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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 Painlevé-II 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 Painlevé-II equation with parameter $\alpha$ exhibit square-root asymptotics in the maximally-large 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 pole-free 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 principal-value sense. We compute the value of this integral for all such solutions. We also prove that if the Painlevé-II 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: Painlevé-II equation; tronquée solutions.

Funding Agency Grant Number
National Science Foundation DMS-1513054
DMS-1812625
The author's work was supported by the National Science Foundation under grants DMS-1513054 and DMS-1812625.


DOI: https://doi.org/10.3842/SIGMA.2018.125

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Full text: https://www.imath.kiev.ua/~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 Painlevé-II 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\'e-II Equation
\jour SIGMA
\yr 2018
\vol 14
\papernumber 125
\totalpages 38
\mathnet{http://mi.mathnet.ru/sigma1424}
\crossref{https://doi.org/10.3842/SIGMA.2018.125}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000450717100002}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Peter A. Clarkson, “Open Problems for Painlevé Equations”, SIGMA, 15 (2019), 006, 20 pp.  mathnet  crossref
  • Symmetry, Integrability and Geometry: Methods and Applications
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