary differential equations are constructed.