The large-time development of the solution to an initial-boundary value problem for the Korteweg-de Vries equation. I. Steady state solutions

In this paper, we consider an initial-boundary value problem for the Korteweg-de Vries equation on the positive quarter-plane. The normalized Korteweg-de Vries equation considered is given by u tau + uu(x) + u(xxx) = 0, 0 <x <infinity. tau > 0, where x and tau represent dimension less 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)