waves will now interact rather than remain pure traveling waves. A solution will be called *relative periodic* if there exists a time $T$ and phase shift $\theta$ such that $$\label{eq:ts:def}
  \eta(x,t+T) = \eta(x-\theta,t), \qquad\quad
  \varphi(x,t+T) = \varphi(x-\theta,t)$$ for all $t$ and $x$. Time-periodicity is obtained as a special case, with $\theta\in2\pi\mathbb{Z}$. We can save a factor of 2 in computational work by imposing the alternative condition $$\label{eq:even:odd}
  \eta(x+\theta/2,T/2) \, \text{ is even}, \qquad\quad
  \varphi(x+\theta/2,T/2) \, \text{ is odd.}$$ From this, it follows that $$\begin{aligned}
  \eta(x+\theta/2,T/2) &= \eta(-x+\theta/2,T/2) =   \eta(x-\theta/2,-T/2), \\
  \varphi(x+\theta/2,T/2) &= -\varphi(-x+\theta/2,T/2) =   \varphi(x-\theta/2,-T/2).\end{aligned}$$ But then both sides of each equation in (\[eq:ts:def\]) agree at time $t=-T/2$. Thus, (\[eq:ts:def\]) holds for all time.

In the context of traveling-standing waves in deep water [@trav:stand], it is natural to define $T$ as twice the value above, replacing all factors of $T/2$ by $T/4$. That way a pure standing wave returns to its original configuration in time $T$ instead of shifting in space by $\pi$ in time $T$. In the present work, we consider pairs of solitary waves moving to the right at different speeds, so it is more natural to define $T$ as the first (rather than the second) time there 