# Supersolid behavior in one-dimensional self-trapped Bose-Einstein condensate

Mithilesh K. Parit<EMAIL_ADDRESS>Department of Physical Sciences,
Indian Institute of Science Education and Research Kolkata, Mohanpur-741246,
West Bengal, India  Gargi Tyagi<EMAIL_ADDRESS>Institute of Applied
Sciences, Amity University, Sector-125, Noida-201303, Uttar Pradesh, India
Dheerendra Singh<EMAIL_ADDRESS>Department of Physical Sciences,
Indian Institute of Science Education and Research Kolkata, Mohanpur-741246,
West Bengal, India  Prasanta K. Panigrahi<EMAIL_ADDRESS>Department
of Physical Sciences, Indian Institute of Science Education and Research
Kolkata, Mohanpur-741246, West Bengal, India

###### Abstract

Supersolid is an exotic state of matter, showing crystalline order with a
superfluid background, observed recently in dipolar Bose-Einstein condensate
(BEC) in a trap. Here, we present exact solutions of the desired Bloch form in
the self-trapped free-floating quantum fluid. Our general solutions of the
amended nonlinear Schrödinger equation, governing the mean-field and beyond
mean-field dynamics, are obtained through a Möbius transform, connecting a
wide class of supersolid solutions to the ubiquitous cnoidal waves. The
solutions yield the constant condensate, supersolid behavior and the self-
trapped droplet in different parameter domains. The lowest residual condensate
is found to be exactly one-third of the constant background.

##  I Introduction

Possible existence of the supersolid phase of matter has been conjectured
quite some time back gross ; lifs , with the liquid Helium being the one
extensively investigated for its possible realization kim . A supersolid is a
state of matter that manifests a crystalline order with the presence of a
superfluid component, the latter displaying zero viscosity and flow without
resistance. Observation in liquid Helium being inconclusive, alternate avenues
to its potential observation emerged in the area of cold atomic systems 6r3 ;
Prok .

Very recently, the transient supersolid phase has been identified in dipolar
condensates in different experiments chomaz1 ; guo1 ; 6r2 ; 6r3 . Three
distinct regimes of ground state phase diagram$-$ a regular Bose-Einstein
condensate (BEC), coherent and incoherent arrays of quantum droplets have been
revealed. Böttcher et al. 6r3  and Tanzi et al. 6r2  achieved the supersolid
phase in strongly dipolar BECs, arising due to competitive repulsive short-
range and attractive long-range dipolar interactions. This has been
theoretically well supported by Roccuzzo and Ancilotto roccuzzo . In both the
experiments, BEC from the trap is released to expand, with the interference of
matter waves producing crystalline structure, immersed in a superfluid BEC.
The properties of the dipolar supersolid are found to be well described by the
extended Gross-Pitaevskii equation, describing the mean-field dynamics with
the presence of an additional quartic term 6r3 . Chomaz et al. identified the
supersolid properties in the dipolar quantum gases of ${}^{166}{Er}$ and
${}^{164}{Dy}$ chomaz1 .

Guo et al. guo1  observed two low energy Goldstone modes, revealing the phase
rigidity, confirming that array of quantum droplets are in the supersolid
phase. The formation of the supersolid phase is identified through careful
observation of these two different Goldstone modes, corresponding to the
distinct sound modes of the residual condensed phase and the supersolid
vibrations. However, the presence of the trap led to distortion of Goldstone
modes. As mentioned earlier, in the case of the other two experiments, the BEC
from the trap was released for the matter-waves to interfere and form
transient supersolid phase. Hence, it is worth investigating the possibility
of realizing supersolid phase without a trap in ultracold scenario.

Recently, Petrov 1r5 , has identified a parametric domain in a two component
BEC, where self-trapped quantum droplets form 1r5_ptr . They have been
observed in the free-floating condition tarruellt ; 2r2_ptr ; 1r7  in dipolar
Bose gas and magnetic quantum liquid 3r4 , making them an ideal ground to
explore the possibility of the self-trapped quantum liquid in the absence of a
trap. Balance of the mean-field (MF) energy and Lee-Huang-Yang (LHY) beyond
mean-field lee  quantum fluctuations (QFs) leads to these self-trapped quantum
liquid droplets. Chomaz et al., demonstrated that LHY stabilization is a
general feature of strongly dipolar gases and also examined the role of QFs
determining the system properties, particularly its collective mode and
expansion dynamics chomaz2 . Various aspects of the quantum droplets have been
explored: Bose-Bose droplets in dimensional crossover regime 1r12 , Bose-Fermi
droplets of attractive degenerate bosons and spin polarized fermions 1r13 ,
have been investigated, some of which have found experimental verification.
Bright solitons to droplet transitions have also been demonstrated 1r14 . The
case of one dimension is distinct from the two and three dimensions 1r5 ; 1r25
, as mean-field repulsion is required to balance the quantum pressure, unlike
that of attraction in higher dimensions.

The cigar shaped self-trapped quantum droplets have been modelled by the
amended nonlinear Schrödinger equation (NLSE), which has a repulsive mean-
field nonlinearity like regular BEC and a quadratic nonlinearity arising from
QFs. It has an exact flat-top large droplet solutions 1r5_ptr  and may have
multiple droplet solutions indicated from variational approach 1r5_boris .

Here, we have identified exact Bloch wave solutions of the amended nonlinear
Schrödinger equation, necessarily accompanied by a non-vanishing condensate
density, showing clearly the supersolid phase of quantum droplet system. We
employ Möbius transform panigrahi2 ; vivek1  to connect a wide class of
desired solutions with the cnoidal functions, representing nonlinear periodic
density waves, which manifest in diverse physical systems. Our procedure
yields a unified picture of the uniform condensate, self-trapped and periodic
supersolid phases in different parameter domains.

The paper is organized as follows. In Sec. II, theory of quantum droplets is
briefly described, leading to the amended NLSE, governing its dynamics in one
dimension. In Sec. III, we present supersolid solutions; a Bloch wave,
necessarily possesing a constant background. Their properties in various
configurations are highlighted. Finally, we conclude with the summary of
results and future directions for investigation.

##  II Dynamics of quantum droplets in one dimension

Quantum droplets have been shown to occur in the binary BECs, with repulsive
intra- ($g_{\uparrow\uparrow},~{}g_{\downarrow\downarrow}$) and attractive
inter-component interaction ($g_{\uparrow\downarrow}$), in the vicinity of the
MF collapse instability point. Specifically, they appear in the regime,
$0<\delta
g=g_{\uparrow\downarrow}+\sqrt{g_{\uparrow\uparrow}~{}g_{\downarrow\downarrow}}\ll\sqrt{g_{\uparrow\uparrow}~{}g_{\downarrow\downarrow}}$,
with attractive inter-component interaction $g_{\uparrow\downarrow}<0$ and
repulsive intra-component average interaction
$g=\sqrt{g_{\uparrow\uparrow}~{}g_{\downarrow\downarrow}}>0$. The energy
density of such homogeneous mixture has been obtained 1r5_ptr ,

$\mathcal{E}=\frac{(g_{\uparrow\uparrow}^{1/2}n_{\uparrow}-g_{\downarrow\downarrow}^{1/2}n_{\downarrow})^{2}}{2}\\\
+\frac{\sqrt{g_{\uparrow\uparrow}~{}g_{\downarrow\downarrow}}~{}(g_{\uparrow\downarrow}+\sqrt{g_{\uparrow\uparrow}~{}g_{\downarrow\downarrow}})~{}(g_{\uparrow\uparrow}^{1/2}n_{\uparrow}+g_{\downarrow\downarrow}^{1/2}n_{\downarrow})^{2}}{(g_{\uparrow\uparrow}+g_{\downarrow\downarrow})^{2}}\\\
-\frac{2\sqrt{M}}{3\pi\hbar}(g_{\uparrow\uparrow}n_{\uparrow}+g_{\downarrow\downarrow}n_{\downarrow})^{3/2},$
(1)

where $n_{\uparrow}$ and $n_{\downarrow}$ are densities of the two components,
related by
$n=n_{\uparrow}=n_{\downarrow}\sqrt{g_{\uparrow\uparrow}/g_{\downarrow\downarrow}}$.
The first two terms in Eq. (1) are the MF contribution and the last term
represents the LHY-BMF correction. With the assumption,
$g=g_{\uparrow\uparrow}\sim g_{\downarrow\downarrow}$, we have
$n=n_{\uparrow}=n_{\downarrow}$ and $0<\delta g\ll g_{\uparrow\uparrow}\sim
g_{\downarrow\downarrow}$. The energy density reduces to,

$\mathcal{E}=\frac{\delta
gn^{2}}{2}-\frac{2\sqrt{M}}{3\pi\hbar}\left(gn\right)^{3/2},$ (2)

having the equilibrium density $n_{0}=8g^{3}/(9\pi^{2}\delta g^{2})$ and
chemical potential $\mu_{0}=-\delta gn_{0}/2$. The validity of $\mathcal{E}$
has been justified by diffusion Monte-Carlo simulation 1r5_ptr .

The modified Gross-Pitaevskii equation, with cubic MF and quadratic BMF
nonlinearty, has the form,

$i\hbar~{}\psi_{t}=-\frac{\hbar^{2}}{2M}~{}\psi_{xx}+\delta
g~{}\left|\psi\right|^{2}\psi-\frac{\sqrt{2M}}{\pi\hbar}g^{3/2}\left|\psi\right|\psi.$
(3)

It has constant solution:
$\psi_{const}=\frac{\sqrt{2M}}{\pi\hbar}\frac{g^{3/2}}{\delta g}$, arising
from the balance of MF and BMF contributions. The plane wave solution,
$\psi=\sqrt{P}\exp{(ikx-i\frac{\mu}{\hbar}t)}$, which is prone to modulational
instability (MI), with

$\sqrt{P}_{\pm}=\frac{\sqrt{2M}g^{3/2}}{2\pi\hbar~{}\delta
g}\pm\sqrt{\frac{Mg^{3}}{2\pi^{2}\hbar^{2}~{}\delta
g^{2}}+\frac{\bar{\mu}}{\delta g}},$ (4)

where $\bar{\mu}=\left(\mu-\frac{\hbar^{2}k^{2}}{2M}\right)$. The steady-state
solution is perturbed by small amplitude and the resulting MI has been
investigated in mi1 ; mi2 . MI manifests at wave number $k_{P}$, where
$\Omega$ becomes complex

$\Omega=\pm~{}\frac{1}{2M}~{}\sqrt{k_{P}^{2}\left(\hbar^{2}k_{P}^{2}-4M\left(\frac{\sqrt{2M}}{\pi\hbar}g^{3/2}\frac{\sqrt{P}}{2}-\delta
gP\right)\right)},$ (5)

with $k_{P}$ is the wavenumber of the small perturbation. MI occurs for
$g>\left(\frac{2\pi^{2}\hbar^{2}P}{M}\delta g^{2}\right)^{1/3}$ and

$\left|k_{P}\right|<2\sqrt{\left(\frac{\sqrt{2M}}{\pi\hbar}g^{3/2}\frac{\sqrt{P}}{2}-\delta
gP\right)}\equiv k_{0}.$ (6)

MI is maximum at $k_{P}=k_{0}/\sqrt{2}$ and ${k_{0}}$ is maximum at
$\mu=-\frac{3M}{8~{}\pi^{2}\hbar^{2}}\frac{g^{3}}{\delta g}$. The regime in
the parameter domain, where MI occurs is conducive for inhomogeneous
solutions. In the following section, we investigate the possible inhomogeneous
solutions for the amended NLSE, explicitly showing the presence of a
supersolid phase.

##  III Supersolid phase

We consider propagating Bloch function type solutions kittel :

$\psi=\psi_{0}\left(\frac{x-vt}{\xi},m\right)\exp\left(ikx-i\frac{\mu}{\hbar}t\right).$
(7)

Here $v=\frac{\hbar k}{M}$ and $\psi_{0}$ is a periodic function with m being
the modulus parameter:

$-\frac{\hbar^{2}}{2M}~{}\partial^{2}_{x}\psi_{0}+\delta
g\psi_{0}^{3}-\frac{\sqrt{2M}}{\pi\hbar}g^{3/2}\psi_{0}^{2}-\bar{\mu}\psi_{0}=0.$
(8)

Henceforth, we take $\psi_{0}$ to be a real periodic function without loss of
generality, taking advantage of the global $U(1)$ symmetry of the system. For
finding general solutions, appropriate Möbius transformations are employed, to
connect the solution space to the ubiquitous cnoidal waves hancock ,
satisfying: $f"\pm af\pm\lambda f^{3}=0$ panigrahi2 . The general solution has
the form,

$\psi_{0}(x,t)=\frac{A+Bf^{\delta}(\frac{x-vt}{\xi})}{1+Df^{\delta}(\frac{x-vt}{\xi})}$
(9)

Balancing of nonlinearity with dispersion leads to $\delta=1$ and $\delta=2$
as possible choices. Here, we consider $\delta=1$ as it has richer structure,
which allows for diverse boundary conditions to be satisfied. We now start
with the periodic solution,

$\psi_{0}\left(x,t\right)=A+B~{}\text{sn}\left(\frac{x-vt}{\xi},m\right),$
(10)

and find that it is necessarily accompanied by positive background
$A=\frac{\sqrt{2M}}{3\pi\hbar}\frac{g^{3/2}}{\delta g}$, having one-third the
value of the uniform background. It is worth emphasizing that the supersolid
solutions does not smoothly trace it to the constant condensate in the limit
$m\to 0$ when $B=0$. Its amplitude, $B=\pm\sqrt{\frac{2m}{m+1}}A$, never
exceeds that of the background in the allowed energy range of the modulus
parameter $0<m<1$. The healing length is obtained as

$\xi=\frac{3\hbar^{2}\pi}{2M}\sqrt{\frac{(m+1)~{}\delta g}{g^{3}}},$ (11)

It is evident that, the existence of the Bloch type solutions with a
superfluid background crucially depends upon the cubic non-linearity, BMF
correction and dispersion. For a moving supersolid,
$\frac{\hbar^{2}k^{2}}{2M}-\mu=\frac{4M}{9\pi^{2}\hbar^{2}}\frac{g^{3}}{\delta
g}$, revealing that the chemical potential is bounded below,
$\mu_{min}=\mu_{0}=-\frac{4M}{9\pi^{2}\hbar^{2}}\frac{g^{3}}{\delta g}<0$,
identical to that of the self-trapped droplet 1r5_ptr . Here,
$\psi_{min/max}=A\pm B=A\left(1\pm\sqrt{\frac{2m}{m+1}}\right)>0$, showing
clearly that for the quantum supersolid immersed in a residual BEC 6r2 ; 6r3 ;
roccuzzo . The density of the residual BEC
$n_{res}=A^{2}\left(1-\sqrt{\frac{2m}{m+1}}\right)^{2}>0$. This diffused
matter-wave density rules out the scenario of one atom per site thereby
overtaking the Penrose and Onsagar criterion PenOn ; chester ; Dy1994 .

Figure 1: (Color online) The density of supersolid phase as a function of
scaled position is depicted for different values of intra-component
interaction ($g$) and modulus parameter, with $\delta g=0.1$ (black: $m=0.2$
and red: $m=0.5$), for both the panels. The value of intra-component
interaction, $g=10$ for panel (a) and $g=12$ for panel (b). Panel (c) shows
the pictorial representation of several quantum droplets immersed in a
residual BEC, with boundaries from the both sides (solid vertical lines),
confined in a box of finite length.

The density of the supersolid phase is shown in Fig. 1, as a function of
scaled position variable $x/\xi$, in the comoving frame. It is evident that
inter-droplet spacing (peak-to-peak distance) and number of quantum droplets
are inter-related. With increasing repulsive intra-component interaction $g$,
number of droplets increases and their size decreases for a fixed value of
$x/\xi$. Further, keeping all the physical parameters constant, increase in
the modulus parameter increases the inter-droplet spacing.

The second exact solution is of the sinusoidal form:

$\psi_{0}=\frac{A}{1+D~{}sin\left(\frac{x-vt}{\xi}\right)},$ (12)

with the amplitudes, $A=\frac{3\pi\hbar\bar{\mu}}{\sqrt{2M}g^{3/2}}$ and
$D=\sqrt{1-\frac{9\pi^{2}\hbar^{2}}{4M}\frac{\delta g}{g^{3}}\bar{\mu}}$. The
healing length $\xi=\sqrt{-\frac{\hbar^{2}}{2M\bar{\mu}}}$ and
$\mu_{min}=-\frac{4M}{9\pi^{2}\hbar^{2}}\frac{g^{3}}{\delta g}$. Hence, the
minimum chemical potential is again identical to that of the self-trapped
droplet 1r5_ptr .

We now proceed to study the general periodic solution of the amended NLSE,

$\psi_{0}=\frac{A+B~{}sn\left(\frac{x-vt}{\xi},m\right)}{1+D~{}sn\left(\frac{x-vt}{\xi},m\right)},$
(13)

A straightforward but tedious calculation yields, $D=\pm\sqrt{2m/(m+1)}$,
$B=DX_{\pm}$, with
$X_{\pm}=\left(g_{2}\pm\sqrt{g_{2}^{2}-4g_{1}\mu}\right)/2g_{1}$. The value of
$D$, $0<|D|<1$, shows the non-singular nature of the general solutions. The
possible solutions of $A$ and $\xi$ ($\frac{1}{\xi^{2}}=\frac{ML}{\hbar^{2}}$)
are listed below:

$\displaystyle A_{\pm}^{1}$ $\displaystyle=\frac{g_{2}X_{+}\pm
6\sqrt{\left(g_{2}^{2}/4g_{1}-\bar{\mu}\right)\left(g_{2}X_{+}-g_{2}^{2}/2g_{1}-\bar{\mu}\right)}}{g_{2}+3g_{1}\sqrt{g_{2}^{2}-4g_{1}\bar{\mu}}},$
(14a) $\displaystyle A_{\pm}^{2}$ $\displaystyle=\frac{g_{2}X_{-}\pm
6\sqrt{\left(g_{2}^{2}/4g_{1}-\bar{\mu}\right)\left(g_{2}X_{-}-\bar{\mu}\right)}}{g_{2}-3g_{1}\sqrt{g_{2}^{2}-4g_{1}\bar{\mu}}},$
(14b) $\displaystyle L_{\pm}$ $\displaystyle=-\frac{g_{2}X_{+}/2-\bar{\mu}\pm
3\sqrt{\left(g_{2}^{2}/4g_{1}-\bar{\mu}\right)\left(g_{2}X_{+}-\bar{\mu}\right)}}{(m+1)},$
(14c) $\displaystyle L_{\pm}$ $\displaystyle=\frac{g_{2}X_{+}/2-\bar{\mu}\pm
3\sqrt{\left(g_{2}^{2}/4g_{1}-\bar{\mu}\right)\left(g_{2}X_{-}-\bar{\mu}\right)}}{(m+1)}.$
(14d)

The chemical potential: (i) $\mu>-\frac{g_{2}^{2}}{4g_{1}}$ and (ii)
$-\frac{0.414~{}g_{2}^{2}}{g_{1}}<\mu<0$. The minimum chemical potential is
less than that of obtained for the self-trapped supersolids [Eq. (10) and
(12)] or quantum droplets 1r5_ptr .

It is worth mentioning that, for the case of cigar-shaped BEC, the Gross-
Pitaevskii (GP) equation does not yield a constant background with periodic
modulation. The presence of both mean-field and BMF energies are crucial for
the existence of these solutions here. The soliton trains observed in the
attractive BEC are composed of BEC droplets in the absence of a superfluid
background Partridge . For the case of NLSE with a phase locked source,
periodic solutions with constant background are permissible panigrahi2 . In
optical fibers analogous periodic solutions in the temporal domain have been
experimentally observed as frequency combs (Gorodetsky, ). Recently, NLSE with
cubic-quadratic nonlinearity and phase locked with a source has been
investigated, wherein the parity even and kink type solutions have been
identified Kumar . The cubic-quadratic equation with a phase-locked source
have also been investigated through the fractional linear transformations, for
attractive Kerr nonlinearity and repulsive BMF correction belicd . The
behavior of solitons under nonlinearity management has been explored malomed2
. In the ensuing sections, the dispersion relation for supersolid is
investigated, distinctly showing that increase of repulsive inter-component
interaction decreases the energy.

##  IV Energy and momentum of the supersolid phase

We now investigate the dispersion relation of the supersolid phase. The energy
and momentum density pethick ; 1r27  can be computed for a supersolid in a box
of finite length. The expressions for the momentum and energy are provided in
the supplementary information.

Figure 2: (Color online) The energy per atom of the supersolid as a function
of $k$ is depicted for different values of repulsive intra-component
interaction and modulus parameter, with $m=0.5$. It shows that increasing
repulsive interaction while keeping the competitive intra- and inter-component
interaction constant, $\delta g=0.1$, decreases the total energy per atom. The
value of intra-component interaction is $g=10$ for panel (b). Increasing the
value of modulus parameter leads to broader dispersion, due to periodicity of
the solution.

The dispersion relation for supersolid phase is demonstrated in Fig. (2), for
different repulsive inter-component interaction in panel $a$. It exhibits the
decrease in energy per atom with increasing $g$. In panel $b$, the dispersion
relation is shown for different moduli parameter, showing comparative
flattening of the energy per atom for greater moduli. This happens due to
formation of front-like droplet, with infinite period when modulus parameter
approaches unity.

##  V Conclusion

In conclusion, we have obtained the exact supersolid solutions in one-
dimensional self-trapped BEC. The supersolid phase occurs in a free-floating
condition, unlike the case of dipolar BEC, wherein the condensate with
appropriately tuned interactions needs to be released, and the interference of
matter-wave results in supersolid formation with superfluid background in the
exact periodic solution. In the present case, exact Bloch waves, quantum
droplets appear in a periodic array with inter-spacing between two droplets
constant, immersed in a residual BEC background, that establishes inter-
droplet coherence 6r3 ; 6r4 . The chemical potential for two solutions is
identical and for the general solution it is less, which is more favourable.

##  VI Acknowledgements

We thank Prof. R. B. MacKenzie for fruitful discussion. M. K. Parit
acknowledges the financial support from the REDX, Center for Artificial
Intelligence, IISER Kolkata, funded by Silicon Valley Community Foundation
through grant number 2018-191175 (5618). G. Tyagi is thankful to IISER Kolkata
for hospitality.

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## Supplementary Information: Supersolid behavior in one-dimensional self-
trapped Bose-Einstein condensate

### Energy and momentum of the supersolid phase

In this section, we have provided the analytical expression for the energy and
momentum. The energy and momentum density pethick ; 1r27  can be computed,
$E=\int\mathcal{H}~{}dx$, where $\mathcal{H}$ is the Hamiltonian density

$\mathcal{H}=\frac{\hbar^{2}}{2M}\left|\partial_{x}\psi\right|^{2}+\frac{\delta
g}{2}~{}\left|\psi\right|^{4}-\frac{2\sqrt{2M}}{3\pi\hbar}g^{3/2}\left|\psi\right|^{3},$
(15)

and momentum
$P=\frac{i\hbar}{2}\int\left(\psi\partial_{x}\psi^{*}-\psi^{*}\partial_{x}\psi\right)dx.$
The analytical expressions for the momentum and energy are presented below for
the solution-I, with $A=\frac{\sqrt{2M}}{3\pi\hbar}\frac{g^{3/2}}{\delta g}$,
$B=\pm\sqrt{\frac{2m}{m+1}}A$,
$\xi=\frac{3\hbar^{2}\pi}{2M}\sqrt{\frac{(m+1)~{}\delta g}{g^{3}}}$, and
$D_{1}=\frac{\hbar^{2}}{2M}$.

  1. 1.

Number of atoms
The number of atoms $N=\int\left|\psi\right|^{2}dx$.

$\mathbf{N(x)}=\left(A^{2}m+B^{2}\right)(x-tv)+B^{2}\xi
E\left(\left.\text{am}\left(\left.\frac{tv-x}{\xi}\right|m\right)\right|m\right)\\\
+2AB\sqrt{m}\xi\log\left(\text{dn}\left(\left.\frac{x-tv}{\xi}\right|m\right)-\sqrt{m}\text{cn}\left(\left.\frac{x-tv}{\xi}\right|m\right)\right).$
(16)

  2. 2.

Energy density

$\displaystyle\mathbf{E(x)}$
$\displaystyle=\frac{E_{1}+E_{2}+E_{3}+E_{4}+E_{5}\times\lambda_{E}}{6m^{2}\xi^{2}}$
(17a) $\displaystyle E_{1}$
$\displaystyle=2\left(3A^{2}m^{2}\xi^{2}(x-tv)\left(A^{2}g_{1}-Ag_{2}+D_{1}k^{2}\right)+B^{4}g_{1}(m+2)\xi^{2}(x-tv)\right)$
(17b) $\displaystyle E_{2}$
$\displaystyle=2\left(B^{2}m\left(9A\xi^{2}(2Ag_{1}-g_{2})(x-tv)+D_{1}tv\left(-3k^{2}\xi^{2}+2m+1\right)+D_{1}x\left(3k^{2}\xi^{2}+m-1\right)\right)\right)$
(17c) $\displaystyle E_{3}$ $\displaystyle=2B^{2}\xi
E\left(\left.\text{am}\left(\left.\frac{x-tv}{\xi}\right|m\right)\right|m\right)\left(\xi^{2}\left(-9Am(2Ag_{1}-g_{2})-2B^{2}g_{1}(m+1)\right)+D_{1}m\left(-3k^{2}\xi^{2}+m+1\right)\right)$
(17d) $\displaystyle E_{4}$
$\displaystyle=B^{2}m\xi\text{cn}\left(\left.\frac{x-tv}{\xi}\right|m\right)\text{dn}\left(\left.\frac{x-tv}{\xi}\right|m\right)\left(3B\xi^{2}(4Ag_{1}-g_{2})+2\left(B^{2}g_{1}\xi^{2}+D_{1}m\right)\text{sn}\left(\left.\frac{x-tv}{\xi}\right|m\right)\right)$
(17e) $\displaystyle E_{5}$
$\displaystyle=3B\sqrt{m}\xi^{3}\left(8A^{3}g_{1}m-6A^{2}g_{2}m+4A\left(B^{2}g_{1}(m+1)+D_{1}k^{2}m\right)-B^{2}g_{2}(m+1)\right)$
(17f) $\displaystyle\lambda_{E}$
$\displaystyle=\log\left(\text{dn}\left(\left.\frac{x-tv}{\xi}\right|m\right)-\sqrt{m}\text{cn}\left(\left.\frac{x-tv}{\xi}\right|m\right)\right),$
(17g)

  3. 3.

Momentum density

$\mathbf{P(x)}=\frac{\hbar k}{m}\left(\left(A^{2}m+B^{2}\right)(x-tv)+B^{2}\xi
E\left(\left.\text{am}\left(\left.\frac{tv-x}{\xi}\right|m\right)\right|m\right)\right.\\\
\left.+2AB\sqrt{m}\xi\log\left(\text{dn}\left(\left.\frac{x-tv}{\xi}\right|m\right)-\sqrt{m}\text{cn}\left(\left.\frac{x-tv}{\xi}\right|m\right)\right)\right)=\frac{\hbar
k}{m}\mathbf{N(x)}.$ (18)

For supersolids confined in a box of finite length $L$ (from $-L/2$ to $L/2$):
number, $N=N(L/2)-N(-L/2)$, energy, $E=E(L/2)-E(-L/2)$, and momentum,
$P=P(L/2)-P(-L/2)$.