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 u τ + u u x + u x x x = 0, − ∞ < x < ∞, τ > 0, where x and τ 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 ⩽ 0 and u ( x, 0 ) = 0 for x > 0 . The method of matched asymptotic coordinate expansions is used to obtain the detailed large‐ τ asymptotic structure of the solution to this problem, which exhibits the formation of a dispersive shock wave.