  We study impurities in integrable models from the viewpoint of generalized
hydrodynamics (GHD). An impurity can be thought of as a boundary condition for
the GHD equation, relating the state on the left and right side. We find that
in interacting models it is not possible to disentangle incoming and outgoing
states, which means that it is not possible to think of scattering as a mapping
which maps the incoming state to the outgoing state. We then introduce a novel
class of impurities, dubbed mesoscopic impurities, whose spatial size is
mesoscopic (i.e.\ their size $L_{\mathrm{micro}} \ll L_{\mathrm{imp}} \ll L$ is
much larger than the microscopic length scale $L_{\mathrm{micro}}$, but much
smaller than the macroscopic scale $L$). Due to their large size it is possible
to describe mesoscopic impurities via GHD. This simplification allows one to
study these impurities both analytically and numerically. These impurities show
interesting non-perturbative scattering behavior, for instance non-uniqueness
of solutions and a non-analytic dependence on the impurity strength. In models
with one quasi-particle species and a scattering phase shift that depends on
the difference of momenta only, we find that one can describe the scattering
using an effective Hamiltonian. This Hamiltonian is dressed due to the
interaction between particles and satisfies a self consistency fixed point
equation. On the example of the hard rods model we demonstrate how this fixed
point equation can be used to find almost explicit solutions to the scattering
problem by reducing it to a two-dimensional system of equations which can be
solved numerically.
