On the integrability of linear and nonlinear partial differential equations

Abstract

A new method for studying boundary value problems for linear and for integrable nonlinear partial differential equations (PDE’s) in two dimensions is reviewed. This method provides a unification as well as a significant extension of the following three seemingly different topics: (a) The classical integral transform method for solving linear PDE’s and several of its variations such as the Wiener–Hopf technique. (b) The integral representation of the solution of linear PDE’s in terms of the Ehrenpreis fundamental principle. (c) The inverse spectral (scattering) method for solving the initial value problem for nonlinear integrable evolution equations. The detailed implementation of the method is presented for: (a) An arbitrary linear dispersive evolution equation on the half line. (b) The nonlinear Schrödinger equation on the half line. (c) The Laplace, Helmholtz and modified Helmholtz equations in an arbitrary convex polygon. In addition, several other applications are briefly considered. The possible extension of this method to multidimensions is also discussed.