The large-time development of the solution to an initial-value problem for the Korteweg-de Vries-Burgers equation: I. Initial data has a discontinuous compressive 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 u(iota) + uu(x) - au(xx) + u(xxx) = 0, - infinity < x < infinity, iota > 0, where alpha > 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 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 complete large-tau asymptotic structure of the solution to this problem, which exhibits for u(0), alpha > 0 the formation of a permanent form travelling wave (PTW) structure. This PTW has wave speed u(0)/2 and its profile is either monotonic when alpha(2) >= 2u(0) or oscillatory when 0 < alpha(2) < 2u(0). Further, the correction to the wave speed and the rate of convergence of the solution of the initial-value problem onto the PTW as tau -> infinity are determined for all values of alpha, u(0) > 0.