Oscillatory and regularized shock waves for a modified Serre–Green–Naghdi system

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 the modified system effectively approximates the classical Serre–Green–Naghdi equations over sufficiently extended time intervals 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.