The piston dispersive shock wave problem

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.