Discrete Quantum Droplets Equation

• Discrete quantum droplets equations are lattice nonlinear wave equations with competing onsite nonlinearities and nearest-neighbor coupling that model bright droplets and dark bubbles.
• Key models like the quadratic-cubic DNLS and discrete Gross–Pitaevskii with LHY correction reveal bistability, front pinning, and homoclinic snaking in discrete systems.
• These equations provide a framework for analyzing stability regimes, modulational instability, and transitions between soliton-like and flat-top droplet profiles in optical lattices.

The discrete quantum droplets equation denotes a class of lattice nonlinear wave equations used to describe self-trapped, liquid-like localized states—usually called quantum droplets for bright states and bubbles for dark states—in discrete media. In the arXiv literature, the most direct realizations are one-dimensional discrete nonlinear Schrödinger-type models with competing onsite nonlinearities and nearest-neighbor coupling, especially the quadratic-cubic DNLS and the discrete Gross–Pitaevskii equation with Lee–Huang–Yang correction derived for deep optical lattices. The phrase is not univocal, however: several papers retrieved under similar terminology are purely continuum GP or generalized GP models, with discreteness entering only through numerical grids, impurity sums, or nonstandard interpretations rather than through a physical lattice Hamiltonian (Kusdiantara et al., 2024, Adriano et al., 18 Jul 2025, Otajonov et al., 2024, Sahu et al., 2021).

1. Terminological scope and what counts as “discrete”

In the strict sense relevant to lattice droplet physics, a discrete quantum droplets equation is a dynamical equation for site amplitudes $\phi_n(t)$ or $\psi_n(t)$, with explicit nearest-neighbor coupling and onsite nonlinear competition. This is the setting of the DNLS-based droplet and bubble literature, where localization is controlled by front pinning, bistability, and lattice-induced multistability (Kusdiantara et al., 2024).

Several adjacent works do not introduce a discrete droplet equation in that sense. The two-dimensional disorder study solves a continuum coupled GP system with LHY correction on a finite-difference grid and states explicitly that the “discrete” aspect is only numerical (Sahu et al., 2021). The Anderson-localization study of a one-dimensional droplet likewise uses a continuum time-dependent generalized GP equation; its only discrete ingredients are the Gaussian disorder spikes used to build the speckle-like potential (Mehri et al., 5 Dec 2025). The low-dimensional rotating-droplet paper uses a continuous logarithmic Gross–Pitaevskii equation and states that it does not derive a discrete lattice equation (Hernández et al., 20 Jun 2025). A different ambiguity appears in the quantum-gravity-inspired droplet work, where the governing model is again continuum, modified by a fourth-order derivative generated by a generalized uncertainty principle rather than by lattice discreteness (Taiba et al., 18 Aug 2025).

This terminological spread matters because the genuine discrete-droplet literature and the continuum droplet literature organize localization by different mechanisms. In the former, lattice pinning and onsite/offsite geometry are intrinsic. In the latter, the dominant mechanisms are MF–LHY balance, disorder, trapping, or higher-order dispersive corrections.

2. Canonical lattice equations

Two discrete equations dominate current arXiv usage for ultracold-gas-inspired discrete droplets, while a third, conceptually distinct construction appears in the quantum liquid-drop/nuclear-surface literature.

Model	Governing equation	Characteristic role
Quadratic-cubic DNLS	$$i\partial_t \phi_n = -\mu \phi_n -\frac{C}{2}\left(\phi_{n+1}-2\phi_n+\phi_{n-1}\right) -|\phi_n|\phi_n +|\phi_n|^2\phi_n$$	Bright droplets, dark bubbles, front pinning, snaking
Discrete GPE with LHY correction	$$i\psi_{n,t} +\kappa(\psi_{n+1}+\psi_{n-1}) +\gamma |\psi_n|^2\psi_n -\delta |\psi_n|^3\psi_n=0$$	Optical-lattice droplets, MI, breathers, PN effects
Quantum NLS chain on a lattice ring	A discretized periodic NLS chain obtained from azimuthal surface dynamics and quantized by ABA	Surface-wave “quantum droplet” spectrum on a ring

In the quadratic-cubic DNLS, the onsite terms $-|\phi_n|\phi_n$ and $+|\phi_n|^2\phi_n$ produce a saturating nonlinearity. In the droplet interpretation adopted there, the quadratic focusing term represents the one-dimensional LHY correction, while the cubic defocusing term models MF repulsion. The balance supports localized states without collapse and, crucially, creates bistability between a vacuum state and a finite-density state (Adriano et al., 18 Jul 2025).

In the deep-optical-lattice model, the discrete equation is obtained from a quasi-one-dimensional beyond-mean-field GP equation by Wannier expansion. The cubic coefficient $\gamma$ and quartic-amplitude coefficient $\delta$ inherit the effective MF and LHY contributions from the underlying mixture. After a gauge transformation, the working lattice equation is the cubic-quartic discrete GPE written above (Otajonov et al., 2024).

A separate construction arises from a spherical liquid-shell model, where a defocusing NLS for azimuthal envelope waves on a droplet surface is discretized on a polygonal ring and quantized by algebraic Bethe ansatz. There the “discrete quantum droplet equation” is not a DNLS for ultracold-gas self-bound droplets but a quantum NLS chain used to model a periodic many-body spectrum on the azimuthal lattice (Carstea et al., 2021).

3. Bistability, fronts, and the Maxwell-point construction

The quadratic-cubic DNLS gives the clearest structural definition of a discrete quantum droplet. Its stationary real states solve

$$0= -\mu \phi_n -\frac{C}{2}\left(\phi_{n+1}-2\phi_n+\phi_{n-1}\right) -|\phi_n|\phi_n +\phi_n^3,$$

and the associated uniform states generate a bistable regime in which localized fronts can connect the lower-density and upper-density equilibria. The special parameter where the relevant uniform states are energetically balanced is the Maxwell point

$\mu^{(M)}=-\frac{2}{9}.$

Near this value, a localized droplet is constructed as two nearly stationary fronts placed back-to-back; a bubble is the analogous configuration built over the upper-density background (Kusdiantara et al., 2024, Adriano et al., 18 Jul 2025).

In the strong-coupling continuum scaling $\psi_n(t)$0, the leading front satisfies

$\psi_n(t)$1

and the heteroclinic front connecting $\psi_n(t)$2 and $\psi_n(t)$3 is

$\psi_n(t)$4

This front is the elementary domain wall separating vacuum from the finite-density droplet phase. The discrete droplet is therefore not simply a peaked lattice soliton; it is a composite pinned object built from bistable front segments (Adriano et al., 18 Jul 2025).

This front-based viewpoint is central because it ties the droplet problem to discrete Allen–Cahn-type pinning theory rather than to small-amplitude envelope soliton theory alone. A plausible implication is that discrete quantum droplets are most naturally classified by front geometry, plateau length, and lattice registry, not only by norm or peak amplitude.

4. Pinning region, homoclinic snaking, and multistability

The defining global structure of the quadratic-cubic DNLS droplet problem is the pinning region: a finite interval of $\psi_n(t)$5 around the Maxwell point in which fronts remain locked to the lattice and localized states persist. Within this interval, multiple droplets and bubbles of different plateau lengths coexist and are organized by homoclinic snaking. The bifurcation diagram oscillates through repeated saddle-node turns; onsite and offsite branches weave together, and asymmetric ladder states connect them (Kusdiantara et al., 2024).

In the weak-coupling or anti-continuum regime, the pinning width has an algebraic dependence on the coupling, described in the conclusions of the multistability study as being inversely proportional to the coupling strength. In the strong-coupling regime, the width is exponentially small. One asymptotic result gives

$\psi_n(t)$6

with the more detailed derivation

$\psi_n(t)$7

A numerical fit reported for the same regime is

$\psi_n(t)$8

close to the analytical prediction $\psi_n(t)$9, with about a $$i\partial_t \phi_n = -\mu \phi_n -\frac{C}{2}\left(\phi_{n+1}-2\phi_n+\phi_{n-1}\right) -|\phi_n|\phi_n +|\phi_n|^2\phi_n$$0 discrepancy in the exponent estimate (Kusdiantara et al., 2024).

The later exponential-asymptotics analysis refines this picture by showing that beyond-all-orders terms generated by complex singularities of the front determine the pinning width. In that treatment,

$$i\partial_t \phi_n = -\mu \phi_n -\frac{C}{2}\left(\phi_{n+1}-2\phi_n+\phi_{n-1}\right) -|\phi_n|\phi_n +|\phi_n|^2\phi_n$$1

and the stationary-front condition is obtained from the exponentially small solvability term that survives optimal truncation of the divergent asymptotic series (Adriano et al., 18 Jul 2025).

These results place discrete quantum droplets in the same general class as other pinned localized structures in bistable lattices, but with a nonlinear closure tailored to droplet and bubble phenomenology.

5. Modulational instability, optical lattices, and discrete breather regimes

A second major branch of the literature studies discrete droplets in a deep quasi-one-dimensional optical lattice via the cubic-quartic discrete GPE

$$i\partial_t \phi_n = -\mu \phi_n -\frac{C}{2}\left(\phi_{n+1}-2\phi_n+\phi_{n-1}\right) -|\phi_n|\phi_n +|\phi_n|^2\phi_n$$2

This equation is derived from a quasi-one-dimensional reduction of the beyond-mean-field GP equation followed by a Wannier expansion in a deep lattice. The plane-wave solution

$$i\partial_t \phi_n = -\mu \phi_n -\frac{C}{2}\left(\phi_{n+1}-2\phi_n+\phi_{n-1}\right) -|\phi_n|\phi_n +|\phi_n|^2\phi_n$$3

has nonlinear dispersion relation

$$i\partial_t \phi_n = -\mu \phi_n -\frac{C}{2}\left(\phi_{n+1}-2\phi_n+\phi_{n-1}\right) -|\phi_n|\phi_n +|\phi_n|^2\phi_n$$4

Linearizing around this state yields the perturbation eigenvalues $$i\partial_t \phi_n = -\mu \phi_n -\frac{C}{2}\left(\phi_{n+1}-2\phi_n+\phi_{n-1}\right) -|\phi_n|\phi_n +|\phi_n|^2\phi_n$$5 and the MI gain $$i\partial_t \phi_n = -\mu \phi_n -\frac{C}{2}\left(\phi_{n+1}-2\phi_n+\phi_{n-1}\right) -|\phi_n|\phi_n +|\phi_n|^2\phi_n$$6, with instability occurring when

$$i\partial_t \phi_n = -\mu \phi_n -\frac{C}{2}\left(\phi_{n+1}-2\phi_n+\phi_{n-1}\right) -|\phi_n|\phi_n +|\phi_n|^2\phi_n$$7

For the staggered case $$i\partial_t \phi_n = -\mu \phi_n -\frac{C}{2}\left(\phi_{n+1}-2\phi_n+\phi_{n-1}\right) -|\phi_n|\phi_n +|\phi_n|^2\phi_n$$8, the MI gain simplifies to

$$i\partial_t \phi_n = -\mu \phi_n -\frac{C}{2}\left(\phi_{n+1}-2\phi_n+\phi_{n-1}\right) -|\phi_n|\phi_n +|\phi_n|^2\phi_n$$9

The paper reports that the system is always unstable for $$i\psi_{n,t} +\kappa(\psi_{n+1}+\psi_{n-1}) +\gamma |\psi_n|^2\psi_n -\delta |\psi_n|^3\psi_n=0$$0 and always stable for $$i\psi_{n,t} +\kappa(\psi_{n+1}+\psi_{n-1}) +\gamma |\psi_n|^2\psi_n -\delta |\psi_n|^3\psi_n=0$$1; for other sign combinations the stability domain depends on the parameter values (Otajonov et al., 2024).

Stationary localized modes are sought from

$$i\psi_{n,t} +\kappa(\psi_{n+1}+\psi_{n-1}) +\gamma |\psi_n|^2\psi_n -\delta |\psi_n|^3\psi_n=0$$2

which gives

$$i\psi_{n,t} +\kappa(\psi_{n+1}+\psi_{n-1}) +\gamma |\psi_n|^2\psi_n -\delta |\psi_n|^3\psi_n=0$$3

Using the Page method and variational approximations, the study identifies intersite, onsite, front-like, flat-top, and dark localized modes. In the strongly localized regime, both symmetric and antisymmetric even modes are reported unstable, whereas symmetric and antisymmetric odd modes are reported stable. In the purely LHY-driven case $$i\psi_{n,t} +\kappa(\psi_{n+1}+\psi_{n-1}) +\gamma |\psi_n|^2\psi_n -\delta |\psi_n|^3\psi_n=0$$4, odd modes remain stable while even and flat-top modes are unstable (Otajonov et al., 2024).

For stronger coupling, the same work passes to a quasi-continuous limit with a super-Gaussian ansatz. There the discrete droplet crosses over from bell-shaped to flat-top as the norm increases, the Thomas–Fermi limit is characterized by

$$i\psi_{n,t} +\kappa(\psi_{n+1}+\psi_{n-1}) +\gamma |\psi_n|^2\psi_n -\delta |\psi_n|^3\psi_n=0$$5

and interaction dynamics show that in-phase droplets repel, out-of-phase droplets with $$i\psi_{n,t} +\kappa(\psi_{n+1}+\psi_{n-1}) +\gamma |\psi_n|^2\psi_n -\delta |\psi_n|^3\psi_n=0$$6 attract and merge, large-$$i\psi_{n,t} +\kappa(\psi_{n+1}+\psi_{n-1}) +\gamma |\psi_n|^2\psi_n -\delta |\psi_n|^3\psi_n=0$$7 flat-top droplets move almost like continuous droplets, and small-$$i\psi_{n,t} +\kappa(\psi_{n+1}+\psi_{n-1}) +\gamma |\psi_n|^2\psi_n -\delta |\psi_n|^3\psi_n=0$$8 soliton-like droplets are hindered by the Peierls–Nabarro barrier (Otajonov et al., 2024).

6. Stability taxonomy and relation to continuum droplet theory

Stability results are not universal across discrete droplet equations. In the exponential-asymptotics analysis of the quadratic-cubic DNLS, the leading eigenvalue of the Maxwell front is exponentially small and changes sign with front position. The asymptotic formula

$$i\psi_{n,t} +\kappa(\psi_{n+1}+\psi_{n-1}) +\gamma |\psi_n|^2\psi_n -\delta |\psi_n|^3\psi_n=0$$9

implies that the onsite front ($-|\phi_n|\phi_n$0) is unstable, while the intersite front ($-|\phi_n|\phi_n$1) is stable. The paper emphasizes that this is opposite to the familiar cubic-discrete dark-soliton case (Adriano et al., 18 Jul 2025).

By contrast, in the deep-optical-lattice cubic-quartic model, the reported robust localized states in the strongly discrete regime are the odd onsite modes, while even intersite and flat-top modes are unstable in the parameter sets studied (Otajonov et al., 2024). This suggests that the stability ordering of onsite and intersite droplets is model-dependent and tied to the precise nonlinear closure, asymptotic regime, and front composition.

The discrete equations themselves are best understood as descendants of continuum droplet theories in which MF attraction or repulsion competes with beyond-MF corrections. A standard one-dimensional continuum reduced equation is

$-|\phi_n|\phi_n$2

whose exact stationary droplets interpolate between small Gaussian-like states and large flat-top puddles (Astrakharchik et al., 2018). A microscopic derivation later showed that the phenomenological extended GP equation used for binary-mixture droplets should be interpreted as the equation of motion for a pairing field rather than for the atomic condensate itself, while the condensate obeys an ordinary GP equation (Hu et al., 2020). In the optical-lattice setting, the discrete cubic-quartic model is obtained precisely by starting from such an extended beyond-mean-field GP description and projecting onto Wannier modes (Otajonov et al., 2024).

The discrete quantum droplets equation is therefore not a single canonical formula but a structured family of lattice models. Its most characteristic features are the coexistence of vacuum and finite-density states, the construction of droplets from pinned fronts, the emergence of homoclinic snaking inside an exponentially or algebraically small pinning interval, and a stability theory governed by lattice registry as much as by nonlinearity.