Front-based construction of quantum droplets, bubbles, and hole states in a Bose mixture

Abstract

We study nonlinear holes, bubbles, kink fronts, and quantum-droplet-like states in an elongated Bose mixture governed by an extended Gross-Pitaevskii equation with attractive cubic mean-field interactions and repulsive quartic Lee-Huang-Yang contribution. We classify the constant-amplitude backgrounds: the Bogoliubov-de Gennes spectrum shows that the upper branch is modulationally stable, whereas the lower branch is unstable to long-wavelength perturbations. Grand-potential and pressure determine the vacuum-finite-density coexistence point and the unique background supporting stationary kink and antikink fronts. Symmetric holes on the stable background have a density deficit and integral width that diverge near coexistence as the hole separates into two nonlinear fronts. A multiplicative kink-antikink construction connects bell-shaped states to broad flat-top droplets and approaches the stationary quantum-droplet family as the front separation increases. An antikink-kink sum with equal signs generates bubble-like density depletions without a phase jump, whereas their difference produces dark-hole-like profiles with a $π$ phase difference between the asymptotic backgrounds. Bubble depth increases with front separation, distinguishing gray-like and dark-like profiles. Real-time simulations show that deep bubbles persist under weak random perturbations, demonstrating finite-time robustness without establishing spectral stability. In the asymmetric two-component model, localized relative-density modes can induce spinor-driven breakup on a modulationally stable background, while robust holes persist near the kink limit. Collisions of kink-antikink droplet-like states display phase-dependent merging, asymmetric particle transfer, and effective reflection. These results connect nonlinear fronts with bubbles, phase-jump holes, and self-bound droplets in beyond-mean-field Bose mixtures.