The Large-Time Development of the Solution to an Initial-Boundary Value Problem for the Korteweg-De Vries Equation. II. Expansion Wave Solutions

In this paper, we consider an initial-boundary value problem for the Korteweg-de Vries equation on the positive quarter-plane. The normalised Korteweg-de Vries equation considered is given by u(tau) + uu(x) + u(xx x) = 0, 0 <x <infinity, 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 0 <x <infinity and u(0, tau) = u(b) for tau > 0, respectively. Here, the initial value u(i) > 0 and we restrict attention to values of the boundary value, u(b), for which u(b) <u(i). We consider the three cases (u(i) > 0, u(b) <0), (u(i) > 0, u(b) = 0) and (u(i) > 0, 0 <u(b) <u(i)). In each case, 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 an expansion wave solution.