This talk will describe some work on the bouncing of particle-like (“kink”) solutions to a nonlinear wave equation, called the sine-Gordon equation, against a fixed boundary. Away from the boundary, this equation has a property known as integrability, making the scattering of the kinks particularly simple. However, if this integrability is broken at the boundary, then the scattering becomes surprisingly complicated, in ways that will be outlined in the talk with the help of some movies.