The KdV equation on the half-line: the Dirichlet to Neumann map

Abstract

We consider initial-boundary value problems for the KdV equation ut + ux + 6uux + uxxx = 0 on the half-line x ⩾ 0. For a well-posed problem, the initial data u(x, 0) as well as one of the three boundary values {u(0, t), ux(0, t), uxx(0, t)} can be prescribed; the other two boundary values remain unknown. We provide a characterization of the unknown boundary values for the Dirichlet as well as the two Neumann problems in terms of a system of nonlinear integral equations. The characterizations are effective in the sense that the integral equations can be solved perturbatively to all orders in a well-defined recursive scheme.