  We investigate the interface coupling between the 2D sine-Gordon equation and
the 2D wave equation in the context of a Josephson window junction using a
finite volume numerical method and soliton perturbation theory. The geometry of
the domain as well as the electrical coupling parameters are considered. When
the linear region is located at each end of the nonlinear domain, we derive an
effective 1D model, and using soliton perturbation theory, compute the fixed
points that can trap either a kink or antikink at an interface between two
sine-Gordon media. This approximate analysis is validated by comparing with the
solution of the partial differential equation and describes kink motion in the
1D window junction. Using this, we analyze steady state kink motion and derive
values for the average speed in the 1D and 2D systems. Finally we show how
geometry and the coupling parameters can destabilize kink motion.
