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Mat. Zametki, 2008, Volume 83, Issue 1, Pages 119–128 (Mi mz4339)  

This article is cited in 2 scientific papers (total in 2 papers)

Cauchy Problem for the Korteweg–de Vries Equation in the Case of a Nonsmooth Unbounded Initial Function

A. V. Faminskii

Peoples Friendship University of Russia

Abstract: In the strip $\Pi=(-1,0)\times\mathbb R$, we establish the existence of solutions of the Cauchy problem for the Korteweg–de Vries equation $u_t+u_{xxx}+uu_x=0$ with initial condition either 1) $u(-1,x)=-x\theta(x)$, or 2) $u(-1,x)=-x\theta(-x)$, where $\theta$ is the Heaviside function. The solutions constructed in this paper are infinitely smooth for $t\in(-1,0)$ and rapidly decreasing as $x\to+\infty$. For the case of the first initial condition, we also establish uniqueness in a certain class. Similar special solutions of the KdV equation arise in the study of the asymptotic behavior with respect to small dispersion of the solutions of certain model problems in a neighborhood of lines of weak discontinuity.

Keywords: Korteweg–de Vries equation, Cauchy problem, Burgers equation, Banach space, gas-dynamic problem, line of weak discontinuity, Bochner measurable mapping

DOI: https://doi.org/10.4213/mzm4339

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English version:
Mathematical Notes, 2008, 83:1, 107–115

Bibliographic databases:

UDC: 517.957
Received: 31.05.2006

Citation: A. V. Faminskii, “Cauchy Problem for the Korteweg–de Vries Equation in the Case of a Nonsmooth Unbounded Initial Function”, Mat. Zametki, 83:1 (2008), 119–128; Math. Notes, 83:1 (2008), 107–115

Citation in format AMSBIB
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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. Garifullin R.N., Suleimanov B.I., “From weak discontinuities to nondissipative shock waves”, Journal of Experimental and Theoretical Physics, 110:1 (2010), 133–146  crossref  adsnasa  isi  elib  scopus
    2. R. N. Garifullin, “Sdvig fazy dlya sovmestnogo resheniya uravneniya KDV i differentsialnogo uravneniya pyatogo poryadka”, Ufimsk. matem. zhurn., 4:2 (2012), 80–86  mathnet  mathscinet
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