  We present a new and complete analysis of the n-bounce resonance and chaotic
scattering in solitary wave collisions. In these phenomena, the speed at which
a wave exits a collision depends in a complicated fractal way on its input
speed. We present a new asymptotic analysis of collective-coordinate ODEs,
reduced models that reproduce the dynamics of these systems. We reduce the ODEs
to discrete-time iterated separatrix maps and obtain new quantitative results
unraveling the fractal structure of the scattering behavior. These phenomena
have been observed repeatedly in many solitary-wave systems over 25 years.
