Dispersive shocks in diffusive-dispersive approximations of elasticity and quantum-hydrodynamics

TL;DR

This paper investigates how diffusion and dispersion influence shock waves in elasticity and quantum hydrodynamics, demonstrating convergence of traveling waves to shocks under specific conditions in the moderate dispersion regime.

Contribution

It provides new convergence results for traveling waves to shocks in diffusive-dispersive models of elasticity and quantum hydrodynamics, extending understanding in the moderate dispersion regime.

Findings

Convergence of traveling waves to shocks under Liu E-condition

Analysis of oscillation length in Hamiltonian systems with small friction

Validation of convergence in quantum hydrodynamics and undular bore models

Abstract

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.