continuous function, i.e. $U\in C\big(\mathbb{T}^N\big)$, and $\vec\kappa$, $\vec\omega$, $\vec\alpha\in\mathbb{R}^N$. Throughout the manuscript, we will use the phrase “solitary waves” in a broad sense to describe waves that, most of the time, remain well-separated from one another and propagate with nearly constant speed and shape. “Stokes waves” will refer to periodic progressive solutions of the free-surface Euler equations of permanent form, or waves that began at $t=0$ as a linear superposition of such traveling waves. They comprise a special class of solitary waves. “Solitons” will refer specifically to superpositions of ${\operatorname}{sech}^2$ solutions of the KdV equation on the whole line, while “cnoidal solutions” will refer to their spatially periodic, multi-phase counterparts; see §\[sec:kdv\] for elaboration.

It was found in [@water2] that decreasing the fluid depth causes standing waves to transition from large-scale symmetric sloshing behavior in deep water to pairs of counter-propagating solitary waves that collide repeatedly in shallow water. In the present work, we consider unidirectional waves of different amplitude that collide due to taller waves moving faster than shorter ones. We present two examples of solutions of this type: one where the resulting dynamics is fully time-periodic; and one where it is relative-periodic, returning to a spatial phase shift 