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Mat. Sb., 2006, Volume 197, Number 1, Pages 55–70 (Mi msb1119)  

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

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
Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/sm1119

Full text: PDF file (485 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2006, 197:1, 53–67

Bibliographic databases:

UDC: 517.928
MSC: Primary 34E05, 35B40; Secondary 35B25
Received: 20.06.2005

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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\pages 55--70
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  • https://doi.org/10.4213/sm1119
  • http://mi.mathnet.ru/eng/msb/v197/i1/p55

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. M. Il'in, B. I. Suleimanov, “Asymptotic behaviour of a special solution of Abel's equation relating to a cusp catastrophe. II. Large values of the parameter $t$”, Sb. Math., 198:9 (2007), 1299–1324  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    2. Elias U, “Existence of global solutions of some ordinary differential equations”, J. Math. Anal. Appl., 340:1 (2008), 739–745  crossref  mathscinet  zmath  isi  elib  scopus
    3. “Arlen Mikhailovich Ilin (k vosmidesyatiletiyu so dnya rozhdeniya)”, Ufimsk. matem. zhurn., 4:2 (2012), 3–12  mathnet  mathscinet
    4. “Arlen Mikhailovich Il'in. On the occasion of his 80th birsday”, Proc. Steklov Inst. Math. (Suppl.), 281, suppl. 1 (2013), 1–4  mathnet  crossref  isi
  • Математический сборник Sbornik: Mathematics (from 1967)
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