The large-time development of the solution to an initial-boundary value problem for the Korteweg-de Vries equation on the negative quarter-plane

In this paper, we consider an initial-boundary value problem for the Korteweg-de Vries equation on the negative quarter-plane. The normalized Korteweg-de Vries equation considered is given by u(tau) + uu(x) + u(xxx) = 0, x <0, tau > 0, where x and tau represent dimensionless distance and time, respectively. In particular, we consider the case when the initial and boundary conditions are given by u(x, 0) = u(i) for x <0 and u(0, tau) = u(b), partial derivative/partial derivative xu(0, tau) = u(bx) for tau > 0. Here the initial value u(i) <0 and we restrict attention to boundary values u(b) and u(bx) in the ranges 0 <u(b) <-2u(i) and vertical bar u(bx)vertical bar <1/root 3 (u(b)-u(i))(-u(b)-2u(i))(1/2), respectively. 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 a steady state solution. A brief discussion is also given of the large-tau asymptotic structure to this problem when u(i) <0, u(i) <u(b) <0 and u(bx) = 0. (C) 2009 Elsevier Inc. All rights reserved.