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 Kortewegde Vries equation on the negative quarter-plane. The normalized Korteweg-de Vries equation considered is given by uτ + uux + uxxx = 0, x < 0, τ > 0, where x and τ represent dimensionless distance and time respectively. In particular, we consider the case when the initial and boundary conditions are given by u(x, 0) = ui for x < 0 and u(0, τ ) = ub, ∂ ∂x u(0, τ ) = ubx for τ > 0. Here the initial value ui < 0 and we restrict attention to boundary values ub and ubx in the ranges 0 < ub < −2ui and |ubx| ≥ 1 √ 3 (ub − ui)(−ub − 2ui) 1 2 respectively. The method of matched asymptotic coordinate expansions is used to obtain the large-τ asymptotic structure of the solution to this problem, which exhibits the formation of a dispersive shock wave when |ubx| > 1