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THE LARGE-TIME DEVELOPMENT OF THE SOLUTION TO AN INITIAL-VALUE PROBLEM FOR THE KORTEWEG–DE VRIES EQUATION. II. INITIAL DATA HAS A DISCONTINUOUS COMPRESSIVE STEP

Part of: Infinite-dimensional Hamiltonian systems

Published online by Cambridge University Press:  14 May 2014

J. A. Leach  and
D. J. Needham
J. A. Leach
Affiliation:
School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT,U.K. email D.J.Needham@bham.ac.uk
D. J. Needham
Affiliation:
School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT,U.K. email J.A.Leach@bham.ac.uk
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Abstract

In this paper, we consider an initial-value problem for the Korteweg–de Vries equation. The normalized Korteweg–de Vries equation considered is given by

$$\begin{equation*} u_{\tau }+u u_{x}+u_{xxx}=0, \quad -\infty <x<\infty ,\ \tau >0, \end{equation*}$$
where $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}x$ and $\tau $ represent dimensionless distance and time, respectively. In particular, we consider the case when the initial data has a discontinuous compressive step, where $u(x,0) =u_{0}>0$ for $x \le 0$ and $u(x,0)=0$ for $x>0$. The method of matched asymptotic coordinate expansions is used to obtain the detailed large-$\tau $ asymptotic structure of the solution to this problem, which exhibits the formation of a dispersive shock wave.

MSC classification

Secondary: 37K40: Soliton theory, asymptotic behavior of solutions
Type
Research Article
Information
Mathematika , Volume 60 , Issue 2 , July 2014 , pp. 391 - 414
Copyright
Copyright © University College London 2014 

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