Abstract Solitonic excitations of the one-dimensional quantum droplets are obtained, which smoothly connect vacuum with the flat-top droplet, akin to compactons in classical liquids. These solitons are of the kink type, necessarily residing on a constant pedestal, determined by the mean-field (MF) repulsion and beyond mean field quantum correction and having exactly one-third of the uniform condensate amplitude. Akin to the kinks, the propagating modes occur in pairs and are phase-locked with the background. The lowest chemical potential and condensate amplitude at the flat-top boundary matches with the self-trapped quantum droplet. More general excitations of analogous kind are obtained through the Möbius transform, which connect the required solutions to elliptic functions in general.
Abstract Supersolid is an exotic state of matter, showing crystalline order with a superfluid background, observed recently in dipolar Bose–Einstein condensate in a trap. Here, we present exact Bloch wave function of the self-trapped supersolid phase, in the presence of mean-field and beyond mean-field interaction. Our general solutions of the amended nonlinear Schrödinger equation are obtained through Möbius transform, connecting a wide class of supersolid solutions to the ubiquitous cnoidal waves. The solutions yield the supersolid phase in the self-trapped quantum matter, where an array of quantum droplets exist, accompanied by a constant condensate. For the supersolid phase, the chemical potential for one class of solutions is the same as that of self-trapped quantum droplets, and is lower for the general non-perturbative solution. Due to the destabilizing effects of fluctuations on long range order in one dimension, the realization of the supersolid phase may be possible in a finite system.
We investigate the existence and dynamics of two-dimensional solitary waves in a quantum droplet environment described by the extended Gross-Pitaevskii equation featuring logarithmic mean-field and Lee-Huang-Yang interactions. In the modulationally stable regime of the background, we employ suitable multiscale asymptotic methods to derive effective nonlinear integrable models corresponding to the Kadomtsev-Petviashvili and Davey-Stewartson equations. Based on these reduced models, we construct approximate analytical solutions describing line solitons, algebraically localized lump solitons, ring solitons, and exponentially localized dromions embedded on the droplet background. The dynamical robustness of these solutions is monitored through numerical simulations. Line, lump and ring solitons stay closest to the theoretical predictions, although progressively deviate due to the emergence of small-amplitude radiation, while dromions depart from their analytical waveform the most, although they roughly maintain their shape. Our results unveil unprecedented multidimensional soliton solutions in models featuring the competition of mean-field and quantum fluctuations and as such are amenable to current ultracold atom experiments.
By using the F-expansion method associated with four auxiliary equations, i.e., the Bernoulli equation, the Riccati equation, the Lenard equation, and the hyperbolic equation, we present exact explicit solutions describing the dynamics of matter-wave condensates with time-varying two- and three-body nonlinearities. Condensates are trapped in a harmonic potential and they exchange atoms with the thermal cloud. These solutions include the generalized Jacobi elliptic function solutions, hyperbolic function solutions, and trigonometric function solutions. In addition, we have also found rational function solutions. Solutions constructed here have many free parameters that can be used to manipulate and control some important features of the condensate, such as the position, width, velocity, acceleration, and homogeneous phase. The stability of the solutions is confirmed by their long-time numerical behavior.
It was recently found that the Lee-Huang-Yang (LHY) correction to the mean-field Hamiltonian of binary atomic boson condensates suppresses the collapse and creates stable localized modes (two-component quantum droplets, QDs) in two and three dimensions (2D and 3D). In particular, the LHY effect modifies the effective Gross-Pitaevskii equation (GPE) in 2D by adding a logarithmic factor to the usual cubic term. In the framework of the accordingly modified two-component GPE system, we construct 2D self-trapped modes in the form of QDs with vorticity
S embedded into each component. Because of the effect of the logarithmic factor, the QDs feature a flat-top shape, which expands with the increase of
S and norm
N. An essential finding, produced by a systematic numerical investigation and analytical estimates, is that the vortical QDs are stable (which is a critical issue for vortex solitons in nonlinear models) up to
S=5, for
N exceeding a certain threshold value, which is predicted to scale as ${N}_{\mathrm{th}}\ensuremath{\sim}{S}^{4}$ for large
S (for three-dimensional QDs, the scaling is ${N}_{\mathrm{th}}\ensuremath{\sim}{S}^{6}$). The prediction is corroborated by numerical findings. Pivots of QDs with $S\ensuremath{\ge}2$ are subject to structural instability, as specially selected perturbations can split the single pivot in a set of
S or
S+2 pivots corresponding to unitary vortices; however, the structural instability remains virtually invisible, as it occurs in a broad central ``hole'' of the vortex soliton, where values of fields are very small, and it does not cause any dynamical instability. In the condensate of
39K atoms, in which QDs with
S=0 and a quasi-2D shape were created recently, the vortical droplets may have radial size $\ensuremath{\lesssim}30\phantom{\rule{0.16em}{0ex}}\ensuremath{\mu}\mathrm{m}$, with the number of atoms in the range of
104−−105. The role of three-body losses is considered too, demonstrating that they do not prevent the creation of the vortex droplets but may produce a noteworthy effect, leading to sudden splitting of ``light'' droplets. In addition, hidden-vorticity (HV) states in QDs, with topological charges ${S}_{+}=\ensuremath{-}{S}_{\ensuremath{-}}=1$ in their components, which are prone to strong instability in other settings, have their stability region too. Unstable HV states tend to spontaneously merge into zero-vorticity solitons. Collisions of QDs, which may lead to their merger, and dynamics of elliptically deformed QDs (which form rotating elongated patterns or ones with oscillations of the eccentricity) are briefly considered too.
We study the ground states of the extended Gross--Pitaevskii equation with the Lee--Huang--Yang correction from both theoretical and numerical perspectives. Starting from the three-dimensional model, we derive reduced one- and two-dimensional equations through nondimensionalization and dimensional reduction. We establish existence and nonexistence results for ground states in different spatial dimensions, both in free space and under confining external potentials. For the numerical computation of ground states, we propose a normalized gradient flow method with a Lagrange multiplier. The numerical results show how the model parameters affect the ground-state profiles, and reveal different regimes in the free-space parameter plane, including no-ground-state, soliton-like, and droplet-like regions. We also introduce a simple flat-top approximation for the droplet regime and present two- and three-dimensional computations to illustrate more general localized structures.