Abstract
For the Painlevé equations (I) through (V) near the point at infinity we present several families of two-parameter solutions. Our solutions are expressed by asymptotic series with coefficients polynomial in exponential terms, and also by convergent power series in exponential terms with coefficients expanded into asymptotic series. Both expressions are valid without a restriction on integration constants. We propose a direct method to derive asymptotic solutions, which is also applicable to more general nonlinear equations. As applications of our results, for general solutions of the third and the fifth Painlevé equations, we give estimates for the number of a-points including poles in given sectors.
| Original language | English |
|---|---|
| Pages (from-to) | 277-319 |
| Number of pages | 43 |
| Journal | Funkcialaj Ekvacioj |
| Volume | 58 |
| Issue number | 2 |
| DOIs | |
| Publication status | Published - 2015 Aug 13 |
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Keywords
- Asymptotic solutions
- Hamiltonian system
- Painlevé equations
ASJC Scopus subject areas
- Algebra and Number Theory
- Analysis
- Geometry and Topology
Cite this
Series expansions of painlevé transcendents near the point at infinity. / Shimomura, Shun.
In: Funkcialaj Ekvacioj, Vol. 58, No. 2, 13.08.2015, p. 277-319.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Series expansions of painlevé transcendents near the point at infinity
AU - Shimomura, Shun
PY - 2015/8/13
Y1 - 2015/8/13
N2 - For the Painlevé equations (I) through (V) near the point at infinity we present several families of two-parameter solutions. Our solutions are expressed by asymptotic series with coefficients polynomial in exponential terms, and also by convergent power series in exponential terms with coefficients expanded into asymptotic series. Both expressions are valid without a restriction on integration constants. We propose a direct method to derive asymptotic solutions, which is also applicable to more general nonlinear equations. As applications of our results, for general solutions of the third and the fifth Painlevé equations, we give estimates for the number of a-points including poles in given sectors.
AB - For the Painlevé equations (I) through (V) near the point at infinity we present several families of two-parameter solutions. Our solutions are expressed by asymptotic series with coefficients polynomial in exponential terms, and also by convergent power series in exponential terms with coefficients expanded into asymptotic series. Both expressions are valid without a restriction on integration constants. We propose a direct method to derive asymptotic solutions, which is also applicable to more general nonlinear equations. As applications of our results, for general solutions of the third and the fifth Painlevé equations, we give estimates for the number of a-points including poles in given sectors.
KW - Asymptotic solutions
KW - Hamiltonian system
KW - Painlevé equations
UR - http://www.scopus.com/inward/record.url?scp=84939250508&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84939250508&partnerID=8YFLogxK
U2 - 10.1619/fesi.58.277
DO - 10.1619/fesi.58.277
M3 - Article
AN - SCOPUS:84939250508
VL - 58
SP - 277
EP - 319
JO - Funkcialaj Ekvacioj
JF - Funkcialaj Ekvacioj
SN - 0532-8721
IS - 2
ER -