Integrable Dynamics of Charges Related to Bilinear Hypergeometric Equation

TL;DR

This paper explores integrable charge dynamics linked to a bilinear hypergeometric equation, connecting polynomial root evolution with Coulomb charge equilibria, and relates these to Calogero-Moser systems and orthogonal polynomials.

Contribution

It introduces a novel framework connecting polynomial root dynamics with charge equilibria and integrable systems, extending classical polynomial theories to new bilinear hypergeometric equations.

Findings

Established a connection between root evolution and Coulomb charge dynamics.

Identified fixed points as equilibria related to classical polynomials.

Linked the system to Calogero-Moser models and Lie-algebraic operators.

Abstract

A family of systems related to a linear and bilinear evolution of roots of polynomials in the complex plane is considered. Restricted to the line, the evolution induces dynamics of the Coulomb charges in external potentials, while its fixed points correspond to equilibria of charges (or point vortices in hydrodynamics) in the plane. The construction reveals a direct connection with the theories of the Calogero-Moser systems and Lie-algebraic differential operators. A study of the equilibrium configurations amounts in a construction (bilinear hypergeometric equation) for which the classical orthogonal and the Adler-Moser polynomials represent some particular cases