Exponential asymptotics of quantum droplets and bubbles

This research investigates the formation and stability of localized states, known as quantum droplets and bubbles, in the quadratic-cubic discrete nonlinear Schr\"odinger equation. Near a Maxwell point, these states emerge from two fronts connecting the bistable equilibria. By adjusting a control parameter, we identify a "pinning region" where multiple stable states coexist and are interconnected through homoclinic snaking. We analyze the system's behavior to uncover the underlying mechanisms under strong coupling conditions. Using exponential asymptotics, we determine the pinning region's width and its dependence on coupling strength, revealing an exponentially small relationship between them. Additionally, we employ eigenvalue counting to establish the stability of these states by computing the critical eigenvalue of their corresponding linearization operator, proving onsite fronts unstable and intersite fronts stable. These theoretical results are validated through numerical simulations, which show excellent agreement with our analytical predictions.