  The aim is to assess the combined effect of diffusion and dispersion on
shocks in the moderate dispersion regime. For a diffusive dispersive
approximation of the equations of one-dimensional elasticity (or p-system), we
study convergence of traveling waves to shocks. The problem is recast as a
Hamiltonian system with small friction, and an analysis of the length of
oscillations yields convergence in the moderate dispersion regime $\eps, \delta
\to 0$ with $\delta = o(\eps)$, under hypotheses that the limiting shock is
admissible according to the Liu E-condition and is not a contact discontinuity
at either end state. A similar convergence result is proved for traveling waves
of the quantum hydrodynamic system with artificial viscosity as well as for a
viscous Peregrine-Boussinesq system where traveling waves model undular bores,
in all cases in the moderate dispersion regime.
