Abstract
For the Schlesinger equation associated with the sixth Painlevé equation (PVI) near the critical point, we present families of solutions expanded into convergent series with matrix coefficients. These families yield four basic solutions of (PVI) in Guzzetti's table describing the critical behaviours of the sixth Painleve transcendents; two of the basic solutions are of complex power type, and two are of logarithmic type. Consequently, the convergence of the logarithmic solutions is verified. Furthermore we obtain more information on these basic solutions as well as on inverse logarithmic solutions. For complex power solutions, examining related inverse oscillatory ones, we discuss sequences of zeros and poles, non-decaying exponential oscillation and the analytic continuation around the critical point, and show the spiral distribution of poles conjectured by Guzzetti.
| Original language | English |
|---|---|
| Pages (from-to) | 417-463 |
| Number of pages | 47 |
| Journal | Publications of the Research Institute for Mathematical Sciences |
| Volume | 51 |
| Issue number | 3 |
| DOIs | |
| Publication status | Published - 2015 Aug 28 |
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Keywords
- Distribution of poles
- Isomonodromy deformation
- Painlevé transcendents
- Schlesinger equation
ASJC Scopus subject areas
- Mathematics(all)
Cite this
The sixth painlevé transcendents and the associated schlesinger equation. / Shimqmura, Shun.
In: Publications of the Research Institute for Mathematical Sciences, Vol. 51, No. 3, 28.08.2015, p. 417-463.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - The sixth painlevé transcendents and the associated schlesinger equation
AU - Shimqmura, Shun
PY - 2015/8/28
Y1 - 2015/8/28
N2 - For the Schlesinger equation associated with the sixth Painlevé equation (PVI) near the critical point, we present families of solutions expanded into convergent series with matrix coefficients. These families yield four basic solutions of (PVI) in Guzzetti's table describing the critical behaviours of the sixth Painleve transcendents; two of the basic solutions are of complex power type, and two are of logarithmic type. Consequently, the convergence of the logarithmic solutions is verified. Furthermore we obtain more information on these basic solutions as well as on inverse logarithmic solutions. For complex power solutions, examining related inverse oscillatory ones, we discuss sequences of zeros and poles, non-decaying exponential oscillation and the analytic continuation around the critical point, and show the spiral distribution of poles conjectured by Guzzetti.
AB - For the Schlesinger equation associated with the sixth Painlevé equation (PVI) near the critical point, we present families of solutions expanded into convergent series with matrix coefficients. These families yield four basic solutions of (PVI) in Guzzetti's table describing the critical behaviours of the sixth Painleve transcendents; two of the basic solutions are of complex power type, and two are of logarithmic type. Consequently, the convergence of the logarithmic solutions is verified. Furthermore we obtain more information on these basic solutions as well as on inverse logarithmic solutions. For complex power solutions, examining related inverse oscillatory ones, we discuss sequences of zeros and poles, non-decaying exponential oscillation and the analytic continuation around the critical point, and show the spiral distribution of poles conjectured by Guzzetti.
KW - Distribution of poles
KW - Isomonodromy deformation
KW - Painlevé transcendents
KW - Schlesinger equation
UR - http://www.scopus.com/inward/record.url?scp=84940370590&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84940370590&partnerID=8YFLogxK
U2 - 10.4171/PRIMS/160
DO - 10.4171/PRIMS/160
M3 - Article
AN - SCOPUS:84940370590
VL - 51
SP - 417
EP - 463
JO - Publications of the Research Institute for Mathematical Sciences
JF - Publications of the Research Institute for Mathematical Sciences
SN - 0034-5318
IS - 3
ER -