Bertola, M. On the location of poles for the Ablowitz-Segur family of solutions to the second Painlevé equation. (English) Zbl 1254.34123 Nonlinearity 25, No. 4, 1179-1185 (2012). The author proves that all solutions to the second Painlevé equation \(u_{ss}=su+2u^3\) satisfying the asymptotic condition \(u(s)\simeq k\, \text{Ai}(s)\), \(s\to+\infty\), \(k\in{\mathbb C}\), including the famous Ablowitz-Segur and Hastings-McLeod solutions, have no one pole in the region \(\text{Re}(s^{3/2})>\tfrac{3}{2}\ln|k|\). The proof is based on the fact that the poles of the second Painlevé transcendent coincide with zeros of the Fredholm determinant \(\det(\text{Id}_{L^2([s,\infty))}-k^2K_{\text{Ai}}|_{[s,\infty)})\) (\(\tau\)-function). Here, \(K_{\text{Ai}}\bigr|_{[s,\infty)}\) is the operator with the Airy kernel acting on the semi-infinite interval \([s,\infty)\). The author observes that, in the Fourier space, this operator appears to be a composition of multiplication and Cauchy operators, and thus its norm is easy to estimate. In the interior of the above indicated region in the \(s\)-complex plane, the author finds that \(\|k^2K_{\text{Ai}}|_{[s,\infty)}\|<1\) and therefore the determinant does not vanish. Reviewer: Andrei A. Kapaev (Trieste) Cited in 12 Documents MSC: 34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies 34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms 33E17 Painlevé-type functions Keywords:Painlevé equation; movable singular points; Fredholm determinant; Airy kernel PDF BibTeX XML Cite \textit{M. Bertola}, Nonlinearity 25, No. 4, 1179--1185 (2012; Zbl 1254.34123) Full Text: DOI arXiv OpenURL