  Motivated by theoretical studies of gravitational clustering in the Universe,
we compute propagators (response functions) in the adhesion model. This model,
which is able to reproduce the skeleton of the cosmic web and includes
nonlinear effects in both Eulerian and Lagrangian frameworks, also corresponds
to the Burgers equation of hydrodynamics. Focusing on the one-dimensional case
with power-law initial conditions, we obtain exact results for Eulerian and
Lagrangian propagators. We find that Eulerian propagators can be expressed in
terms of the one-point velocity probability distribution and show a strong
decay at late times and high wavenumbers, interpreted as a "sweeping effect"
but not a genuine damping of small-scale structures. By contrast, Lagrangian
propagators can be written in terms of the shock mass function -- which would
correspond to the halo mass function in cosmology -- and saturate to a constant
value at late times. Moreover, they show a power-law dependence on scale or
wavenumber which depends on the initial power-spectrum index and is directly
related to the low-mass tail of the shock mass function. These results strongly
suggest that Lagrangian propagators are much more sensitive probes of nonlinear
structures in the underlying density field and of relaxation processes than
their Eulerian counterparts.
