overtaking collisions for the water wave: one in which the larger wave completely subsumes the smaller wave for a time, and one where the two waves remain distinct throughout the interaction. Similar behavior has of course been observed for the Korteweg-de Vries equation, which was part of our motivation for looking for such solutions. Lax [@lax:1968] classified overtaking collisions of two KdV solitons as bimodal, mixed, or unimodal. Unimodal and bimodal waves are analogous to the ones we computed above, while mixed mode collisions have the larger wave mostly subsume the smaller wave at the beginning and end of the interaction, but with a two-peaked structure re-emerging midway through the interaction. Lax showed that if $1<c_1/c_2<A=(3+\sqrt{5})/2$, the collision is bimodal; if $c_1/c_2>3$, the collision is unimodal; and if $A<c_1/c_2<3$, the collision is mixed. Here $c_1$ and $c_2$ are the wave speeds of the trailing and leading waves, respectively, at $t=-\infty$. Leveque [@leveque:87] has studied the asymptotic dynamics of the interaction of two solitons of nearly equal amplitude. Zou and Su [@zou:su] performed a computational study of overtaking water wave collisions, compared the results to KdV interactions, and found that the water wave collisions ceased to be elastic at third order. 