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Mat. Sb. (N.S.), 1976, Volume 99(141), Number 2, Pages 261–281 (Mi msb2751)  

This article is cited in 64 scientific papers (total in 64 papers)

Asymptotics of the solution of the Cauchy problem for the Korteweg–de Vries equation with initial data of step type

E. Ya. Khruslov


Abstract: The method of the inverse scattering problem is used to solve the Cauchy problem for the Korteweg–deVries equation with initial data of step type: $u(x,0)\to-c^2$ ($x\to-\infty$), $u(x,0)\to0$ ($x\to\infty$). Formulas are obtained for transforming the scattering data with respect to the time, making it possible to obtain a solution $u(x,t)$ of the problem for arbitrary $t$ with the aid of linear integral equations of scattering theory. The asymptotic behavior of the solution as $t\to+\infty$ is investigated in a neighborhood of the wave front $(x>4c^2t-\frac1{2c}\ln t^N)$. It is shown that in this region the solution splits up into solitons, the distance between which increases as $\ln t^{1/c}$, and an explicit form for these solitons is derived.
Bibliography: 12 titles.

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English version:
Mathematics of the USSR-Sbornik, 1976, 28:2, 229–248

Bibliographic databases:

UDC: 517.946
MSC: Primary 35B40, 35Q99; Secondary 76B25
Received: 21.05.1975

Citation: E. Ya. Khruslov, “Asymptotics of the solution of the Cauchy problem for the Korteweg–de Vries equation with initial data of step type”, Mat. Sb. (N.S.), 99(141):2 (1976), 261–281; Math. USSR-Sb., 28:2 (1976), 229–248

Citation in format AMSBIB
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\by E.~Ya.~Khruslov
\paper Asymptotics of the solution of the Cauchy problem for the Korteweg--de~Vries equation with initial data of step type
\jour Mat. Sb. (N.S.)
\yr 1976
\vol 99(141)
\issue 2
\pages 261--281
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\mathscinet{http://www.ams.org/mathscinet-getitem?mr=487088}
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\transl
\jour Math. USSR-Sb.
\yr 1976
\vol 28
\issue 2
\pages 229--248
\crossref{https://doi.org/10.1070/SM1976v028n02ABEH001649}
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    This publication is cited in the following articles:
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  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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