DISPERSIVE SHOCKS IN DIFFUSIVE-DISPERSIVE APPROXIMATIONS OF ELASTICITY AND QUANTUM-HYDRODYNAMICS

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 (Formula presented) with (Formula presented), 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

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

Oscillatory and regularized shock waves for a dissipative Peregrine-Boussinesq system

We consider a dissipative, dispersive system of the Boussinesq type, which describes wave phenomena in scenarios where dissipation plays a significant role. Examples include undular bores in rivers or oceans, where turbulence-induced dissipation significantly influences their behavior. In this study, we demonstrate that the proposed system admits traveling wave solutions known as diffusive-dispersive shock waves. These solutions can be categorized as oscillatory and regularized shock waves, depending on the interplay between dispersion and dissipation effects. By comparing numerically computed solutions with laboratory data, we observe that the proposed model accurately captures the behavior of undular bores over a broad range of phase speeds. Traditionally, undular bores have been approximated using the original Peregrine system, which, even though it doesn't possess these as traveling wave solutions, tends to offer accurate approximations within suitable time scales. To shed light on this phenomenon, we demonstrate that the discrepancy between the solutions of the dissipative Peregrine system and the non-dissipative counterpart is proportional to the product of the dissipation coefficient and the observation time