The Kadomtsev–Petviashvili II equation on the half-plane

The KPII equation is an integrable nonlinear PDE in 2+1 dimensions (two spatial and one temporal), which arises in several physical circumstances, including fluid mechanics, where it describes waves in shallow water. It provides a multidimensional generalisation of the renowned KdV equation. In this work, we employ a novel approach recently introduced by one of the authors in connection with the Davey–Stewartson equation (Fokas (2009) [13]), in order to analyse the initial-boundary value problem for the KPII equation formulated on the half-plane. The analysis makes crucial use of the so-called dd-bar formalism, as well as of the so-called global relation. A novel feature of boundary as opposed to initial value problems in 2+1 is that the dd-bar formalism now involves a function in the complex plane which is discontinuous across the real axis.

Research highlights
► IBV problem for KPII in 2+1 is discussed via the simultaneous analysis of a Lax pair.
► Fourier transform gives an expression for the auxiliary function of the Lax pair.
► Fourier transform and Green’s theorem yield the so-called global relation.
► dd-bar formalism and global relation give integral representation for the solution.