Critical behaviours of the fifth Painlevé transcendents and the monodromy data

2016/02/29 by Shimomura, Shun
#34M35 #34M55 #34M56 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1602.08808

Abstract

For the fifth Painlevé equation, we present families of convergent series solutions near the origin and the corresponding monodromy data for the associated isomonodromy linear system. These solutions are of complex power type, of inverse logarithmic type and of Taylor series type. It is also possible to compute the monodromy data in non-generic cases. Solutions of logarithmic type are derived from those of inverse logarithmic type through a Bäcklund transformation found by Gromak. In a special case the complex power type of solutions have relatively simple oscillatory expressions. For the complex power type of solutions in the generic case, we clarify the structure of the analytic continuation on the universal covering around the origin, and examine the distribution of zeros, poles and 1-points. It is shown that two kinds of spiral domains including a sector as a special case are alternately arrayed; the domains of one kind contain sequences both of zeros and of poles, and those of the other kind sequences of 1-points.

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