Equilibrium of Charges and Differential Equations Solved by Polynomials II

Igor Loutsenko, Oksana Yermolayeva

Published 2024-10-02, updated 2025-01-02Version 2

We continue study of equilibrium of two species of 2d coulomb charges (or point vortices in 2d ideal fluid) started in [20]. 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.