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We explore the dynamics and interactions of multiple bright droplets and bubbles, as well as the interactions of kinks with droplets and with antikinks, in the extended one-dimensional Gross–Pitaevskii model including the Lee–Huang–Yang correction. Existence regions are identified for the one-dimensional droplets and bubbles in terms of their chemical potential, verifying the stability of the droplets and exposing the instability of the bubbles. The limiting case of the droplet family is a stable kink. The interactions between droplets demonstrate in-phase (out-of-phase) attraction (repulsion), with the so-called Manton’s method explicating the observed dynamical response, and mixed behavior for intermediate values of the phase shift. Droplets bearing different chemical potentials experience mass-exchange phenomena. Individual bubbles exhibit core expansion and mutual attraction prior to their destabilization. Droplets interacting with kinks are absorbed by them, a process accompanied by the emission of dispersive shock waves and gray solitons. Kink–antikink interactions are repulsive, generating counter-propagating shock waves. Our findings reveal dynamical features of droplets and kinks that can be detected in current experiments. 

Abstract In the seminal work (Weinstein 1999Nonlinearity12673), Weinstein considered the question of the ground states for discrete Schrödinger equations with power law nonlinearities, posed on ${\mathbb{Z}}^d$. More specifically, he constructed the so-called normalised waves, by minimising the Hamiltonian functional, for fixed powerP(i.e.l²mass). This type of variational method allows one to claim, in a straightforward manner, set stability for such waves. In this work, we revisit these questions and build upon Weinstein’s work, as well as the innovative variational methods introduced for this problem in (Laedkeet al1994Phys. Rev. Lett.731055 and Laedkeet al1996Phys. Rev.E544299) in several directions. First, for the normalised waves, we show that they are in fact spectrally stable as solutions of the corresponding discrete nonlinear Schroedinger equation (NLS) evolution equation. Next, we construct the so-called homogeneous waves, by using a different constrained optimisation problem. Importantly, this construction works for all values of the parameters, e.g.l²supercritical problems. We establish a rigorous criterion for stability, which decides the stability on the homogeneous waves, based on the classical Grillakis–Shatah–Strauss/Vakhitov–Kolokolov (GSS/VK) quantity ${\mathrm{\partial }}_{\omega }\parallel {\phi }_{\omega }{\parallel }_{l^2}^2$. In addition, we provide some symmetry results for the solitons. Finally, we complement our results with numerical computations, which showcase the full agreement between the conclusion from the GSS/VK criterion vis-á-vis with the linearised problem. In particular, one observes that it is possible for the stability of the wave to change as the spectral parameterωvaries, in contrast with the corresponding continuous NLS model.