  We study the scattering of a soliton-like propagating particle with a wall of
bound particles, in several strongly interacting one-dimensional lattice models
with discrete degrees of freedom. We consider spin-polarized fermions
(anisotropic Heisenberg spin chain), the fermionic Hubbard model, and the Bose
Hubbard model, using precise numerical time dependent Density Matrix
Renormalization Group techniques. We show that in all integrable models
studied, there is no reflection. Instead, an incoming particle experiences
particle-hole transmutation upon entry and exit of the wall, and travels inside
the wall as a hole, analoguous to Klein tunneling, even though the dispersion
is highly nonlinear and there is no external potential. {\em Two} particles are
added to the wall on the incoming side and removed on the opposite side. For
spin-polarized fermions a single transmitted particle thus shifts the wall by
two lattice sites, in complete contrast to classical physics. For both Hubbard
models, the wall shifts by one doubly occupied single site. In the
nonintegrable models studied, the same process occurs in linear superposition
with backscattering events. We demonstrate a corresponding fermionic quantum
Newton's cradle and a metamaterial with "tachyonic" modes travelling faster
than in an empty system. We present a possible atomic scale signal counter for
spintronics. Our scenario should be realizable in future cold atom experiments.
