The Korteweg–de Vries equation on the half-line

Abstract

The initial-boundary value problem (ibvp) for the Korteweg–de Vries (KdV) equation on the half-line with data in Sobolev spaces is analysed by combining the unified transform method with a contraction mapping approach. First, the linear KdV ibvp with initial and boundary data in Sobolev spaces is solved and the basic space and time estimates of the solution are derived. Then, further linear estimates in a new norm, motivated by the KdV bilinear term, are obtained. Finally, well-posedness of the KdV ibvp with data (u(x, 0), u(0, t)) in , , is established via a fixed point argument in an appropriate solution space.