  We compute time-periodic and relative-periodic solutions of the free-surface
Euler equations that take the form of overtaking collisions of unidirectional
solitary waves of different amplitude on a periodic domain. As a starting
guess, we superpose two Stokes waves offset by half the spatial period. Using
an overdetermined shooting method, the background radiation generated by
collisions of the Stokes waves is tuned to be identical before and after each
collision. In some cases, the radiation is effectively eliminated in this
procedure, yielding smooth soliton-like solutions that interact elastically
forever. We find examples in which the larger wave subsumes the smaller wave
each time they collide, and others in which the trailing wave bumps into the
leading wave, transferring energy without fully merging. Similarities
notwithstanding, these solutions are found quantitatively to lie outside of the
Korteweg-de Vries regime. We conclude that quasi-periodic elastic collisions
are not unique to integrable model water wave equations when the domain is
periodic.
