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 by u_τ +u u_x+u_xxx=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) =u₀>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.