Premium
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
Author(s) -
Leach J. A.,
Needham D. J.
Publication year - 2014
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579313000284
Subject(s) - mathematics , dimensionless quantity , initial value problem , mathematical analysis , korteweg–de vries equation , value (mathematics) , statistics , mechanics , physics , nonlinear system , quantum mechanics
In this paper, we consider an initial‐value problem for the Korteweg–de Vries equation. The normalized Korteweg–de Vries equation considered is given byu 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.