as the Korteweg-deVries equation, the Benjamin-Ono equation, and the nonlinear Schrödinger equation is that they interact elastically, leading to time-periodic, relative-periodic, or quasi-periodic dynamics. By contrast, the interaction of solitary waves for the free-surface Euler equations is inelastic. However, it has been observed many times in the literature [@chan:street:70; @cooker:97; @maxworthy:76; @su:mirie; @mirie:su; @zou:su; @craig:guyenne:06; @milewski:11] that the residual radiation after a collision of such solitary waves can be remarkably small. In the present paper we explore the possibility of finding nearby time-periodic and relative-periodic solutions of the Euler equations using a collision of unidirectional Stokes waves as a starting guess. Such solutions demonstrate that recurrent elastic collisions of solitary waves in the spatially periodic case do not necessarily indicate that the underlying system is integrable.

A relative-periodic solution is one that returns to a spatial phase shift of its initial condition at a later time. This only makes sense on a periodic domain, where the waves collide repeatedly at regular intervals in both time and space, with the locations of the collisions drifting steadily in time. They are special cases (with $N=2$) of quasi-periodic solutions, which have the form $u(x,t)=U(\vec\kappa
x+\vec \omega t + \vec\alpha)$ with $U$ an $N$-periodic 