RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
 General information Latest issue Archive Impact factor Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 SIGMA: Year: Volume: Issue: Page: Find

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

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-1513054DMS-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

Full text: PDF file (3595 kB)
Full text: https://www.imath.kiev.ua/~sigma/2018/125/
References: PDF file   HTML file

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
\Bibitem{Mil18} \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} 

• http://mi.mathnet.ru/eng/sigma1424
• http://mi.mathnet.ru/eng/sigma/v14/p125

 SHARE:

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.