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-cle Vries-Burgers equation. The normalized Korteweg-de Vries-Burgers equation considered is given by U-r + uu(x) - alpha u(xx) + u(xxx) = 0, -infinity <x <infinity, tau > 0, where alpha > 0 is a parameter and x and tau 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) (>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-tau asymptotic structure of the solution to this problem, which exhibits the formation of an expansion 2 wave in x >= 0, while the solution is oscillatory when x <-alpha(2)/3 tau as tau -> infinity, with the oscillatory envelope being exponentially small in tau, as tau -> infinity. (C) 2009 Elsevier Ltd. All rights reserved.