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Location of Poles for the Hastings–McLeod Solution to the Second Painlevé Equation

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Abstract

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.

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Acknowledgments

We would like to express our sincere gratitude and appreciation to Roderick Wong and Dan Dai for organizing a series of conferences and seminars that facilitated our collaboration on this project, as well as other related discussions. Min Huang was supported by the Early Career Scheme 21300114 of Research Grants Council of Hong Kong. Shuai-Xia Xu was supported in part by the National Natural Science Foundation of China under Grant No. 11201493 and 11571376, GuangDong Natural Science Foundation under Grant No. S2012040007824 and 2014A030313176, Postdoctoral Science Foundation of China under Grant No. 2012M521638, and the Fundamental Research Funds for the Central Universities under Grant No. 13lgpy41. Lun Zhang was partially supported by The Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning (No. SHH1411007), by National Natural Science Foundation of China (No.11501120) and by Grant SGST 12DZ 2272800, EZH1411513 from Fudan University.

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Correspondence to Lun Zhang.

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Communicated by Percy A. Deift.

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Huang, M., Xu, SX. & Zhang, L. Location of Poles for the Hastings–McLeod Solution to the Second Painlevé Equation. Constr Approx 43, 463–494 (2016). https://doi.org/10.1007/s00365-015-9307-1

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  • DOI: https://doi.org/10.1007/s00365-015-9307-1

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