Volume 52, pp. 88-99, 2020.

The swallowtail integral in the highly oscillatory region II

Chelo Ferreira, José L. López, and Ester Pérez Sinusía

Abstract

We analyze the asymptotic behavior of the swallowtail integral ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{i(t^5+x{t}^3+y{t}^2+zt)}dt$ for large values of $|y|$ and bounded values of $|x|$ and $|z|$. We use the simplified saddle point method introduced in [López et al., J. Math. Anal. Appl., 354 (2009), pp. 347–359]. With this method, the analysis is more straightforward than with the standard saddle point method, and it is possible to derive complete asymptotic expansions of the integral for large $|y|$ and fixed $x$ and $z$. There are four Stokes lines in the sector $(-\pi ,\pi ]$ that divide the complex $y$-plane into four sectors in which the swallowtail integral behaves differently when $|y|$ is large. The asymptotic approximation is the sum of two asymptotic series whose terms are elementary functions of $x$, $y$, and $z$. One of them is of Poincaré type and is given in terms of inverse powers of $y^{1/2}$. The other one is given in terms of an asymptotic sequence whose terms are of the order of inverse powers of $y^{1/9}$ when $|y|\to \mathrm{\infty }$, and it is multiplied by an exponential factor that behaves differently in the four mentioned sectors. Some numerical experiments illustrate the accuracy of the approximation.

Full Text (PDF) [409 KB], BibTeX , DOI: 10.1553/etna_vol52s88

Key words

swallowtail integral, asymptotic expansions, modified saddle point method

AMS subject classifications

33E20, 41A60