Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 117, 14 pages      arXiv:1804.11273      https://doi.org/10.3842/SIGMA.2018.117
Contribution to the Special Issue on Painlevé Equations and Applications in Memory of Andrei Kapaev

Truncated Solutions of Painlevé Equation ${\rm P}_{\rm V}$

Rodica D. Costin
The Ohio State University, 231 W 18th Ave, Columbus, OH 43210, USA

Received May 01, 2018, in final form October 25, 2018; Published online October 31, 2018

Abstract
We obtain convergent representations (as Borel summed transseries) for the five one-parameter families of truncated solutions of the fifth Painlevé 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 Painlevé equations are also included.

Key words: Painlevé trascendents; the fifth Painlevé equation; truncated solutions; poles of truncated solutions.

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