Why is my rational Painlev\'e V solution not unique?

TL;DR

This paper investigates the conditions under which rational solutions to the Painlevé V equation are not unique, identifying seed solutions and providing a systematic method to derive multiple solutions under special parameter conditions.

Contribution

It introduces a formalism linking affine Weyl group orbits to rational solutions and derives explicit conditions and methods for generating non-unique solutions.

Findings

Identified conditions for non-uniqueness of rational Painlevé V solutions.

Derived explicit seed solutions leading to multiple solutions.

Developed a systematic method to obtain closed-form solutions from seed solutions.

Abstract

Under special conditions the Painlev\'e V equation has more than one rational solution solving it with the same parameters. In the setting of formalism that identifies points on orbits of the fundamental shift operators of $A^{(1)}_{3}$ affine Weyl group with rational solutions we derive conditions for such non-uniqueness to occur. We identify the seed solutions from which the non-unique solutions are generated and put forward a method to systematically obtain their closed expressions from the underlying seed solutions.