A question about solitary waves to test your intuition before I show what we found.
The family of long-wave equations
\[ u_t - u_{xxt} = -u_x - u\,u_x + \alpha\,(u\,u_{xxx} + 2u_x u_{xx}) \]
contains the Benjamin-Bona-Mahony (BBM) equation of 1972 at \(\alpha = 0\). Every member has solitary waves \(u(x - ct)\), and in every member the crest of such a wave has height \(3(c-1)\), whatever \(\alpha\) is: at the crest, the first integral \((c - \alpha u)\,u'^2 = (c-1)\,u^2 - u^3/3\) leaves only \(u = 3(c-1)\).
Now let a tall solitary wave (height 1.2, speed 1.4) overtake a small one (height 0.3, speed 1.1) on a periodic channel. In most members the two waves come out of the collision slightly changed, and a train of small ripples stays behind. For which \(\alpha\) do they come out clean, with no ripples at all?
Vote with your intuition. The answer, with pictures and the theorem behind it, comes on Tuesday.