An asymptotic expansion of the hyberbolic umbilic catastrophe integral

Abstract

AbstractWe obtain an asymptotic expansion of the hyperbolic umbilic catastrophe integral $$\Psi ^{(H)}(x,y,z):= \int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\exp (i(s^3+t^3+zst$$ Ψ ( H ) ( x , y , z ) : = ∫ - ∞ ∞ ∫ - ∞ ∞ exp ( i ( s 3 + t 3 + z s t $$+yt+xs))\mathrm{{d}}s\,\mathrm{{d}}t$$ + y t + x s ) ) d s 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 $$\arg x=\pi $$ arg x = π . 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.