Equilibrium of Charges and Differential Equations Solved by Polynomials II

TL;DR

This paper investigates equilibrium configurations of two-dimensional Coulomb charges and point vortices, revealing that third-order differential operators and Darboux transformations are key to understanding complex multi-species equilibria.

Contribution

It extends previous work by establishing the role of third-order Darboux transformations in modeling multi-species charge equilibria and introduces new configurations involving a third species.

Findings

Third-order Darboux transformations relate to multi-species equilibria.

Equilibrium configurations with a third species are generated by these transformations.

Connections to integrable hierarchies are briefly discussed.

Abstract

We continue study of equilibrium of two species of 2d coulomb charges (or point vortices in 2d ideal fluid) started in (Igor Loutsenko, J. Phys. A: Math. Gen. 37, 1309, 2004). Although for two species of vortices with circulation ratio -1 the relationship between the equilibria and the factorization/Darboux transformation of the Schrodinger operator was established a long ago, the question about similar relationship for the ratio -2 remained unanswered. Here we present the answer: One has to consider Darboux-type transformations of third order differential operators rather than second order Schrodinger operators. Furthermore, we show that such transformations can also generate equilibrium configurations where an additional charge of a third specie is present. Relationship with integrable hierarchies is briefly discussed.