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Sbornik: Mathematics, 2006, Volume 197, Issue 1, Pages 53–67
DOI: https://doi.org/10.1070/SM2006v197n01ABEH003746 (Mi sm1119) 		 
This article is cited in 2 scientific papers (total in 4 papers)

Asymptotic behaviour of a special solution of Abel's equation relating to a cusp catastrophe

A. M. Il'ina, B. I. Suleimanovb

a Chelyabinsk State University
b Institute of Mathematics with Computing Centre, Ufa Science Centre, Russian Academy of Sciences
English version PDF (260 kB) Citations (4)
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DOI: https://doi.org/10.1070/SM2006v197n01ABEH003746
Abstract: A special solution of Abel's ordinary differential equation of the first kind $u'_{x}+u^3-tu-x=0$ is considered, which describes the behaviour of a broad spectrum of solutions of partial differential equations with a small parameter in the neighbourhood of cusp points of their slowly varying equilibrium positions. The existence of this special solution is demonstrated; an asymptotic formula for it as $|x|\to\infty$, $t\to-\infty$ is constructed and substantiated.
Bibliography: 4 titles.
Keywords: asymptotics, singular perturbations, small parameter, cusp catastrophe.
Received: 20.06.2005
Russian version:
Matematicheskii Sbornik, 2006, Volume 197, Number 1, Pages 55–70
DOI: https://doi.org/10.4213/sm1119
Bibliographic databases:
UDC: 517.928
MSC: Primary 34E05, 35B40; Secondary 35B25
Language: English
Original paper language: Russian
Citation: A. M. Il'in, B. I. Suleimanov, “Asymptotic behaviour of a special solution of Abel's equation relating to a cusp catastrophe”, Mat. Sb., 197:1 (2006), 55–70; Sb. Math., 197:1 (2006), 53–67
Citation in format AMSBIB
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  • https://www.mathnet.ru/eng/sm/v197/i1/p55
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    This publication is cited in the following 4 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Abstract page:1608
    Russian version PDF:259
    English version PDF:7
    References:72
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