increases the decay rate of the Fourier modes (slope of $\ln|\hat\eta_k|$ vs $k$) by a factor of 1.24 in this example, compared to 3.36 in the previous example, where it is definitely worthwhile.

![\[fig:evol:kdv3\] A relative-periodic solution found using a superposition of the Stokes waves labeled B and C in Figure \[fig:bif:stokes\] as a starting guess. Unlike the previous case, the waves do not fully merge at $t=T/2$. ](figs/evol_kdv3){width="\linewidth"}

The large change from $T_0/2 = 717.93$ to $T/2=554.053$ is due to nonlinear interaction of the waves. There are two main factors contributing to this change in period. The first is that the waves do not fully combine when they collide. Instead, the trailing wave runs into the leading wave, passing on much of its amplitude and speed. The peaks remain separated by a distance of $d=0.52462$ at $t=T/2$, the transition point where the waves have the same amplitude. Thus, the peak separation changes by $\pi-d$ rather than $\pi$ in half a period. The second effect is that the larger wave slows down the smaller wave more than the smaller slows the larger. Recall from Fig. \[fig:AandC\] that each wave induces a negative potential gradient across the other wave that generates 