Piston dispersive shock wave problem

The piston shock problem is a classical result of shock wave theory. In this work, the analogous dispersive shock wave (DSW) problem for a fluid described by the nonlinear Schr?dinger equation is analyzed. Asymptotic solutions are calculated for a piston (step potential) moving with uniform speed into a dispersive fluid at rest. In contrast to the classical case, there is a bifurcation of shock behavior where, for large enough piston velocities, the DSW develops a periodic wave train in its wake with vacuum points and a maximum density that remains fixed as the piston velocity is increased further. These results have application to Bose-Einstein condensates and nonlinear optics.