The large-time development of the solution to an initial-value problem for the Korteweg–de Vries equation: I. Initial data has a discontinuous expansive 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 where x and τ represent dimensionless distance and time, respectively. In particular, we consider the case when the initial data has a discontinuous expansive step, where u ( x , 0) = u 0 for x ⩾ 0 and u ( x , 0) = 0 for x < 0. The method of matched asymptotic coordinate expansions is used to obtain the large-τ asymptotic structure of the solution to this problem, which exhibits the formation of an expansion wave in x ⩾ 0, while the solution is oscillatory in x < 0, with the oscillatory envelope being of as τ → ∞.