Temporal dynamics of tunneling: Hydrodynamic approach

We use the hydrodynamic representation of the Gross-Pitaevskii and nonlinear Schrödinger equations in order to analyze the dynamics of macroscopic tunneling processes. We observe a tendency to wave breaking and shock formation during the early stages of the tunneling process. A blip in the density distribution appears on the outskirts of the barrier and under proper conditions it may transform into a bright soliton. Our approach, based on the theory of shock formation in solutions of the Burgers equation, allows us to find the parameters of the ejected blip (or soliton if formed), including the velocity of its propagation. The blip in the density is formed regardless of the value and sign of the nonlinearity parameter. However, a soliton may be formed only if this parameter is negative (attraction) and large enough. A criterion is proposed. An ejection of a soliton is also observed numerically. We demonstrate, theoretically and numerically, controlled formation of a soliton through tunneling. The mass of the ejected soliton is controlled by the initial state.