The D-bar method, inversion of certain integrals and integrability in 4 + 2 and 3 + 1 dimensions

Abstract

We first review a method for deriving linear and nonlinear transform pairs, which is based on the spectral analysis of an eigenvalue equation and on the formulation of a d-bar problem. Then, we present two applications of this method: (a) we derive a certain linear transform pair in one dimension, which appears in the characterization of the Dirichlet-to-Neumann map of the Laplace equation in the interior of a convex two-dimensional curvilinear domain. (b) We derive a nonlinear Fourier transform pair in four dimensions, which can be used for the solution of the Cauchy problem of an integrable generalization of the Kadomtsev–Petviashvilli equation in 4 + 2, i.e. in four spatial and two temporal dimensions. The question of reducing this equation form 4 + 2 to 3 + 1 dimensions is also discussed.