  We study a third-order nonlinear ordinary differential equation whose
solutions, under certain specific conditions, are individual pulses. These
correspond to homoclinic orbits in the phase space of the equation and we study
the possible pulse types in some detail. Sufficiently close to the conditions
under which a homoclinic orbit exists, the solutions take the form of trains of
well-separated pulses. A measure of closeness to homoclinic conditions provides
a small parameter for the development of an asymptotic solution consisting of
superposed, isolated pulses. The solvability condition in the resulting
singular perturbation theory is a {\its timing map} relating successive pulse
spacings. This map of the real line onto itself, together with the known form
of the homoclinic orbit, provides a concise and accurate solution of the
equation.
