The Kadomtsev-Petviashvili II Equation on the Half-Plane

TL;DR

This paper develops a new analytical approach using d-bar formalism and global relations to solve the initial-boundary value problem for the KPII equation on the half-plane, advancing understanding of multidimensional integrable PDEs with boundary conditions.

Contribution

It introduces a novel method employing d-bar formalism and global relations to analyze the KPII equation on a half-plane, addressing boundary value challenges in 2+1 dimensional integrable systems.

Findings

Successful formulation of the initial-boundary value problem for KPII

Introduction of a complex-plane function discontinuous across the real axis

Application of the d-bar formalism to boundary problems in 2+1 dimensions

Abstract

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 \cite{FDS2009}, 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 d-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 d-bar formalism now involves a function in the complex plane which is discontinuous across the real axis.