Shock Waves in Dispersive Eulerian Fluids
Abstract
The longtime behavior of an initial step resulting in a dispersive shock wave (DSW) for the onedimensional isentropic Euler equations regularized by generic, thirdorder 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 (convexity or concavity) 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 Schrödinger equation. These include superfluids (BoseEinstein condensates and ultracold fermions) and "optical fluids." Estimates of breaking times for smooth initial data and the longtime behavior of the shock tube problem are presented. Detailed 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.
 Publication:

Journal of NonLinear Science
 Pub Date:
 June 2014
 DOI:
 10.1007/s0033201491994
 arXiv:
 arXiv:1303.2541
 Bibcode:
 2014JNS....24..525H
 Keywords:

 Dispersive shock wave;
 Riemann problem;
 Whitham averaging;
 Dispersive Euler equations;
 Generalized nonlinear Schrödinger equation;
 Nonlinear Sciences  Pattern Formation and Solitons;
 Physics  Fluid Dynamics
 EPrint:
 updates to introducation, readability, references