Integrable dynamics of charges related to bilinear hypergeometric equation
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 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 of bilinear hypergeometric equation
for which the classical orthogonal and the Adler-Moser polynomials represent
some particular cases.