The sliding-frequency filter equation is shown to have similarity solutions which travel with steady profile but with constant acceleration. Over a wide range of the gain, filter strength and sliding-rate parameters, the pulse envelope is very well approximated by a sech profile. However, when the sliding rate is large, the chirp differs greatly from the usually assumed linear variation of frequency through the pulse. The amplitude and chirp are found for small and moderate sliding rate by a perturbation analysis and, for larger sliding rates, by solving a nonlinear eigenvalue problem for a nonautonomous differential equation.
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