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Huang, Min; Xu, Shuai-Xia; Zhang, Lun

Location of poles for the Hastings-McLeod solution to the second Painlevé equation. (English) Zbl 1342.30026

Constr. Approx. 43, No. 3, 463-494 (2016).
Summary: We show that the well-known Hastings-McLeod solution to the second Painlevé equation is pole-free in the region \(\arg x \in [-\frac{\pi }{3},\frac{\pi }{3}]\cup [\frac{2\pi }{3},\frac{ 4 \pi }{3}]\), which proves an important special case of a general conjecture concerning pole distributions of Painlevé transcedents proposed by Novokshenov. Our strategy is to construct explicit quasi-solutions approximating the Hastings-McLeod solution in different regions of the complex plane and estimate the errors rigorously. The main idea is very similar to the one used to prove Dubrovin’s conjecture for the first Painlevé equation, but there are various technical improvements.

Cited in 10 Documents

MSC:

30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
30E10 Approximation in the complex plane
34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies

Keywords:

second Painlevé equation; Hastings-McLeod solutions; pole distribution

Software:

DLMF
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Full Text: DOI arXiv

References:

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