left panel of Figure \[fig:evol:kdv1\], the small wave advances by $\pi$ units to the right while the large wave advances by $1.2\pi$ units. The waves collide at $t=T/2$. This generates a small amount of radiation, which can be seen at $t=T$ in the right panel of Figure \[fig:align:stokes\]. Some radiation behind the large wave is present for all $t>0$, as shown in Figure \[fig:pov:kdv1\].

Before minimizing $\tilde f$, we advance the two Stokes waves to the time of the first collision, $t=T/2$. At this time, the larger solitary wave has traversed the domain 3 times and the smaller one 2.5 times, so their peaks lie on top of each other at $x=0$. The reason to do this is that when the waves merge, the combined wave is shorter, wider, and smoother than at any other time during the evolution. Quantitatively, the Fourier modes of $\hat\eta_k(t)$ and $\hat\varphi_k(t)$ decay below $10^{-15}$ for $k\ge600$ at $t=0$, and $k\ge200$ when $t=T/2$. Thus, the number of columns needed in the Jacobian is reduced by a factor of 3, and the problem becomes more overdetermined, hence more robust. For the calculation of a time-periodic solution, we let $t=0$ correspond to this merged state, which affects 