Collisions and umbilic catastrophes

Abstract

Two procedures to evaluate the hyperbolic umbilic canonical diffraction integral, a function that occurs in many applications of semiclassical collision theory and optics, are described. The first is the integration of the various differential equations satisfied by this function. We present schemes to perform the numerical integration and investigate their range of validity. A second algorithm, which evaluates the integral directly by quadrature, is given. Here, attention is focused on the very rapidly oscillating regions of the integrand. The direct quadrature method is shown to be more widely applicable and better suited to numerical computation.