ABSTRACT
The Inverse Scattering Transform (1ST) is a nonlinear analog of the Fourier transform which has been used to solve certain nonlinear evolution equations in the same way that the Fourier transform can be used to solve linear partial differential evolution equations. The current understanding of the 1ST is based on the complex inhomogeneous Cauchy- Riemann equation. This works fine in one space dimension, and has been extended (with some modifications) to two space dimensions, but there are fundamental difficulties which have yet to be overcome in order to have substantial higher-dimensional applications of the method. This situation is to be contrasted with that for the Fourier transform, which works equally well in all space dimensions. Leading researchers in [1] have called the problem of finding a satisfactory extension of 1ST to several space dimensions the most important open problem in this field. Here it will be shown how prototype forward and inverse scattering transforms (which reduce to the forward and inverse Fourier transforms) for the linear hierarchy in an arbitrary number of space dimensions can be constructed based on the inhomogeneous Dirac equation and Clifford analysis.
