
SIGMA, 2018, Volume 14, 125, 38 pages
(Mi sigma1424)




This article is cited in 1 scientific paper (total in 1 paper)
On the Increasing Tritronquée Solutions of the PainlevéII Equation
Peter D. Miller^{} ^{} Department of Mathematics, University of Michigan, East Hall, 530 Church St., Ann Arbor, MI 48109, USA
Abstract:
The increasing tritronquée solutions of the PainlevéII equation with parameter $\alpha$ exhibit squareroot asymptotics in the maximallylarge sector $\arg(x)<\tfrac{2}{3}\pi$ and have recently appeared in applications where it is necessary to understand the behavior of these solutions for complex values of $\alpha$. Here these solutions are investigated from the point of view of a Riemann–Hilbert representation related to the Lax pair of Jimbo and Miwa, which naturally arises in the analysis of rogue waves of infinite order. We show that for generic complex $\alpha$, all such solutions are asymptotically polefree along the bisecting ray of the complementary sector $\arg(x)<\tfrac{1}{3}\pi$ that contains the poles far from the origin. This allows the definition of a total integral of the solution along the axis containing the bisecting ray, in which certain algebraic terms are subtracted at infinity and the poles are dealt with in the principalvalue sense. We compute the value of this integral for all such solutions. We also prove that if the PainlevéII parameter $\alpha$ is of the form $\alpha=\pm\tfrac{1}{2}+\mathrm{i} p$, $p\in\mathbb{R}\setminus\{0\}$, one of the increasing tritronquée solutions has no poles or zeros whatsoever along the bisecting axis.
Keywords:
PainlevéII equation; tronquée solutions.
Funding Agency 
Grant Number 
National Science Foundation 
DMS1513054 DMS1812625 
The author's work was supported by the National Science Foundation under grants DMS1513054
and DMS1812625. 
DOI:
https://doi.org/10.3842/SIGMA.2018.125
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Bibliographic databases:
ArXiv:
1804.03173
Document Type:
Article
MSC: 33E17; 34M40; 34M55; 35Q15 Received: April 11, 2018; in final form November 12, 2018; Published online November 15, 2018
Language: English
Citation:
Peter D. Miller, “On the Increasing Tritronquée Solutions of the PainlevéII Equation”, SIGMA, 14 (2018), 125, 38 pp.
Citation in format AMSBIB
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\by Peter~D.~Miller
\paper On the Increasing Tritronqu\'ee Solutions of the Painlev\'eII Equation
\jour SIGMA
\yr 2018
\vol 14
\papernumber 125
\totalpages 38
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\crossref{https://doi.org/10.3842/SIGMA.2018.125}
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This publication is cited in the following articles:

Peter A. Clarkson, “Open Problems for Painlevé Equations”, SIGMA, 15 (2019), 006, 20 pp.

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