An asymptotic expansion of the hyberbolic umbilic catastrophe integral

Abstract

We obtain an asymptotic expansion of the hyperbolic umbilic catastrophe integral Psi ^{(H)}(x,y,z):= int _{-infty }^{infty }int _{-infty }^{infty }exp (i(s^3+t^3+zst+yt+xs))mathrm{{d}}s,mathrm{{d}}t for large values of |x| and bounded values of |y| and |z|. The expansion is given in terms of Airy functions and inverse powers of x. There is only one Stokes ray at arg x=pi . We use the modified saddle point method introduced in (López et al. J Math Anal Appl 354(1):347–359, 2009). The accuracy and the asymptotic character of the approximations are illustrated with numerical experiments.