The research on the quantum droplet is enriched via two main tributaries: 1) the quantum droplet in binary BEC and 2) the quantum droplet in dipolar BEC. The latter is the result of balance among the short-range repulsive interaction born from a two-body contact pseudo-potential, namely, the mean-field (MF) interaction, the repulsive beyond mean-field (BMF) effect, which can be noted as a typical attribute of quantum fluctuation, and attractive long-rang dipolar interaction. However, the self-bound nature of the former comes from the opposite nature of MF and BMF effects, where the interplay of inter-species and intra-species plays a crucial role. In three dimensions, the quantum droplet is balanced and shows instability against collapse by attractive effective MF and a repulsive BMF interaction, respectively. However, in the lower dimension, specifically in one dimension, the role is reversed; that is, effective MF energy is repulsive, whereas the BMF energy is attractive. In this review article, we plan to explicate this unique role reversal phenomena and elaborate on the consequences.

Since the experimental observation of the liquid-like state, it has become the subject of intense research activity. In a recent self-consistent theoretical formulation, LHY correction is incorporated in the GP equation via quantum fluctuation [29]. We have also noted theoretical assertions on the collective modes in a droplet-soliton crossover [30], the existence of vortices in droplets [31–34], dynamics of purely one-dimensional droplets [35], its collective excitations [36], and the effect of external artificial coupling [37] along with comprehensive reviews [38]. At this juncture, we also recall a numerical investigation of quantum liquids for the dipolar BEC in quasi-one-dimensional (Q1D) geometry [39]. Apart from that, we have also seen considerable interest in analyzing the origin of the droplets in lower dimensions [33,40]. A contemporary study also notes a possible connection between the droplets and modulational instability in a one-dimensional (1D) system [41].

The primary focus of this review article is quantum droplets in lower dimensions emerging from a binary condensate. It has already been demonstrated that the coupled equations can be projected into an effective single component equation if both components share the same spatial mode while neglecting the spin excitation [20]. Hence, we start from an extended GP equation, which contains an additional non-linear contribution of the quartic exponent. We then summarize the well-known mathematical prescription to reduce the 3 + 1-dimensional system to 1 + 1-dimension [42]. In this case, the transverse confinement is much stronger than the longitudinal trap frequency. As BEC in one dimension is not realizable, this Q1D geometry is akin to the interest of the experimental community as it can be amenable experimentally. Nevertheless, we summarize the recent theoretical developments in understanding quantum droplets in 1D, and then we move to Q1D. In the 1D system, the GP equation carries an additional quadratic non-linearity, which takes care of the BMF contribution. Therefore, we name this equation quadratic-cubic NLSE or QCNLSE. On the contrary, in Q1D, the governing dynamical equation is cubic quadratic NLSE or CQNLSE. One can also note a unique role reversal of MF and BMF interaction while shifting from Q1D to 1D system geometry.

The review is arranged in the following way: we summarize the theoretical formulation to analyze the liquid-like state in Section 2, which allows us to construct an extended GP equation in 1D. We analyze the existence of the droplets at 1D in Section 3. In Section 4, we present our recent results in Q1D geometry. We brief some of the very recent theoretical developments to describe the recently observed exotic state of the supersolid in Section 5. It has been reported that in between droplet arrays and solitonic state, it is possible to observe phase-coherent periodic waves as if the droplets are immersed in the constant background of a liquid-like state. This kind of paradoxical state of matter that combines the friction-less flow of a superfluid with the crystal-like periodic density modulation of a solid is noted as a supersolid [43]. Experiments were mainly performed using dipolar quantum gases, proving the supersolid nature of arrays of quantum droplets, and establishing the coexistence of spatial order and global phase coherence [44–46]. We also explicate recent theoretical contributions in this aspect in the lower dimension. We draw our conclusion in Section 6.

In the 1D setting, for the symmetric case g = g ₁₁ = g ₂₂ and n = n ₁ = n ₂, we end up with the following effective energy density [21]: ε 1 D = δ g n 2 − 4 2 3 π g n 3 / 2 , (5)

From Eq. 5, we can obtain the equilibrium density n ₀ = 8mg ³/π ² ℏ ² δg ², which denotes a stable liquid when the pressure is zero, and the corresponding chemical potential is noted as ν ₀ = −δgn ₀/2.

The above energy leads to an eGP equation: i ℏ ∂ ϕ ∂ t = − ℏ 2 2 m ϕ x x + δ g | ϕ | 2 ϕ − 2 m π ℏ g 3 / 2 | ϕ | ϕ , (6)

The dynamics of quantum droplets governing Eq. 6 is given by [35]. Eq. 6 can be carried over from weak to strong coupling regime by tunning δg, previously discussed in [52].

Reference [51] explicitly demonstrates kink-like solitons in the droplet regime, that is, 0 < δg ≪ g. The final form of the soliton reads [51] as follows: ϕ ± = g 2 3 g 1 e i k x − ν ℏ t 1 ± tanh x ± v t ξ , (7)

Here, g ₁ = δg, g 2 = 2 m π ℏ g 3 / 2 . The healing length is ξ = ℏ m δ g 1 A where amplitude A = g 2 3 g 1 . Thus, the sharper the healing, the larger the amplitude larger. These propagating modes share the same phase as the background condensate. A denotes the constant background present in the healing length. It is worth noting that the value of A is exactly one-third of the uniform condensate amplitude 2 m π ℏ g 3 / 2 / δ g . When ϕ ₊ has maximum value, then ϕ ₋ shows its minimum value and vice versa [51]. These pairs, ϕ _±, vanish asymptotically at one end and connect the normal vacuum with the quantum droplets located at the origin, which is analogous to kink/antikinks in the ϕ ⁴-theory [53].

Figure 1 illustrates the density profiles for BMF repulsion g = 5. The blue solid line denotes kink, and the red dashed line describes antikink. These solitons, ϕ _±, travel in the opposite directions with equal velocity. The amplitude of the solitons is directly proportional to the repulsive interaction.

Eq. 6 can also lead to flat-headed and localized solution as noted in [35]. Here, we can observe a clear transition from a soliton-like state to a liquid-like state from low-to-moderate particle number. This type of solution is noted for τ > 0, where τ = δg/g, as a result of the fine balance between the repulsive MF and attractive BMF interaction: ϕ = A e − i ν t 1 + B cosh − 2 ν x , (8) where A = n 0 ν ν 0 , B = 1 − ν ν 0 . The solution in Eq. 8 suggests negative values of the chemical potential and is bound in ν ₀ < ν < 0. It features the flat top shape at 0 < ν − ν ₀ ≪|ν ₀|, with size of the droplet L ≈ ( − 2 ν 0 ) − 1 / 2 ⁡ ln [ ( 1 − ν / ν 0 ) − 1 ] [35,50].

From the norm of Eq. 8 [41], N = n 0 − 2 ν 0 ln 1 + ν / ν 0 1 − ν / ν 0 − ν / ν 0 , (9)

In Figure 2, density profiles are plotted for different particle numbers. For N ≪ 1, the kinetic term in Eq. 6 is significant, and a non-uniform localized wave appears. To a certain extent, this profile resembles a bright soliton-like profile where the kinetic energy and the potential energy balance each other. As N increases, for instance, at N = 10, we observe a monotonous growth of the density distribution until it reaches the equilibrium density n ₀. Further increase in the particle number does not make any substantial difference as the system has already attained an equilibrium density. This suggests a delocalize wave with uniform density n ₀. Adding more particles will lead to special growth of the wave keeping the density of the flat-topped condensate restricted at n ₀. The condensate showing this incompressibility may be considered fluid. The situation is analogous to regular liquid, where density remains constant over a wider spatial dimension.

Solitons are in a bound state similar to droplets, except their stabilization process is different. We can distinguish soliton from droplets by analyzing their property while they collide, as the former does not alter its shape upon collision, but the latter behaves differently depending on their momentum, particle number, and phase. We summarize a few recent reports discussing the dynamics theoretically and experimentally [19,35,39].

Dynamical analysis of droplets can be done by simulating Eq. 6 using the split-step method based on the fast-Fourier transform [35]. The initial wave function was taken as ϕ(x, t = 0) = exp(ikx + iφ)ϕ(x + x ₀) + exp( − ikx)ϕ(x − x ₀), where x ₀ is the initial position, k is the initial momentum, and φ is the relative phase.

We note the dynamics in Figure 3, where we observe that the droplets will separate or merge depending on their velocity [19]. For lower initial momentum k = 0.1, droplets with particle numbers N ₁ = N ₂ = 20 marge into one droplet, and the presence of a sound wave states that the newly formed droplet is in an excited state (Figure 3A). The sound wave can be seen oscillating inside the merged droplet. The effect of the phase is shown in Figures 3B,C. In case the collisions are out-of-phase, that is, φ = π, small excitation can be seen at the time of reflection. On the contrary, population transfer can be seen for phase difference φ = π/2. Population transfer is different for different phase differences. The effect of higher momentum k = 1 leads to fission of the droplet into multiple smaller droplets (Figure 3D), which can be seen occurring in classical liquid [39].

The exciting developments in droplet formation and recent developments in artificial couplings motivate us to summarize the recent theoretical prediction of droplet formation in Rabi-coupled bosonic atoms in one dimension [37]. We consider the effect of Rabi-coupling between two hyperfine states where effective intra-species interactions are weakly repulsive, whereas the inter-species interaction is attractive. In this scenario, the beyond mean-field interaction and external Rabi coupling play a crucial role in stabilizing the droplets. The transition between the two states is induced by an external coherent Rabi coupling of frequency Ω _R .

The scaled energy density of ultra-dilute with uniform Rabi-coupled mixtures can be written as [37] ε R s = 1 2 υ − n s 2 − Ω R s n s − 4 2 3 π n s 3 / 2 − 2 2 π Ω R s υ − n s 1 / 2 , (12)

Here, υ ₋ = 1 − |υ|, υ = g 12 g , and suffix s denotes scaled form. In the absence of Rabi coupling, that is, Ω _Rs = 0, Eq. 12 becomes analogous to Eq. 5. The shift in the ground state energy density in the presence of Rabi-coupling leads to quantum mechanical instability. Equilibrium density n _0s is [37] n 0 s = 128 81 π 2 υ − 2 cos 1 3 arccos 1 − 729 π 2 Ω R s υ − 128 + 1 2 2 , (13)

The mixture is stable for 0 < Ω R s υ − ≤ 256 729 π 2 as plotted in Figure 4A. The white band shows instability at |υ|∼ 1. Figure 4B shows that local minima in ɛ _Rs disappears above the critical value of Ω R s c ≃ 0.356 [37] as predicted by Eq. 13. Beyond Ω R s c , the ground state becomes unstable due to effective attraction between atoms.

The NLSE describes a homogeneous second-order non-linear differential equation, which can be mapped to the Jacobi Elliptic equation and solved analytically [58]. This mapping allows us to use the 12 Jacobi elliptic or Cnoidal functions as solutions [59]. The Jacobi elliptic functions can be derived from the amplitude function of Jacobi elliptic integrals [58]. These solutions can be constant, periodic, or localized based on the parameter q as 0 ≤ q ≤ 1. We examine the recently reported cnoidal solutions for QCNLSE with different natures of interaction strength.

Here, we are interested in discussing the static solutions, and for that purpose, ϕ(x, t) is taken as ψ(x)e ^−iνt . Therefore, taking into account the above consideration, Eq. 20 leads to [63] d 2 ψ x d x 2 + α ψ x − β | ψ x | 2 ψ x − γ | ψ x | 3 ψ x − δ x ψ x = 0 , (21) where α = 2ν, β = 2g ₁, γ = 2g ₂, and δ ( x ) = 2 D ( x ) . D ( x ) is the cnoidal potential. At this point, interactions MF and BMF are considered repulsive in nature. However, their exact characteristics can be understood by analyzing the exact solution.

In the experimental setup of cigar-shaped (Q1D) BEC, ω _⊥ is typically more than 10 times stronger than ω ₀. The suitable choice of the transverse (ω _⊥) and longitudinal (ω ₀) trapping frequency ensures the interaction energy of the atoms is much less than the kinetic energy in the transverse direction. Since the trapping frequencies can be controlled quite efficiently, it is also possible to tune the transverse confinement much stronger than the longitudinal counterpart (ω ₀ ≪ ω _⊥), resulting in K → 0. The system can now be viewed as quasi-homogeneous. Here, we brief the obtained analytical solution for Eq. 20, assuming K = 0.

Using ϕ ( x , t ) = ρ ( x , t ) exp i χ ( x , t ) + ν t where ρ(x, t) leads to the amplitude contribution and χ(x, t) is the non-trivial phase, ν is the chemical potential, and with the required assumptions mentioned in [62], we finally end up with the solution [62] ρ ζ = 1 + 12 ν g 1 + 12 ν g cosh − | g | ζ f o r | g | = 3 g ′ . (26)

Here, ν _g = ν/g which implies that the localized structures can only be sustained iff g < 0 or the effective MF interaction is attractive and for real solution, ν _g > 0. Hence, the chemical potential must be negative.

We define the relationship between normalization N and chemical potential ν as [62] N = 1 + ν I 2 g 1 − ν I ln 1 − 2 ν I 1 − ν I + 1 − 2 . (27)

We have denoted 12ν _g = ν _I . N can also be noted as the number of particles associated with the formation of localized wave and scaled by N ₀ where N ₀ defines the particle number obtained from the constant background density solution such that N 0 = 2 a B 15 π a ⊥ 64 2 δ a δ a ′ 2 . Solving Eq. 27 numerically for ν, we can create the density profile following Eq. 26. The density profile is presented in Figure 7. It clearly suggests that, for the very low particle number, the system tends to produce localized modes, whereas, for the reasonably large number, we observe the density is constant over a finite spatial dimension, thereby suggesting the formation of a liquid-like state.

The droplet formation results from mutual competition between the MF and BMF interaction. It has been argued that, at a lower particle number, the system supports the bright soliton-like modes. In the region where the effective interaction energy (E _eff = E _MF + E _BMF ) is negative, the transition from localized to droplet state happens when the effective pressure is nullified (i.e., P = E eff − n d E eff d n ). The corresponding density is known as equilibrium density. The droplets can sustain below the critical density defined by E _eff = 0. Beyond this point, E _eff > 0 yields gradual evolution of the localized modes. The chemical potential in equilibrium turns out to be −0.02, which is even true for a weakly trapped system, as observed in the following section.

The analytical solution of the CQNLSE motivates us now to examine and validate the result numerically. For that purpose, we implement the imaginary time propagation of the split-step Crank–Nicolson (CN) method [64] over small time steps (5 × 10⁻⁶) for a total of 20,000 steps and solve the homogeneous CQNLSE [62]. Then, we consider K ≠ 0 for several trap frequencies and different BMF interaction strengths κ, where κ is defined as g′/|g|. We use three arbitrary values of κ, as 0.1, 0.5, and 0.9 [65]. Figure 8 depicts the density distribution for K = 0.00001 (blue dashed-dotted line), K = 0.001 (red dotted line), and K = 0.1 (grey dashed-double-dotted line), respectively. The density profile suggests a transition from a droplet-like state to a localized density distribution as we increase the trap frequency. At K = 0.00001, the flat top density distribution is distinctly visible, indicating the droplet-like state, whereas tighter confinement leads to the generation of a bright soliton-like state as described by the grey dashed-double-dotted line. In the intermediate region with moderate trap frequency (say at K = 0.001), a bell-like structure emerges. Though Figure 8 is prepared for κ = 0.1, we do not observe any significant deviation in nature for larger κ values.

To understand the possible droplet-soliton transition, we examine the chemical potential. Assuming ϕ(x, t) = ϕ(x)e ^−iνt and applying in Eq. 20, we obtain ν ψ x = − 1 2 d 2 d x 2 + 1 2 K x 2 + g ψ 2 x + g ′ ψ 3 x ψ x . (28)

Further incorporating the normalization condition ∫ − ∞ ∞ ψ 2 ( x ) d x = 1 , we yield ν = ∫ − ∞ ∞ 1 2 d ψ d x 2 + ψ 2 x 1 2 K x 2 + g ψ 2 x + g ′ ψ 3 x d x . (29)

The variation of chemical potential while trap frequency is tuned is noted in Figure 9A. We repeat this calculation for different interaction strengths (κ = 0.1, 0.5, and 0.9). For a quite loos trap, the system can be treated as quasi-homogeneous, leading to the formation of droplets signified by the negative value of the chemical potential. Furthermore, we note that, in this situation, the chemical potential converges to ∼ − 0.02 for different interaction strengths. It must be noted that a similar result has already been reported for κ = 0.33 in the previous section, where we discussed the analytical calculation for a homogeneous system [62]. Figure 9A also reveals that the stronger confinement leads to the solitonic state with ν > 0 (right side of the grey shaded region of Figure 9A). Curiously, we observe a non-monotonic behavior of the chemical potential in the shaded region where the confining potential strength is in the regions 0.001 and 0.1. A comparison with Figure 8 encourages us to consider this frequency window as the transition region.

Now, it is important to understand the nature of the transition. The pertaining question is whether the transition is a quantum phase transition (QPT) or a crossover. In QPT, one can identify an abrupt change in the ground state of the system when a Hamiltonian parameter crosses a critical value [66,67]. Here, we recognize K or the square of the longitudinal trapping frequency as the Hamiltonian parameter. Because we do not observe any abrupt change in ν, we calculate the first derivative of the chemical potential with respect to K but again fail to identify any abrupt change. Thus, we conclude the transition to be a crossover rather than a QPT, and we recognize a loose crossover boundary as 0.0001 ≲ K ≲ 0.1 (described in the light-magenta shaded region in the figures).

To verify our analysis, we also calculate the total energy E = ∫ − ∞ ∞ 1 2 d ψ d x 2 + ψ 2 x 1 2 K x 2 + 1 2 g ψ 2 x + 2 5 g ′ ψ 3 x d x , (30) and capture its variation in Figure 9B. The energy description also supports our assertion as we observe negative energy for weak confinement leading to the accumulation of droplet-like bound pairs. The energy crosses the zero line in the vicinity of K = 0.001, which might point to the breakdown of droplet-like bound pairs. However, the energy flattens in the shaded area might be suggestive of an equilibrium of the dropleton-soliton mixture. Further increase in the trapping potential destroys the bound pairs, and soliton-like state emerges.

We also observe a very smooth transition of root-mean-square (rms) ( ⟨ x 2 ⟩ ) size of the condensate, which is depicted in Figure 10. The rms size is normalized by the rms size ( ⟨ x 0 2 ⟩ ) for K = 0, and we introduce ⟨ X 2 ⟩ = ⟨ x 2 ⟩ ⟨ x 0 2 ⟩ . According to Figure 10fig10, we can conclude that the weak trapping potential allows the larger pair size, suggesting the droplet formation. However, gradually and smoothly, the pair size falls off as we increase the longitudinal frequency. We also observe that the pair size drops much faster in the shaded region, defined as the crossover region. The rms curve flattens thereafter and asymptotically reaches 1/10 of the droplet size for a tightly confined one-dimensional geometry.

The role of particle number in the transition is also a matter of deep interest. We calculate the energy and chemical potential for different particle numbers (from 25 to 100). We draw a contour phase diagram based on our findings depicted in Figure 11. The bluish region corresponds to the negative energy region, thereby suggesting the existence of the bound pairs and droplet formation. The red region describes the unbound solitonic region, and the greenish region corresponds to the region of crossover where the droplets are breaking down into solitons. Moreover, one can also conclude that the larger particle number hinders the droplet formation even if the confinement is very weak. It is similar to the fact mentioned in the previous section regarding the existence of a critical density beyond which droplets cannot form in a homogeneous system.

A recent analytical study has highlighted the generation of supersolid-like phase in a self-trapped 1D system [70]. The analysis comes up with several analytical solutions by considering a propagating Bloch function type solutions [71]. The self-trapped dynamical system can be described as i ℏ ∂ Φ 1 D ∂ t = − ℏ 2 2 m Φ x x 1 D + δ g | Φ 1 D | 2 Φ 1 D − 2 m π ℏ g 3 / 2 | Φ 1 D | Φ 1 D , (31)

Here, Φ^1D defines the wavefunction in a strictly 1D system. For notational convenience let us replace Φ^1D as ϕ, and we assume ϕ represents a propagating Bloch function such that ϕ = ϕ 0 x − v t ξ exp i k x − i ν ℏ t , (32)

Here, ϕ ₀ is taken as a conoidal function with modulus parameter q. We have already noted that, for q = 0, the cnoidal function manifests sinusoidal nature: ϕ 0 = A + B s n x − v t ξ , q , (33) where A = 2 m 3 π ℏ g 3 / 2 δ g , B = ± 2 q q + 1 A , ξ = 3 ℏ 2 π 2 m ( q + 1 ) δ g g 3 , and 0 < q < 1.

It is evident that the existence of the Bloch type solutions with a superfluid background crucially depends upon the MF term, BMF correction, and dispersion. Here, ϕ min/max = A ± B = A 1 ± 2 q q + 1 > 0 shows that the quantum supersolid immersed in a residual BEC [44,46,68]. The density of the residual BEC, n res = A 2 1 − 2 q q + 1 2 > 0 . This diffused matter-wave density rules out the scenario of one atom per site thereby overtaking the Penrose and Onsager criterion [72–74].

The density of the supersolid phase is shown in Figure 12 in the comoving frame. With increasing repulsive intra-component interaction g, the number of droplets increases and their size decreases to a fixed value of x/ξ. Furthermore, keeping all the physical parameters constant, an increase in the modulus parameter increases the interdroplet spacing [70].

Moving toward the Q1D geometry, we recognize that it is impossible to yield any analytical solution from a self-trapped system. Hence, we employ a bi-chromatic lattice in CQNLSE. The application of bi-chromatic lattice was motivated by the recent observations of supersolid-like features in a spin-orbit coupled BEC, where the experiment was performed in the presence of two optical lattices of different frequencies in the same spatial dimension where the effective lattice potential was described as a superlattice [75]. However, the manifestation of supersolidity emerges naturally in a spin-orbit or dipolar BEC. To mimic these additional contributions, we add a driving force that is periodic in nature to compete with the two-body mean-field interaction. The use of external driving force in ultracold atomic systems is not alien [76,77]. There are suggestions for generating and controlling the transport of BEC atoms from a reservoir to the waveguide via a source/driving force [76–79].

The knowledge of Eq. 20 allows us to write a generic time-dependent DCQNLSE as [80] − 1 2 ∂ 2 Φ ∂ x 2 + V 2 ⁡ sin ⁡ ζ x − V 1 ⁡ sin 3 ⁡ ζ x Φ + g 1 | Φ | 2 Φ + g 2 | Φ | 3 Φ − i ∂ Φ ∂ t = F ′ x , t . (34)

Focusing on the static solution we assume, Φ(x, t) = ϕ(x)e ^−iνt , where ν is the chemical potential. To extract the static behavior, it is also necessary to lock the temporal phase part of the driving force with the solution ansatz. Also, we assume the spatial variation in sinusoidal in nature. Hence, we denote F′(x, t) = F e ^−iνt sin ζx. The resulting time independent DCQNLSE reduces to [80] − 1 2 d 2 ϕ d x 2 + V 2 ⁡ sin ⁡ ζ x − V 1 ⁡ sin 3 ⁡ ζ x ϕ + g 1 | ϕ | 2 ϕ + g 2 | ϕ | 3 ϕ − ν ϕ = F sin ⁡ ζ x . (35)

Here, the strength of odd (cubic) and even (quartic) exponents is described via g ₁ and g ₂, respectively. The coherence length can be recognized as 1/ζ. The bi-chromatic lattice potential depths are described via V ₁ and V ₂. A suitable modulation of the lattice depth can create a superlattice [75]. As the cubic exponent of the potential can be written in terms of triple angle representation, the wavenumber of one laser is required to be thrice the second laser. F is the amplitude of the external force. To obtain a sinusoidal mode, we assume an ansatz such that ϕ(x) = A + B sin ζx. From the recent investigation described in [46], we learn that the density distribution of the supersolid phase can be described adequately via the sinusoidal function similar to the ansatz. Applying the ansatz in Eq. 35, we encounter a set of consistency conditions as noted in [80].

The solutions parameters such as A and B can be solved, and we can write the final solution as ϕ x = ϕ Q1D x = − g 1 3 g 2 + V 1 g 2 1 / 3 ⁡ sin ⁡ ζ x . (36)

Here, A = −g ₁/3g ₂ and B = ( V 1 / g 2 ) 1 / 3 . The coherence length shows a function of external driving force as ζ = ± 2 F B . The expression for B suggests that B ∈ R , thus avoiding the occurrence of the nonphysical situation such as complex coherence length. This also implies that if V ₁ > 0 then g ₂ > 0 or vice versa. However, g ₁ does not have any such constraints. Therefore, the MF interaction strength can be attractive and repulsive, whereas the nature of BMF will depend on the nature of optical lattice potential amplitude. We also note that it is necessary to apply the driving force in the same direction as the displacement.

Figure 13 depicts the spatial variation of density (|ϕ(x)|²) via a red dashed line. The blue solid line describes the spatial variation of the BOL. As shown in Figure 13, the existence of the density wave is quite evident, with density maxima coinciding with the potential minima.

The stability of these modes can be discussed in the light of the VK criterion [54]. Here, calculating the particle number (N) in a unit cell of length L we find that ∂ N / ∂ ν = L / g 1 . Hence, for a stable solution, the MF interaction should be positive, which physically implies a repulsive nature.

Till now, we have discussed the 1D and Q1D systems. The 1D and Q1D systems have already been characterized by the one-dimensional scattering length ( a ̃ ) and density ( n ̃ ) . For a 1D system Σ = 1 / | n ̃ a ̃ | < 1 , whereas Σ > 1 for Q1D geometry. Then, what happens if Σ ∼ 1 or at the dimensional crossover? From a mathematical point of view, we understand that the even exponent in the BMF contribution is two for 1D and four for Q1D. However, we lack the clarity of the dynamical equation at the dimensional crossover. We attempt the problem by assuming a mixture of BMF contributions from 1D and Q1D, which implies that the dynamical equation contains both the quadratic and quartic non-linearities; thus, the NLSE can now be termed as quadratic-cubic-quartic NLSE or QCQNLSE. We solve the dynamical equation following the already described methodology and look for sinusoidal solutions. For brevity, we do not explicate the detailed calculation. Nevertheless, one can easily obtain the analytical results following the same prescription of Section 5.2. If we denote ϕ _1D , ϕ _Q1D as analytically obtained sinusoidal wave functions and ϕ _Q1D−1D describe the wave function in the crossover; then, we observe that ϕ _Q1D = ϕ _Q1D−1D ≠ ϕ _1D . Thus, the continuity of the wave function is retained in the region Σ < 1 and Σ ∼ 1. However, there is a clear discontinuity in moving toward the 1D regime [80]. We have also noted that, for Q1D and dimensional crossover, we require a super lattice-like trapping geometry, whereas a 1D system can be analyzed in a regular optical lattice.