  To study the propagation of nonlinear waves across Y- and T-type junctions,
we consider the 2D sine--Gordon equation as a model and study the dynamics of
kinks and breathers in such geometries. The comparison of the energies reveals
that the angle of the fork plays no role. Motivated by this, we introduce a 1D
effective equation whose solutions agree well with the 2D simulations for kink
and breather solutions. For branches of equal width, breather crossing occurs
approximately when $v > 1 - \omega$, where $v$ is the breather celerity and
$\omega$ is its frequency. We then characterize the breathers in the two upper
branches by estimating their velocity and frequency. These new breathers are
slower than the initial breather and up-shifted in frequency. In perspective,
this study could be generalized to more complex nonlinear waves.
