  The Serre-Green-Naghdi equations of water wave theory have been widely
employed to study undular bores. In this study, we introduce a modified
Serre-Green-Naghdi system incorporating the effect of an artificial term that
results in dispersive and dissipative dynamics. We show that, over sufficiently
extended time intervals, effectively approximates the classical
Serre-Green-Naghdi equations and admits dispersive-diffusive shock waves as
traveling wave solutions. The traveling waves converge to the entropic shock
wave solution of the shallow water equations when the dispersion and diffusion
approach zero in a moderate dispersion regime. These findings contribute to an
understanding of the formation of dispersive shock waves in the classical
Serre-Green-Naghdi equations and the effects of diffusion in the generation and
propagation of undular bores.
