  We consider the semiclassical limit of systems of autonomous PDE's in 1+1
space-time dimensions in a scattering regime. We assume the matrix valued
coefficients are analytic in the space variable and we further suppose that the
corresponding dispersion relation admits real-valued modes only with
one-dimensional polarization subspaces. Hence a BKW-type analysis of the
solutions is possible. We typically consider time-dependent solutions to the
PDE which are carried asymptotically in the past and as $x\to -\infty$ along
one mode only and determine the piece of the solution that is carried for $x\to
+\infty$ along some other mode in the future. Because of the assumed
non-degeneracy of the modes, such transitions between modes are exponentially
small in the semiclassical parameter; this is an expression of the Landau-Zener
mechanism. We completely elucidate the space-time properties of the leading
term of this exponentially small wave, when the semiclassical parameter is
small, for large values of $x$ and $t$, when some avoided crossing of finite
width takes place between the involved modes.
