Wave tunneling and hysteresis in nonlinear junctions.

We consider the nonlinear tunneling of a plane wave through a small barrier potential in a medium with self-defocusing, or repulsive, interactions. We show that nonlinearity can either suppress or enhance transmission rates, determined by whether the initial kinetic energy is above or below the barrier height. Associated with this threshold is the appearance of two distinct hysteresis loops, going clockwise or counterclockwise, respectively. Spatial dynamics upon reflection and transmission reveals the formation of dispersive shock waves (dark soliton trains) due to phase jumps at the interfaces and wave steepening during propagation. The results are demonstrated experimentally for optical wave tunneling through a refractive index defect but will hold for any Schrödinger system that contains a nonlinear junction.