Semi-classical determination of exponentially small intermode transitions for 1+1 space-time scattering systems

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.