Geodesic
Tronqu\'ee solutions of the~Painlev\'e II equation
Teoretičeskaâ i matematičeskaâ fizika, Tome 172 (2012) no. 2, pp. 296-307

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We study special solutions of the Painlevé II (PII) equation called tronquée solutions, i.e., those having no poles along one or more critical rays in the complex plane. They are parameterized by special monodromy data of the Lax pair equations. The manifold of the monodromy data for a general solution is a two-dimensional complex manifold with one- and zero-dimensional singularities, which arise because there is no global parameterization of the manifold. We show that these and only these singularities (together with zeros of the parameterization) are related to the tronquée solutions of the PII equation. As an illustration, we consider the known Hastings–McLeod and Ablowitz–Segur solutions and some other solutions to show that they belong to the class of tronquée solutions and correspond to one or another type of singularity of the monodromy data.
Keywords: Painlevé equation, Riemann–Hilbert problem, anharmonic oscillator, Bohr–Sommerfeld quantization, complex WKB method.
Mots-clés : tronquée solution, distribution of poles 
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     author = {V. Yu. Novokshenov},
     title = {Tronqu\'ee solutions of {the~Painlev\'e} {II} equation},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {296--307},
     publisher = {mathdoc},
     volume = {172},
     number = {2},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2012_172_2_a9/}
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V. Yu. Novokshenov. Tronqu\'ee solutions of the~Painlev\'e II equation. Teoretičeskaâ i matematičeskaâ fizika, Tome 172 (2012) no. 2, pp. 296-307. http://geodesic.mathdoc.fr/item/TMF_2012_172_2_a9/