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Mat. Sb., 2004, Volume 195, Number 12, Pages 27–46 (Mi msb863)  

This article is cited in 6 scientific papers (total in 8 papers)

Birth of step-like contrast structures connected with 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 the ordinary differential equation $u"_{xx}=u^3-tu-x$ is considered which relates to solutions of a broad spectrum of partial differential equations with a small parameter. The function $u(x,t)$ is the dominant term of asymptotic expressions with respect to the small parameter for these solutions near cusp points of the limiting solution. The existence of this special function $u(x,t)$ is proved; its uniform asymptotics at infinity are constructed and substantiated.

DOI: https://doi.org/10.4213/sm863

Full text: PDF file (309 kB)
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English version:
Sbornik: Mathematics, 2004, 195:12, 1727–1746

Bibliographic databases:

UDC: 517.928
MSC: 34E05, 35B40, 35B25
Received: 20.05.2004

Citation: A. M. Il'in, B. I. Suleimanov, “Birth of step-like contrast structures connected with a cusp catastrophe”, Mat. Sb., 195:12 (2004), 27–46; Sb. Math., 195:12 (2004), 1727–1746

Citation in format AMSBIB
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\paper Birth of step-like contrast structures connected with a cusp catastrophe
\jour Mat. Sb.
\yr 2004
\vol 195
\issue 12
\pages 27--46
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    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”, Sb. Math., 197:1 (2006), 53–67  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    2. 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
    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
    5. Oleg Yu. Khachay, Pavel A. Nosov, “On some numerical integration curves for PDE in neighborhood of “butterfly” catastrophe point”, Ural Math. J., 2:2 (2016), 127–140  mathnet  crossref  zmath
    6. O. Yu. Khachay, “Asymptotics of a solution of a three-dimensional nonlinear wave equation near a butterfly catastrophe point”, Proc. Steklov Inst. Math. (Suppl.), 301, suppl. 1 (2018), 72–87  mathnet  crossref  crossref  isi  elib
    7. Konopelchenko B.G., Ortenzi G., “Parabolic Regularization of the Gradient Catastrophes For the Burgers-Hopf Equation and Jordan Chain”, J. Phys. A-Math. Theor., 51:27 (2018), 275201  crossref  mathscinet  zmath  isi
    8. O. Yu. Khachai, “Ob odnoi asimptoticheskoi zadache dlya obyknovennogo differentsialnogo uravneniya vtorogo poryadka s nelineinostyu, sootvetstvuyuschei katastrofe tipa «babochka»”, Matematicheskaya fizika, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 152, VINITI RAN, M., 2018, 125–142  mathnet  mathscinet
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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