Truncated Solutions of Painlev\'e Equation ${\rm P}_{\rm V}$

TL;DR

This paper derives convergent Borel summed transseries representations for truncated solutions of the fifth Painlevé equation, identifying pole positions and special cases with near-entire analyticity.

Contribution

It provides the first convergent representations for these solutions and characterizes their pole structure, extending understanding of Painlevé V solutions.

Findings

Convergent Borel summed transseries for truncated solutions

Pole locations bordering analyticity regions

Special tri-truncated solutions nearly entire in the complex plane

Abstract

We obtain convergent representations (as Borel summed transseries) for the five one-parameter families of truncated solutions of the fifth Painlev\'e equation with nonzero parameters, valid in half planes, for large independent variable. We also find the position of the first array of poles, bordering the region of analyticity. For a special value of this parameter they represent tri-truncated solutions, analytic in almost the full complex plane, for large independent variable. A brief historical note, and references on truncated solutions of the other Painlev\'e equations are also included.