################ KdV Collisions ##################

The most interesting property of the KdV equation is the fashion in which two soliton solutions of the form

   S(x,t)=(a/2)^2*sech^2(0.5(ax-(a+a^3)t+d)

collide.  For a collision to occur, Solitons must have different values of a (since waves can only move in one direction a taller wave has to catch up with a shorter wave).

As in the linear wave equation Utt=Uxx, travelling waves emerge from a collision retaining their separate structure and identities.  What is interesting is that during the collision process, the waves to not superimpose in a linear fashion - the usual 'Amplitude of the sum is the sum of the amplitudes' law does not apply.  As you will see from watching the videos, the superposition of a tall wave and a short wave is actually shorter in amplitude than the tall wave by itself.  In addition, each wave suffers a change in phase - the shorter wave emerges from the collision behind where it would have been if no tall wave existed, and conversely the tall wave emerges ahead of where it would have been if no short wave existed.  This phase phenomena is shown in detail in the video "Phase Shift".