The asymptotic expansion of the swallowtail integral in the highly oscillatory region

We consider the swallowtail integral Ψ(x,y,z):=∫−∞∞ei(t5+xt3+yt2+zt)dt for large values of |x| and bounded values of |y| and |z|. The integrand of the swallowtail integral oscillates wildly in this region and the asymptotic analysis is subtle. The standard saddle point method is complicated and then we use the simplified saddle point method introduced in (López et al., 2009). The analysis is more straightforward with this method and it is possible to derive complete asymptotic expansions of Ψ(x, y, z) for large |x| and fixed y and z. The asymptotic analysis requires the study of three different regions for argx separated by three Stokes lines. The expansion is given in terms of inverse powers of x13 and x12 and the coefficients are elementary functions of y and z. The accuracy and the asymptotic character of the approximations is illustrated with some numerical experiments.