Point vortices and polynomials of the Sawada-Kotera and Kaup-Kupershmidt equations

TL;DR

This paper explores the special polynomials linked to the generalized K_2 hierarchy, revealing their properties, differential relations, and connections to point vortex configurations in fluid dynamics.

Contribution

It introduces new differential-difference relations and algebraic properties of polynomials associated with integrable equations like Sawada-Kotera and Kaup-Kupershmidt.

Findings

Derived differential-difference relations for the polynomials

Established connection between polynomials and vortex configurations

Analyzed properties and root relations of the polynomials

Abstract

Rational solutions and special polynomials associated with the generalized K_2 hierarchy are studied. This hierarchy is related to the Sawada-Kotera and Kaup-Kupershmidt equations and some other integrable partial differential equations including the Fordy-Gibbons equation. Differential-difference relations and differential equations satisfied by the polynomials are derived. The relationship between these special polynomials and stationary configurations of point vortices with circulations Gamma and -2Gamma is established. Properties of the polynomials are studied. Differential-difference relations enabling one to construct these polynomials explicitly are derived. Algebraic relations satisfied by the roots of the polynomials are found.