Detection of complex singularities for a function of several variables

A numerical method for investigating singularities in solutions to non-linear evolution equations is presented. The method is based on a complex analytical approach to singularities introduced by Sulem, Sulem and Frisch, which uses analytic continuation of an independent variable and numerical detection of the width of the analyticity strip, defined as the distance δ from the real domain to the nearest complex singularity. Their method, originally formulated for functions of a single variable, is here generalized to problems and functions of several variables. We first analyse the asymptotic behaviour of the multidimensional Fourier transform of an analytic function, and use this to numerically detect the complex singularity surface. The approach allows us to determine the parameters that characterize the singularity surface in a neighbourhood of its closest point to the real domain.