  We analyse the scattering of sine-Gordon breathers on a square potential
well. We show that the scattering process depends not only on the vibration
frequency of the breather and its incoming speed but also on its phase as well
as the depth and width of the well. We show that the breather can pass through
the well and exit with a speed different, sometime larger, from the initial
one. It can also be trapped and very slowly decay inside the well or bounce out
of the well and go back to where it came from. We also show that the breather
can split into a kink and an anti-kink pair when it hits the well.
