“tune” the background radiation generated by the Stokes collisions so that the amount coming out of each collision is identical to what went into it. In the first example of §\[sec:num\], we find that the tuned background radiation takes the form of a train of traveling waves of smaller wavelength moving to the right more slowly than either solitary wave. By contrast, in the counter-propagating case studied in [@water2], it consists of an array of smaller-wavelength standing waves oscillating rapidly relative to the time between collisions of the primary waves. In the second example of §\[sec:num\], the background radiation is essentially absent, which is to say that the optimized solution is free from high-frequency, low-amplitude disturbances in the trough, and closely resembles a relative-periodic cnoidal solution of KdV. We call the collisions in this solution “elastic” as they repeat forever, unchanged up to spatial translation, and there are no features to distinguish radiation from the waves themselves. This process of tuning parameters to minimize or eliminate small-amplitude oscillations in the wave troughs is reminiscent of Vanden-Broeck’s work [@vandenBroeck91] in which oscillations at infinity could be eliminated from solitary capillary-gravity waves by choosing the amplitude appropriately.

To search for relative periodic 