  To understand an oft-observed but poorly understood phenomenon in which a
solitary wave in a dispersive equation slowly deteriorates due to a persistent
emission of radiation (i.e. a ``radiating solitary wave''), we propose a
bare-bones model which captures many essential features and which we are
capable of analyzing completely by way of the Laplace transform. We find that
wave amplitude decreases at an exponential rate but with a decay constant that
is (in many cases) small beyond all orders of the frequency.
