RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Mat. Zametki: Year: Volume: Issue: Page: Find

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

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

Full text: PDF file (476 kB)
References: PDF file   HTML file

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

Bibliographic databases:

UDC: 517.957

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
\Bibitem{Fam08} \by A.~V.~Faminskii \paper Cauchy Problem for the Korteweg--de~Vries Equation in the Case of a Nonsmooth Unbounded Initial Function \jour Mat. Zametki \yr 2008 \vol 83 \issue 1 \pages 119--128 \mathnet{http://mi.mathnet.ru/mz4339} \crossref{https://doi.org/10.4213/mzm4339} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2400004} \zmath{https://zbmath.org/?q=an:1156.35080} \elib{http://elibrary.ru/item.asp?id=10019485} \transl \jour Math. Notes \yr 2008 \vol 83 \issue 1 \pages 107--115 \crossref{https://doi.org/10.1134/S0001434608010136} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000254056300013} \elib{http://elibrary.ru/item.asp?id=13573911} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-48849085654} 

• http://mi.mathnet.ru/eng/mz4339
• https://doi.org/10.4213/mzm4339
• http://mi.mathnet.ru/eng/mz/v83/i1/p119

 SHARE:

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. Garifullin R.N., Suleimanov B.I., “From weak discontinuities to nondissipative shock waves”, Journal of Experimental and Theoretical Physics, 110:1 (2010), 133–146
2. R. N. Garifullin, “Sdvig fazy dlya sovmestnogo resheniya uravneniya KDV i differentsialnogo uravneniya pyatogo poryadka”, Ufimsk. matem. zhurn., 4:2 (2012), 80–86
•  Number of views: This page: 450 Full text: 136 References: 33 First page: 4