  Analytic modeling of gravitational waves from inspiraling eccentric binaries
poses an interesting mathematical challenge. When constructing analytic
waveforms in the frequency domain, one has to contend with the fact that the
phase of the Fourier integral in non-monotonic, resulting in a breakdown of the
standard stationary phase approximation. In this work, we study this breakdown
within the context of catastrophe theory. We find that the stationary phase
approximation holds in the context of eccentric Keplerian orbits when the
Fourier frequency satisfies $f_{\rm min} < f < f_{\rm max}$, where $f_{\rm
min/max}$ are integer multiples of the apocenter/pericenter frequencies,
respectively. For values outside of this interval, the phase undergoes a fold
catastrophe, giving rise to an Airy function approximation of the Fourier
integral. Using these two different approximations, we generate a matched
asymptotic expansion that approximates generic Fourier integrals of Keplerian
motion for bound orbits across all frequency values. This asymptotic expansion
is purely analytic and closed-form. We discuss several applications of this
investigation and the resulting approximation, specifically: 1) the development
and improvement of effective fly-by waveforms for binary black holes, 2) the
transition from burst emission in the high eccentricity limit to wave-like
emission in the quasi-circular limit, which results in an analogy between
eccentric gravitational wave bursts and Bose-Einstein condensates, and 3) the
calculation of f-mode amplitudes in eccentric binary neutron stars and black
hole-neutron star binaries in terms of complex Hansen coefficients. The
techniques and approximations developed herein are generic, and will be useful
for future studies of gravitational waves from eccentric binaries within the
context of post-Newtonian theory.
