Theory and experiments on soliton interactions

Abstract

The behaviour of surfacewaves on shallow water is described by nonlinear differential equations. Putting a constraint on these equation, such that they only describe waves moving in a certain direction, one obtains the KdV-equation. This nonlinear partial differential equation has the surprising property that it admits stable solutions. In these solutions dispersive and nonlinear effects balance out, creating what is known as a soliton. For certain initial conditions the KdV-equation can be solved analytically by using the inverse scattering transform. This method shows that, for a certain initial condition, there is a one-to-one correspondence between the solitons that emerge from this condition, and the eigenvalues of the time-independent Schrodinger equation, where the potential is given by this initial condition. Furthermore, it shows that when co-propagating solitons overtake each other they maintain their shape and velocity, but obtain a phase difference. A solution for the system where waves are not constraint to a single direction has not been found yet. However using both numerical and analytic methods it has been shown that for both reflection at a vertical wall and head on collision between solitons, a phaselag occurs. Here experiments on reflection and head-on collisions of solitons are compared to these analytic and numerical results to show that indeed a phaselag occurs at reflection.