An asymptotic expansion of the hyberbolic umbilic catastrophe integral

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