The piston dispersive shock wave problem

TL;DR

This paper analyzes the dispersive shock wave problem in a nonlinear Schrödinger fluid, deriving asymptotic solutions that differ from classical shocks and applying to Bose-Einstein condensates and nonlinear optics.

Contribution

It introduces an asymptotic analysis of dispersive shock waves using Whitham theory for the nonlinear Schrödinger equation, highlighting differences from classical shock behavior.

Findings

Asymptotic solutions match numerical simulations quantitatively.

Shock structure depends on piston speed.

Results applicable to Bose-Einstein condensates and nonlinear optics.

Abstract

The one-dimensional piston shock problem is a classical result of shock wave theory. In this work, the analogous dispersive shock wave (DSW) problem for a dispersive fluid described by the nonlinear Schr\"odinger equation is analyzed. Asymptotic solutions are calculated using Whitham averaging theory for a "piston" (step potential) moving with uniform speed into a dispersive fluid at rest. These asymptotic results agree quantitatively with numerical simulations. It is shown that the behavior of these solutions is quite different from their classical counterparts. In particular, the shock structure depends on the speed of the piston. These results have direct application to Bose-Einstein condensates and the propagation of light through a nonlinear, defocusing medium.