he large-time development of the solution to an initial-value roblemfor the Kortewegde VriesBurgers equation, II: Initial data has a discontinuous expansive step

摘要

In this paper, we consider an initial-value problem for the Kortewegde VriesBurgers equation. The normalized Kortewegde VriesBurgers equation considered is given by uC uux ?? uxx C uxxx D 0; ??1 < x < 1; > 0; where > 0 is a parameter and x and 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/ D u0 .>0/ for x 0 and u.x; 0/ D 0 for x < 0. The method of matched asymptotic coordinate expansions is used to obtain the large- asymptotic structure of the solution to this problem, which exhibits the formation of an expansion wave in x 0, while the solution is oscillatory when x < ??2 3 as ! 1, with the oscillatory envelope being exponentially small in , as ! 1.