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

Bolbot, Daria; Mitsotakis, Dimitrios; Tzavaras, Athanasios

doi:10.1090/qam/1658

Authors: Daria Bolbot, Dimitrios Mitsotakis and Athanasios E. Tzavaras
Journal: Quart. Appl. Math. 81 (2023), 455-481
MSC (2020): Primary 35L65, 35Q40, 74J40, 76N30
DOI: https://doi.org/10.1090/qam/1658
Published electronically: February 17, 2023
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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 $\varepsilon , \delta \to 0$ with $\delta = o(\varepsilon )$, 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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