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arXiv · 2301.00197

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

Daria Bolbot·Dimitrios Mitsotakis·Athanasios E. Tzavaras
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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.

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Daria Bolbot, Dimitrios Mitsotakis, Athanasios E. Tzavaras. 2022-12-31. Dispersive shocks in diffusive-dispersive approximations of elasticity and quantum-hydrodynamics. https://arxiv.org/abs/2301.00197

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