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(r) + uu(x)+ u(xxx) = O, x <O, tau > O, where x and T 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 x u(0, tau) = u(bx) for tau > 0. Here the initial value ui <0 and we restrict attention to boundary values u(b) and u(bx) in the ranges 0 <u(b) <-2u(i) and vertical bar ub(x)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 dispersive shock wave when vertical bar ub(x)vertical bar >= 1/root 3 (u(b) - u(i))(-u(b) - 2u(i))(1/2). We also present detailed numerical simulations of the full initial-boundary value problem which support the asymptotic analysis presented. A brief discussion is also given of the large-tau asymptotic structure to this problem when u(i) <0: u(b) >= -2u(i) and u(bx) is an element of (-infinity: infinity). (C) 2009 Elsevier B.V. All rights reserved,