Two-fold degeneracy of a class of rational Painlev\'e V solutions

TL;DR

This paper constructs a class of rational solutions to the Painlevé V equation that exhibit a two-fold degeneracy, revealing new symmetries and explicit solutions, with potential generalizations to higher Painlevé systems.

Contribution

It introduces a novel construction of degenerate rational solutions for Painlevé V using affine Weyl group symmetries and Bäcklund transformations, extending to higher Painlevé systems.

Findings

Identified conditions for solution degeneracy

Derived explicit expressions for degenerate solutions

Extended formalism to higher Painlevé systems

Abstract

We present a construction of a class of rational solutions of the Painlev\'e V equation that exhibit a two-fold degeneracy, meaning that there exist two distinct solutions that share identical parameters. The fundamental object of our study is the orbit of translation operators of $A^{(1)}_{3}$ affine Weyl group acting on the underlying seed solution that only allows action of some symmetry operations. By linking points on this orbit to rational solutions, we establish conditions for such degeneracy to occur after involving in the construction additional B\"acklund transformations that are inexpressible as translation operators. This approach enables us to derive explicit expressions for these degenerate solutions. An advantage of this formalism is that it easily allows generalization to higher Painlev\'e systems associated with dressing chains of even period $N>4$.