Interaction of one-dimensional quantum droplets with potential wells and barriers

We address static and dynamical properties of one-dimensional (1D) quantum droplets (QDs) under the action of local potentials in the form of narrow wells and barriers. The dynamics of QDs is governed by the 1D Gross–Pitaevskii equation including the mean-field cubic repulsive term and the beyond-mean-field attractive quadratic one. In the case of the well represented by the delta-function potential, three exact stable solutions are found for localized states pinned to the well. The Thomas–Fermi approximation for the well and the adiabatic approximation for the collision of the QD with the barrier are developed too. Collisions of incident QDs with the wells and barriers are analyzed in detail by means of systematic simulations. Outcomes, such as fission of the moving QD into transmitted, reflected, and trapped fragments, are identified in relevant parameter planes. In particular, a counter-intuitive effect of partial or full rebound of the incident QD from the potential well is studied in detail and qualitatively explained.