Shock waves in dispersive Eulerian fluids

TL;DR

This paper analyzes the formation and behavior of dispersive shock waves in one-dimensional Eulerian fluids with third-order dispersion, using Whitham averaging to derive conditions and explore large amplitude regimes.

Contribution

It introduces a characterization of dispersive shock waves in Eulerian fluids with generic dispersion, including conditions for their existence and mechanisms for their breakdown.

Findings

DSW jump conditions depend on dispersion sign and pressure law

Breakdown mechanisms include gradient catastrophe and loss of nonlinearity

Numerical simulations agree with asymptotic theory in moderate regimes

Abstract

The long time behavior of an initial step resulting in a dispersive shock wave (DSW) for the one-dimensional isentropic Euler equations regularized by generic, third order dispersion is considered by use of Whitham averaging. Under modest assumptions, the jump conditions (DSW locus and speeds) for admissible, weak DSWs are characterized and found to depend only upon the sign of dispersion (convex or concave) and a general pressure law. Two mechanisms leading to the breakdown of this simple wave DSW theory for sufficiently large jumps are identified: a change in the sign of dispersion, leading to gradient catastrophe in the modulation equations, and the loss of genuine nonlinearity in the modulation equations. Large amplitude DSWs are constructed for several particular dispersive fluids with differing pressure laws modeled by the generalized nonlinear Schrodinger equation. These include superfluids (Bose-Einstein condensates and ultracold Fermions) and "optical fluids". Estimates of breaking times for smooth initial data and the long time behavior of the shock tube problem are presented. Numerical simulations compare favorably with the asymptotic results in the weak to moderate amplitude regimes. Deviations in the large amplitude regime are identified with breakdown of the simple wave DSW theory.