The large-time development of the solution to an initial-value problem for the Korteweg–de Vries–Burgers equation, II: Initial data has a discontinuous expansive step

In this paper, we consider an initial-value problem for the Korteweg–de Vries–Burgers equation. The normalized Korteweg–de Vries–Burgers equation considered is given by uτ+uux−αuxx+uxxx=0,−∞0, where α>0α>0 is a parameter and xx and ττ 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)=u0(>0) for x≥0x≥0 and u(x,0)=0u(x,0)=0 for x<0x<0. The method of matched asymptotic coordinate expansions is used to obtain the large-ττ asymptotic structure of the solution to this problem, which exhibits the formation of an expansion wave in x≥0x≥0, while the solution is oscillatory when x<−α23τ as τ→∞τ→∞, with the oscillatory envelope being exponentially small in ττ, as τ→∞τ→∞.