Geodesic
Asymptotic behaviour of a special solution of Abel's equation relating to a cusp catastrophe
Sbornik. Mathematics, Tome 197 (2006) no. 1, pp. 53-67 Cet article a éte moissonné depuis la source Math-Net.Ru

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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, small parameter
Mots-clés : singular perturbations, cusp catastrophe. 
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A. M. Il'in; B. I. Suleimanov. Asymptotic behaviour of a special solution of Abel's equation relating to a cusp catastrophe. Sbornik. Mathematics, Tome 197 (2006) no. 1, pp. 53-67. http://geodesic.mathdoc.fr/item/SM_2006_197_1_a3/

[1] A. M. Ilin, B. I. Suleimanov, “O dvukh spetsialnykh funktsiyakh, svyazannykh s osobennostyami tipa sborki”, Dokl. RAN, 387:2 (2002), 156–158 | MR

[2] V. I. Arnold, Teoriya katastrof, Nauka, M., 1980 | MR

[3] A. M. Ilin, B. I. Suleimanov, “Zarozhdenie kontrastnykh struktur tipa stupenki, svyazannoe s katastrofoi sborki”, Matem. sb., 195:12 (2004), 27–46 | MR

[4] S. A. Chaplygin, Novyi metod priblizhennogo integrirovaniya differentsialnykh uravnenii, Gostekhizdat, M.–L., 1950