Junction conditions in stepwise Sine-Gordon equation and soltion collisions

The stepwise Sine-Gordon equation in which the value of the system parameter is different for negative and positive value of the scalar field is investigated. By applying appropriate junction conditions, the relation between solition velocities before and after collision is derived. It turns out that a heavy soliton anti-soliton pair can transform into a light pair moving at right velocities. We call this phenomenon “soliton gun”. Some of the result which were obtained before numerically, are derived analytically in this paper.