Continuous limit of the difference second Painlevé equation and its asymptotic solutions
type: Item
instance of: scholarly article
title: Continuous limit of the difference second Painlevé equation and its asymptotic solutions
zbMATH Open document ID: 1255.39016
author: Shun Shimomura
published in: Journal of the Mathematical Society of Japan
publication date: 2012-10-02T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.jmsj/1343133742
reviewed by: Zhenlai Han
Mathematics Subject Classification ID: 39A45
Mathematics Subject Classification ID: 39A20
Mathematics Subject Classification ID: 34M55
Mathematics Subject Classification ID: 39A12
zbMATH DE Number: 6090091
zbMATH Keywords: difference second Painlevé equation
zbMATH Keywords: second Painlevé equation
zbMATH Keywords: continuous limit
zbMATH Keywords: asymptotic solution
MaRDI profile type: Publication
OpenAlex ID: W2026185435
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DOI: 10.2969/JMSJ/06430733
review text: From the discrete second Painlev\(\acute{\text{e}}\) equation dP\(_{\amalg}\), the author obtains the equation NEWLINE\[NEWLINE v(x+i\varepsilon)+v(x-i\varepsilon)=\frac{(2-\varepsilon^2x)v(x)-\varepsilon^2\alpha}{1+\varepsilon^2v(x)^2}. NEWLINE\]NEWLINE Regarding dP\(_{\amalg}\) as a difference equation, the author presents a certain asymptotic solution that reduces to a triply-truncated solution of P\(_{\amalg}\) in this continuous limit. In a special case, the solution corresponds to a rational one of dP\(_{\amalg}\). Furthermore, the author describes one-parameter families of solutions having sequential limits to truncated solutions of P\(_{\amalg}\). So, the main results in this paper are new and important. These results can be applied to future studies.
description: scientific article; zbMATH DE number 6090091