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

We show that the well-known Hastings–McLeod solution to the second Painlevé equation is pole-free in the region $$\arg x \in [-\frac{\pi }{3},\frac{\pi }{3}]\cup [\frac{2\pi }{3},\frac{ 4 \pi }{3}]$$argx∈[-π3,π3]∪[2π3,4π3], 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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