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. Issue 2 (14th May 2014)

• Record Type: Journal Article
• Authors: Leach, J. A.; Needham, D. J.
• Is Part Of: Mathematika. Volume 60:Issue 2(2014)
• Page Start: 391
• Page End: 414
• Publication Date: 2014-05-14
• Subjects: 37K40 (primary); Mathematics -- Periodicals; 510.5
• DOI: 10.1112/S0025579313000284
• Languages: English
• ISSNs: 0025-5793

Abstract

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.