Geodesic
Pad\'e approximations for Painlev\'e I and II transcendents
Teoretičeskaâ i matematičeskaâ fizika, Tome 159 (2009) no. 3, pp. 515-526

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We use a version of the Fair–Luke algorithm to find the Padé approximate solutions of the Painlevé I and II equations. We find the distributions of poles for the well-known Ablowitz–Segur and Hastings–McLeod solutions of the Painlevé II equation. We show that the Boutroux tritronquée solution of the Painleé I equation has poles only in the critical sector of the complex plane. The algorithm allows checking other analytic properties of the Painlevé transcendents, such as the asymptotic behavior at infinity in the complex plane.
Keywords: Painlevé equation, meromorphic solution, Padé approximation, continued fraction, Riemann–Hilbert problem.
Mots-clés : distribution of poles 
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     author = {V. Yu. Novokshenov},
     title = {Pad\'e approximations for {Painlev\'e} {I} and {II} transcendents},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {515--526},
     publisher = {mathdoc},
     volume = {159},
     number = {3},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2009_159_3_a16/}
}
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V. Yu. Novokshenov. Pad\'e approximations for Painlev\'e I and II transcendents. Teoretičeskaâ i matematičeskaâ fizika, Tome 159 (2009) no. 3, pp. 515-526. http://geodesic.mathdoc.fr/item/TMF_2009_159_3_a16/