The Stokes set of the cusp diffraction catastrophe

Abstract

Stokes and anti-Stokes lines are familiar in the asymptotic approximation of functions of a complex variable. The author generalises this notion and define the Stokes and anti-Stokes sets of a complex function of many (possibly complex) variables, defined by a diffraction-type integral. They are subsets of the Maxwell set of catastrophe theory, extended to complex variables. On the Stokes set the number of complex stationary points contributing to the integral changes by one, whereas on the caustic the number of real stationary points changes by two. Knowledge of the location of the Stokes set is essential to perform a full stationary phase analysis of a diffraction integral, and it imposes a constraint upon the positions of wavefront dislocations. For the canonical cusp diffraction catastrophe the author finds the explicit equation of the Stokes set, which is a broadened mirror image of the cusp caustic.