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 $\int_{-\infty}^\infty e^{i(t^5 +xt^3+yt^2+zt)}dt$ for large values of $\vert y\vert$ and bounded values of $\vert x\vert$ and $\vert z\vert$. 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 $\vert y\vert$ and fixed $x$ and $z$. There are four Stokes lines in the sector $(-\pi,\pi]$ that divide the complex $y$-plane into four sectors in which the swallowtail integral behaves differently when $\vert y\vert$ 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 $\vert y\vert\to\infty$, 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.

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Key words

swallowtail integral, asymptotic expansions, modified saddle point method

33E20, 41A60

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