constraint $\eta(0,0)=0.0707148$, the third case described in §\[sec:method\] for specifying the amplitude. This causes $\tilde f$ to decrease from $2.3\times 10^{-8}$ to $4.26\times 10^{-27}$ using $M=1200$ grid points and $N=1200$ time-steps (to $t=T$). The results are shown in the left panel of Figures \[fig:evol:kdv2\] and \[fig:pov:kdv2\]. The main difference between the Stokes collision and this nearby time-periodic solution is that the Stokes waves generate additional background ripples each time they collide while the time-periodic solution contains an equilibrium background wave configuration that does not grow in amplitude after the collision. While the background waves in the counter-propagating case (studied in [@water2]) look like small-amplitude standing waves, these background waves travel to the right, but slower than either solitary wave. After computing the $h=0.05$ time-periodic solution, we computed 10 other solutions with nearby values of $h$ and $\eta(0,0)$ to try to decrease the amplitude of the background radiation. The best solution we found (in the sense of small background radiation) is shown in the right panel of Figures \[fig:evol:kdv2\] and \[fig:pov:kdv2\], with $h=0.0503$ and $\eta(0,0)=0.0707637$. The amplitude of the background waves of this solution are comparable to that of the Stokes waves after two collisions.

Our second example is a relative 