Verlag der Österreichischen Akademie der Wissenschaften

Abstract:
We analyze the asymptotic behavior of the swallowtail integral $\\int_{-\\infty}^\\infty e^{i(t^5 +xt^3+yt^2+zt)}dt$ for largevalues of $\\vert y\\vert$ and bounded values of $\\vert x\\vert$ and $\\vert z\\vert$. We use the simplified saddle point method introducedin [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 $\\vert y\\vert$ and fixed $x$ and $z$.There are four Stokes lines in the sector $(-\\pi,\\pi]$ that divide the complex $y$-plane into four sectors inwhich the swallowtail integral behaves differently when $\\vert y\\vert$ is large. The asymptotic approximation is the sum oftwo 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 $\\vert y\\vert\\to\\infty$, 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.