  We discuss the long-time behaviour of solutions to Smoluchowski's coagulation
equation with kernels of homogeneity one, combining formal asymptotics,
heuristic arguments based on linearization, and numerical simulations. The case
of what we call diagonally dominant kernels is particularly interesting. Here
one expects that the long-time behaviour is, after a suitable change of
variables, the same as for the Burgers equation. However, for kernels that are
close to the diagonal one we obtain instability of both, constant solutions and
traveling waves and in general no convergence to N-waves for integrable data.
On the other hand, for kernels not close to the diagonal one these structures
are stable, but the traveling waves have strong oscillations. This has
implications on the approach towards an N-wave for integrable data, which is
also characterized by strong oscillations near the shock front.
