Dark Shock Waves in the Nonlinear Schrödinger System with Internal Losses

We demonstrate that the $(1+1)$ dimensional, normal-dispersion, nonlinear Schr\"odinger equation with an ``internal viscosity'' has a stable ``dark'' shock wave (SW) solution, which is the invasion of the empty (dark) domain into the energy-carrying one. It may be interpreted as an optical SW in a loss-compensated nonlinear optical fiber. We predict that it can be created experimentally with a temporal width of a few picoseconds at a carrier-wave background power about 10 W. We develop a theoretical analysis that captures the physics of the SW propagation. The prediction that the SW velocity has a constant value in the limit of small viscosity, and scales as the square root of the viscosity in the large viscosity limit, are confirmed by the full dynamics simulations.