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

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.