Application of connection formula on eliminating plausible asymptotic representation of the painlevé transcendents

Abstract

There are several existing ways to find plausible asymptotic representations of the Painlevé transcendents, but it is very hard, if not impossible, to show that these asymptotic representations are the ones of the transcendents unless the uniqueness is proved. In this aricle, we study the fourth general Painlevé transcendents and use connection formula to eliminate a very plausible asymptotic representation of the fourth Painlevé transcendents.

This aricle also has a very important side product. It is well known that each Painlevé equation

is related to a linear system

and the mondromy data of system (2) is independent of x if y is a solution of (1) and q = y′. There is no doubt that this is a great tool to be used to find the connection formulae of two asymptotics of the Painlevé transcendents. Can we claim that an asymptotic representation is that of a solution of Eq. (1) if the monodromy data of (2) corresponding to this asymptotic representation is independent of x? Our result in this aricle proves that the answer to this question is surprisingly no.

Keywords:

• Painlevé equations
• Isomonodromic data
• Asymptotics
• Connection formula
• 2000 Mathematics Subject Classification: 34E99