A. M. Il'in, B. I. Suleimanov, “Asymptotic behaviour of a special solution of Abel's equation relating to a cusp catastrophe”, Sb. Math., 197:1 (2006), 53

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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'in^a, B. I. Suleimanov^b

^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) Russian version article

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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

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”, Sb. Math., 197:1 (2006), 53–67

Citation in format AMSBIB

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

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\vol 197

\issue 1

\pages 53--67

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Linking options:

https://www.mathnet.ru/eng/sm1119

https://doi.org/10.1070/SM2006v197n01ABEH003746

https://www.mathnet.ru/eng/sm/v197/i1/p55

Cycle of papers

Asymptotic behaviour of a special solution of Abel's equation relating to a cusp catastrophe
A. M. Il'in, B. I. Suleimanov
Mat. Sb., 2006, 197:1, 55–70
Asymptotic behaviour of a special solution of Abel's equation relating to a cusp catastrophe. II. Large values of the parameter $t$
A. M. Il'in, B. I. Suleimanov
Mat. Sb., 2007, 198:9, 81–106

This publication is cited in the following 4 articles:

“Arlen Mikhailovich Ilin (k vosmidesyatiletiyu so dnya rozhdeniya)”, Ufimsk. matem. zhurn., 4:2 (2012), 3–12
“Arlen Mikhailovich Il'in. On the occasion of his 80th birsday”, Proc. Steklov Inst. Math. (Suppl.), 281, suppl. 1 (2013), 1–4
Elias U, “Existence of global solutions of some ordinary differential equations”, J. Math. Anal. Appl., 340:1 (2008), 739–745
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

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

Statistics & downloads:
Abstract page:	1678
Russian version PDF:	276
English version PDF:	24
References:	92
First page:	3

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