# Controlling dark solitons on the healing length scale

Ling-Zheng Meng  School of Physics, Northwest University, Xi’an, 710127, China
Li-Chen Zhao<EMAIL_ADDRESS>School of Physics, Northwest University,
Xi’an, 710127, China  NSFC-SPTP Peng Huanwu Center for Fundamental Theory,
Xi’an 710127, China  Shaanxi Key Laboratory for Theoretical Physics Frontiers,
Xi’an 710127, China  Th. Busch<EMAIL_ADDRESS>Quantum Systems Unit,
OIST Graduate University, Onna, Okinawa 904-0495, Japan  Yongping Zhang
<EMAIL_ADDRESS>Department of Physics, Shanghai University, Shanghai
200444, China

###### Abstract

While usually the optical diffraction limit is setting a limit for the
lengthscales on which a typical alkali Bose-Einstein condensate can be
controlled, we show that in certain situations control via matter waves can
achieve smaller resolutions. For this we consider a small number of impurity
atoms which are trapped inside the density dip of a dark soliton state and
show that any grey soliton can be obtained by just driving the impurity atoms.
By controlling the driving force on the impurity, one can therefore fully
control the position and velocity of the dark soliton, and also study
controlled collisions between these non-linear objects.

##  I Introduction

Dark solitons in atomic Bose-Einstein condensates (BECs) are highly localised
structures where the phase of the wavefunction changes by $\pi$ over a very
short distance Tsuzuki ; Kivshar1 ; Burger1 ; Denschlag . For this to be
possible, the density needs to vanish across this jump, which leads to the
appearance of a localised density dip on the scale of the healing length. Dark
solitons are part of a larger family of structures, so called grey-solitons,
where the phase jump is smaller than $\pi$ and also happens over a wider
region. Grey solitons therefore do not have a point of vanishing density and
they are also not stationary objects Busch1 ; Pelinovsky1 ; Fedichev ; Huang1
; Parker1 ; Parker2 . In fact, dark and grey solitons are known to be
dynamically unstable and develop into accelerating solitons, which vanish when
the speed of sound is reached.

The dynamics of dark solitons is an interesting topic and over the last
decades many works have studied their dynamics Busch1 ; Konotop2 ;
Frantzeskakis ; Becker  as a function of the external and internal parameters
of the condensate supporting them. Of particular interest have been their
scattering properties, however, controlling the dynamics of dark solitons is a
difficult problem, since the healing length is usually much smaller than
optical length scales. Optical potentials can therefore not be used for local
control Becker ; Carr ; Carretero ; Hans , which makes independently
controlling two solitons at close distance very difficult.

In this work we suggest that control of dark solitons can be achieved by using
matter wave potentials instead of optical ones. These can be of significant
shorter wavelength and through scattering directly effect the condensate
itself. The basic idea relies on considering a small number of impurity atoms
immersed into the condensate, which interact repulsively with the BEC.
Assuming a density-density interaction, these atoms will be trapped inside the
dark solitons, where the condensate density is low, and if their number is
small, the soliton shape will be mostly unaffected. Using species-selective
external potentials, one can now control the impurities, which, through their
interaction with the condensate, will in turn allow one to control the
soliton.

##  II Controllable motion of dark solitons

We consider a quasi-one dimensional BEC that can be described in the mean-
field limit by the Gross-Pitaevskii equation. Such a situation can be achieved
by tightly confining a gas in two directions using a harmonic oscillator of
frequency $\omega_{\perp}$, such that the dynamics in these directions is
frozen out Olshanii . The BEC is coupled to a small number of impurity atoms
and in the limit of weak interactions the set of equations describing the
dynamics of the coupled system is given by SI1 ; SI2 ; SI3

$\displaystyle{\rm i}\frac{\partial\psi_{c}}{\partial t}=$
$\displaystyle-\frac{1}{2}\frac{\partial^{2}\psi_{c}}{\partial
x^{2}}+|\psi_{c}|^{2}\psi_{c}+|\psi_{i}|^{2}\psi_{c}+V_{c}\psi_{c},$ (1)
$\displaystyle{\rm i}\frac{\partial\psi_{i}}{\partial t}=$
$\displaystyle-\frac{1}{2}\frac{\partial^{2}\psi_{i}}{\partial
x^{2}}+|\psi_{c}|^{2}\psi_{i}+V_{i}\psi_{i}-Fx\psi_{i}.$ (2)

Here $\psi_{c}$ and $\psi_{i}$ denote the wave functions of condensate and the
impurity atoms, respectively. To make the above equations dimensionless, we
have assumed that the mass $m$ of the atoms in the BEC and in the impurity is
the same and have scaled all lengths in units of the healing length
$\xi=\hbar/\sqrt{mUn_{B}}$, all energies in units of
$E=\hbar^{2}/m\xi^{2}=Un_{B}$ and time in units of
$T=m\xi^{2}/\hbar=\hbar/Un_{B}$. Here the reduced quasi-one-dimensional
interaction coefficient is given by $U=2\hbar a\omega_{\bot}$, where $a$ is
the s-wave scattering length. The density of the BEC, and therefore the
background density of the dark soliton state, is given by $n_{B}$ and the wave
functions $\psi_{c}$ and $\psi_{i}$ are renormalized to this density. Using
typical experimental parameters for a BEC experiment of ${}^{87}\textrm{Rb}$
atoms we have $a=100a_{B}$ vanKempen ; Widera  with $a_{B}$ being the Bohr
radius, $\omega_{\bot}/2\pi=500{\rm Hz}$, and $n_{B}=10^{9}$ per meter. The
scales of units then become $\xi=0.15\upmu{\rm m}$ and $T=0.03{\rm ms}$. Since
the impurity component only contains a very small number of atoms, we neglect
a potential nonlinear interaction within this component in Eq. (2), however we
assume that a selective force $F$ is applied to the impurity atoms only. This
is described by the last term in the Eq. (2). The external potentials that the
BEC and impurity feel are given by $V_{c}$ and $V_{i}$, respectively.

Figure 1: (a) Schematic of a dark soliton with few impurity atoms trapped in
the density dip. The external force $F$ is applied only to the impurity atoms.
(b) Time evolved center of the soliton density dip with the state at $t=0$
being a dark soliton. (c) Velocity of the accelerated soliton as a function of
time. The red circles, blue squares, and purple asterisks are obtained by
switching off the linear external potential at different times and the black
solid line is given by Eq. (6). (d) Snapshots of the phase of the condensate
during the acceleration process, compared to the initial state. The parameters
are $\varepsilon=0.3$ and $F=-0.01$.

We first study the effect of an external force on the impurity for the case of
$V_{c}=V_{i}=0$. If the initial state of the condensate is given by the exact
dark soliton solution, $\psi_{c}=\tanh(x)$, then the ground state of the
impurity can be approximated as the ground state of the mean-field matter wave
potential [see schematic in Fig. 1(a)],
$\psi_{i}=\varepsilon\operatorname{sech}(x)$. Here the normalisation is chosen
such that $\varepsilon^{2}\ll 1$ to ensure that the number of impurity atoms
is much smaller than the number of atoms in the BEC.

While one would naively expect that the dark soliton will be dragged in the
direction of the force by the accelerating impurity atoms, one can see from
the numerical evolution of the GP equations in Fig. 1(b) that the joint system
moves in the direction opposite to the force. This might seem counterintuitive
at first, but it can be easily understood by the concept of negative mass
which is associated with dark and grey solitons Busch1 ; Konotop2 ; Busch2 ;
Aycock . We use a variational approach to get the equation of motion for the
center of the soliton, from which the negative mass effect can been seen. For
this we introduce trial wavefunctions as Kivshar:95

$\displaystyle\psi_{c}(t)$
$\displaystyle=\textrm{i}\sqrt{1-p^{2}(t)}+p(t)\tanh\left[\frac{x-x_{c}(t)}{w(t)}\right],$
(3) $\displaystyle\psi_{i}(t)$
$\displaystyle=\varepsilon(t)\operatorname{sech}\left[\frac{x-x_{c}(t)}{w(t)}\right]\textrm{e}^{\textrm{i}\\{\phi_{0}(t)+[x-x_{c}(t)]\phi_{1}(t)\\}},$
(4)

where $p(t)$ accounts for the depth of the dark soliton, $w(t)$ for its width,
and $x_{c}(t)$ for its position. The amplitude of the impurity is described by
$\varepsilon(t)$. For the initial static dark soliton the parameters are
$p(0)=1,w(0)=1,\varepsilon(0)=\varepsilon$ and $\dot{x}_{c}(0)=0$ and the
evolution is guided by the Lagrangian of the joint system

$\displaystyle\mathcal{L}=$
$\displaystyle\int_{-\infty}^{+\infty}\bigg{[}\frac{\textrm{i}}{2}(\psi_{c}^{*}\partial_{t}\psi_{c}-\psi_{c}\partial_{t}\psi_{c}^{*})(1-\frac{1}{|\psi_{c}|^{2}})-\frac{1}{2}|\partial_{x}\psi_{c}|^{2}$
$\displaystyle+\frac{\textrm{i}}{2}(\psi_{i}^{*}\partial_{t}\psi_{i}-\psi_{i}\partial_{t}\psi_{i}^{*})-\frac{1}{2}|\partial_{x}\psi_{i}|^{2}+Fx|\psi_{i}|^{2}$
$\displaystyle-\frac{1}{2}(|\psi_{c}|^{2}-1)^{2}-|\psi_{i}|^{2}(|\psi_{c}|^{2}-1)\bigg{]}\textrm{d}x.$
(5)

Substituting the trial wavefunctions into the Lagrangian and applying the
Euler-Lagrangian equations
$\frac{\textrm{d}}{\textrm{d}t}(\frac{\partial\mathcal{L}(t)}{\partial\dot{\alpha}})=\frac{\partial\mathcal{L}(t)}{\partial\alpha}$
with $\alpha=\\{w(t),p(t),x_{c}(t)\\}$ and $\dot{\alpha}$ denoting
$\frac{\textrm{d}\alpha}{\textrm{d}t}$, we derive the coupled equations for
the motions of $\alpha$. Since for the system we consider the impurity to be
the minority component ($\varepsilon^{2}(t)\ll 1$), almost all terms that
contain $\varepsilon^{2}(t)$ can be neglected. The exception is the term
describing the potential energy, which can be seen to increase with $x_{c}(t)$
and which can therefore become comparable to the energy of the dark soliton.
In fact, under the local density approximation, the potential energy can be
evaluated as
$E_{p}=\int_{-\infty}^{+\infty}-Fx|\psi_{i}|^{2}\textrm{d}x\approx-
Fx_{c}\int_{-\infty}^{+\infty}|\psi_{i}|^{2}\textrm{d}x=-FN_{i}x_{c}(t)$,
where $N_{i}=2\varepsilon^{2}(t)w(t)=2\varepsilon^{2}$ is the particle number
of the impurity component. The effective force applied to the soliton is
therefore $F_{\textrm{eff}}=-\nabla E_{p}=FN_{i}$. At last, the equation of
motion for the center can be simplified as

$\ddot{x}_{c}=-\frac{F_{\textrm{eff}}}{4\sqrt{1-\dot{x}_{c}^{2}}}.$ (6)

This equation shows that the center of mass motion behaves as a quasi-particle
with the negative mass $m_{\mathrm{eff}}=-4\sqrt{1-\dot{x}_{c}^{2}}$ and it
clearly shows that the soliton moves against the force applied to it. In Fig.
1(c) we show the velocity of the dark soliton calculated from the numerical
evolution of the GP equations and from the quasi-particle equation in Eq. (6).
The two results match very well, which also indicates that the trial
wavefunctions is a good description of the system. We also show the phase of
the BEC component around the center of the dark soliton dip in Fig. 1(d),
where one can see that for the different acceleration times the resulting
condensate states have the appropriate phase changes expected for grey
solitons. The results shown in Figs. 1(c) and 1(d) therefore indicate that
during the evolution the background of the soliton is not significantly
perturbed, which suggests that a dark soliton can be accelerated by applying a
force to an impurity. In that sense, the state of large condensate system is
controlled precisely by driving the small matter wave.

This can also be seen by deriving the equation of motion for the center of the
soliton directly from the dispersion relation of the exact scalar dark soliton
solution, while ignoring the kinetic energy of impurity atoms and the
nonlinear coupling between the impurity and condensate. In this case the total
energy of the system is given by the excitation energy of the dark soliton
Kivshar:95 , i.e., $E_{s}=\frac{4}{3}(1-v^{2})^{3/2}$. With the potential
energy given by $E_{p}\approx-FN_{i}x_{c}$, the trajectory of the dark soliton
can be derived based on the conservation of energy
$E_{s}+E_{p}=E_{s}\big{|}_{v=v_{0}}$, where we assume that the initial center
position of the soliton is at $x_{c}|_{t=0}=0$ Zhao ; Meng . This approach
gives the same equation of motion as the variational one in Eq. (6), which
furthermore indicates that the dark soliton background is kept nearly perfect
during the acceleration process. It should therefore be possible to control
the soliton velocity precisely and without exciting dispersive waves or
generate excitations on the background Becker ; Carr ; Carretero ; Radouani2 ;
Burger2 ; Engels .

Figure 2: The evolution of the (a) impurity component and (b) condensate
density. A force $F=-0.01\sin(\pi t/200)$ is applied to the impurity, which
changes sign at $t=200$. Here $\varepsilon=0.3$.

In addition to accelerating the soliton states to different velocities through
the impurities, it also possible to slow them down and, in fact, return them
to their initial dark state. We show an example of this in Fig. 2, where the
impurity component coupled to an initially stationary dark soliton experiences
a slowly varying force of the form $F=-0.01\sin(\pi t/200)$. One can see that
the dark soliton is accelerated until $t=200$ and then begins to decelerate
and return to be the initially static state at $t=400$. In order to quantify
how close the final state is to a translated dark soliton, we define a
fidelity

$\mathcal{I}=\frac{\left|\int^{L}_{-L}\psi_{v}^{*}\psi_{D}\textrm{dx}\right|^{2}}{\left|\int^{L}_{-L}\psi_{v}^{*}\psi_{v}\textrm{dx}\right|^{2}},$
(7)

where $\psi_{v}$ is the target state given by the exact scalar dark soliton
solution, $\psi_{D}$ is the wavefunction at the end of the acceleration and
de-acceleration process, and $2L$ is a finite area around the soliton center
within which the density has well returned to the bulk value. The fidelity
$\mathcal{I}$ takes value from $0$ to $1$, and a larger $\mathcal{I}$
indicates that the created state is more consistent with the exact solution.
For the example shown in Fig. 2 the fidelity at $t=400$ within a region given
by $L=16$ is given by $\mathcal{I}=0.9975$. This shows that not only the
region around the density dip is close to the exact dark soliton wavefunction,
but also that no significant sound waves were emitted during the whole
dynamical process Parker1 ; Parker2 ; Parker3 ; Radouani1 ; Pelinovsky2 . This
fact is of particular importance for the development of techniques that allow
for highly accurate control over localised structures in condensates Carr ;
Carretero ; Hans ; Radouani2 ; Radouani1 ; Konotop1 ; Hakim ; CRaman ; Bongs ;
Pavloff ; Theocharis1 ; Theocharis2 ; Lyu ; Fritsch .

Figure 3: Fidelities of grey solitons generated by applying different
strengths acceleration potentials $-Fx$ to the impurity component. The blank
area represent parameter regimes in which the force is too strong and the
impurity atoms are pulled out of the traps formed by the condensate density
dips. Here $\varepsilon=0.3$.

Being able to control the motion of dark solitons also can be used to create
grey solitons with high precision. In Fig. 1(c) we show results for two cases
of accelerating dark solitons and suddenly switching off the force acting on
the impurity at a finite time. One can see that it is not only possible to
create grey solitons of specific velocities this way, but also that these
structures move with essentially constant velocities for longer times. In Fig.
3, we show the fidelity of grey solitons generated in this way as a function
of the applied force and resulting velocity of grey solitons. While a too
strong acceleration of the impurity results in it leaving the dark soliton
density dip (white areas in Fig. 3), for all parameters where the trapping
inside the dark soliton remains one can see that the fidelities are
$\mathcal{I}\geq 0.95$. Furthermore, fewer impurity atoms and weaker external
force can be used to broaden the speed regime of grey solitons, but much
longer acceleration (and numerical evolution) times will be needed.

Figure 4: Densities of the (a) condensate dark soliton and (b) impurity
component in the presence of a weak harmonic trap, $\eta=0.005$, under a force
on the impurity. (c) Position of the minimum of the dark soliton dip from the
numerical evolution (circles) compared to the quasi-particle motion given by
Eq. (6) (full line).

##  III The effect of the external Harmonic traps

In real BEC experiments external traps are inevitably involved to confine the
condensate atoms. While box potentials can be realised experimentally, we
study in the following the effect of the much more common harmonic traps on
the controllable motions of dark solitons. For simplicity we assume that the
harmonic trapping strengths seen by the BEC and the impurity components are
the same, $V_{c}=V_{i}=\frac{1}{2}\eta^{2}x^{2}$, where we have defined the
dimensionless quantity $\eta=m\omega_{x}\xi^{2}/\hbar$ with $\omega_{x}$ being
the trapping frequency in the axial direction. We first study a weak harmonic
trap, $\eta=0.005$, which corresponds to $\omega_{x}/2\pi=25{\rm Hz}$, and
prepare the initial dark soliton within a background of Thomas-Fermi shape.
The initial state of the impurity can be chosen the same as the case without a
trap, as it only sees the mean-field potential of the condensate dark soliton
over its healing length extension. The evolution of this initial state when a
force $F=0.01$ is applied to the impurity component is shown in Fig. 4. As
before, the force accelerates the dark soliton and the impurity to move
together in the direction opposite to the force. During the acceleration, the
background stays essentially unperturbed and no phononic excitations in the
BEC component are visible. Since the density of the background condensate
changes slowly on the healing lengths scale, the dynamics of the center of the
dark soliton still matches very well with the expected trajectory in the free
case given by Eq. (6) and for short acceleration times the two trajectories
coincide [see panel (c)]. However, it can also be seen that the change in the
background density leads to a slowing down of the acceleration over time, as
one would expect from a freely oscillating dark-bright soliton as well Busch2
.

It is interesting to wonder if the slowdown of the acceleration due to the
inhomogeneous background density in the harmonic trap can completely stop the
joint system. To explore this we study a system that is more strongly trapped,
$\omega_{x}/2\pi=250{\rm Hz}$ and $\eta=0.05$, in which the inhomogeneity
happens over shorter length scales. For the same, weak strengths of the
accelerating force, $F=-0.01$, the resulting dynamics is shown in Fig. 5 and
one can see that the trap indeed stops the acceleration. However, no
stationary state is obtained, but instead the joint system starts to oscillate
around the traps center. This can be understood by realising that the number
of impurity atoms is small and the strength provided by nonlinear coupling
between them and the condensate is limited. When the force exerted by harmonic
potential becomes stronger than the weak constant force on the impurity, the
well known dynamics of a dark soliton oscillating in a harmonic trap becomes
dominant Busch1 . Therefore, the predictions of Eq. (6) are only valid when
the harmonic trap is weak, whereas when the condensate is strongly trapped,
the oscillatory dynamics is dominant and the constant force only causes a
small position shift of the oscillation center.

Figure 5: Same as these in Fig. 4, but in the presence of a stronger harmonic
trap with $\eta=0.05$.  Figure 6: Densities of the two components for (a) an
elastic and (b) an inelastic collision between two dark solitons. The soliton
initially on the left hand side is accelerated by means of driving the
impurity, and the external force is switched off at (a) $t=200$ for the
elastic case and (b) $t=50$ for the inelastic case. The soliton initially on
the right is impurity free. The parameters are $\varepsilon=0.3$ and
$F=-0.01$.

##  IV Applications to dark solitons collision

Interactions between dark solitons in BECs are an extensive research area
which has produced many beautiful theoretical and experimental results Huang ;
Burger2 ; Weller ; Chang ; Stellmer ; Theocharis3 . Yet, controlling such
collisions to a high degree is quite difficult, with one challenge being the
precise acceleration of the dark solitons without dynamically affecting the
entire BEC system. Two ways previously experimentally used for creating grey
solitons to study interactions between them are matter-wave interference
Weller ; Chang ; Theocharis3  and phase imprinting and density engineering
techniques Burger2 ; Stellmer . However, grey solitons generated by these
method are often accompanied by a significant number of sounds waves, which
makes precise control of their velocity difficult. Creating a stationary dark
solitons can therefore be thought of as a much more well defined problem, with
the acceleration done in a next step through the method discussed here. In
particular, the sound generated when preparing the dark soliton could be
allowed to decay before the soliton gets accelerated.

We therefore investigate a scheme to realize a collision between two dark
solitons by initially preparing two static dark solitons away from each other
and adding a small number of impurity atoms to one of them. Applying a force
to the impurity component for a certain amount of time will accelerate the
dark soliton to the required speed, and have it then collide with the second
soliton subsequently. Two examples of such a protocol are shown in Fig. 6,
where the soliton on the left is accelerated and collides with the one on the
right. If the collisional speed is large (panels (a1) and (a2)) one can see
that the impurity component maintains its momentum and the collision can be
thought of as elastic. This is also indicative of an integrable model. For
collisions at lower velocities (panels (b1) and (b2)) the impurity component
splits between the two dark solitons, and the interaction can be thought of as
inelastic. In general the switch-over between elastic and inelastic collisions
depends on the number of impurity atoms present. As before, in either
scattering regime the BEC background is not affected and this method therefore
allows one to carefully study very clean soliton-soliton collisions Zakharov .

##  V Conclusion

We have shown that the dark soliton state of a large condensate system can be
controlled well by a small matter wave potential that is selectively
accelerated or decelerated. Since the impurity is trapped inside the dark
soliton density dip, it allows to control the condensate through a force
applied over the healing length scale, which can be much shorter than a
typical optical wavelength.

In particular, this allows to turn a stationary dark soliton state into a
moving grey soliton one with high precision and without exciting any
significant phonon modes within the background condensate density. This
process can also be reversed or used to study collisions between solitons in a
highly controlled way. While the creation of stationary dark solitons is still
an experimentally difficult process, decoupling it from the acceleration
process for creating grey solitons is a significant simplification for
studying soliton dynamics in a broad range of settings.

Our work above lays out the foundations of this new idea and demonstrates some
of the advantages, for example the possibility to accelerate excitation free.
Further studies to explore the limits and also higher dimensional settings are
the important next steps.

## Acknowledges

L.-C. Z. was supported by National Natural Science Foundation of China with
Grants No. 12022513 and 12235007\. Y.Z. was supported by National Natural
Science Foundation of China with Grants No.11974235 and 11774219\. This work
was supported by the Okinawa Institute of Science and Technology Graduate
University. Y.Z.and T.B. were also supported by the Japan-China Scientific
Cooperation Program between JSPS and NSFC under Nos. 120227414 and
12211540002.

## References

  * (1) T. Tsuzuki,  Nonlinear waves in the Pitaevskii-Gross equation,  J. Low Temp. Phys. 4, 441 (1971).
  * (2) Y. S. Kivshar, and B. Luther-Davies,  Dark optical solitons: physics and applications,  Phys. Rep. 298, 81 (1998).
  * (3) S. Burger, K. Bongs, S. Dettmer, W. Ertmer, and K. Sengstock,  Dark Solitons in Bose-Einstein Condensates,  Phys. Rev. Lett. 83, 5198 (1999).
  * (4) J. Denschlag, J. E. Simsarian, D. L. Feder, C. W. Clark, L. A. Collins, L. Cubizolles, J. ans Deng, E. W. Hagley, K. Helmerson, W. P. Reinhardt, S. L. Rolston, B. I. Schneider, and W. D. Phillips,  Generating solitons by engineering of a Bose-Einstein Condensate,  Science 287, 97 (2000).
  * (5) Th. Busch and J. R. Anglin,  Motion of Dark Solitons in Trapped Bose-Einstein Condensates,  Phys. Rev. Lett. 84, 2298 (2000).
  * (6) D. E. Pelinovsky, Y. S. Kivshar, and V. V. Afanasjev,  Instability-induced dynamics of dark solitons,  Phys. Rev. E 54, 2015 (1996).
  * (7) P. O. Fedichev, A. E. Muryshev, and G. V. Shlyapnikov,  Dissipative dynamics of a kink state in a Bose-condensed gas,  Phys. Rev. A 60, 3320 (1999).
  * (8) G. Huang, J. Szeftel, and S. Zhu,  Dynamics of dark solitons in quasi-one-dimensional Bose-Einstein condensates,  Phys. Rev. A 65, 053605 (2002).
  * (9) N. G. Parker, N. P. Proukakis, M. Leadbeater, and C. S. Adams,  Soliton-Sound Interactions in Quasi-One-Dimensional Bose-Einstein Condensates,  Phys. Rev. Lett. 90, 220401 (2003).
  * (10) N. G. Parker, N. P. Proukakis, C. F. Barenghi, and C. S. Adams,  Dynamical instability of a dark soliton in a quasi-one-dimensional Bose-Einstein condensate perturbed by an optical lattice,  J. Phys. B: At., Mol. Opt. Phys. 37, S175 (2004).
  * (11) V. V. Konotop and L. P. Pitaevskii,  Landau Dynamics of a Grey Soliton in a Trapped Condensate,  Phys. Rev. Lett. 93, 240403 (2004).
  * (12) D. J. Frantzeskakis,  Dark solitons in atomic Bose-Einstein condensates: from theory to experiments,  J. Phys. A: Math. Theor. 43, 213001 (2010).
  * (13) C. Becker, S. Stellmer, P. Soltan-Panahi, S. Dörscher, M. Baumert, E.-M. Richter, J. Kronjäger, K. Bongs, and K. Sengstock,  Oscillations and interactions of dark and dark-bright solitons in Bose-Einstein condensates,  Nat. Phys. 4, 496 (2008).
  * (14) L. D. Carr, J. Brand, S. Burger, and A. Sanpera,  Dark-soliton creation in Bose-Einstein Condensates,  Phys. Rev. A 63, 051601(R) (2001).
  * (15) R. Carretero-Gonz$\acute{a}$lez, P. G. Kevrekidis, D. J. Frantzeskakis, B. A. Malomed, S. Nandi, and A. R. Bishop,  Soliton trains and vortex streets as a form of Cerenkov radiation in trapped Bose-Einstein condensates,  Math. Comput. Simul. A 74, 361 (2007).
  * (16) I. Hans, J. Stockhofe, and P. Schmelcher,  Generating, dragging, and releasing dark solitons in elongated Bose-Einstein condensates,  Phys. Rev. A 92, 013627 (2015).
  * (17) M. Olshanii,  Atomic scattering in the presence of an external confinement and a gas of impenetrable bosons,  Phys. Rev. Lett. 81, 938 (1998).
  * (18) M. I. Shaukat, E. V. Castro, and H. Tercas,  Entanglement sudden death and revival in quantum dark-soliton qubits,  Phys. Rev. A 98, 022319 (2018).
  * (19) M. J. Edmonds, J. L. Helm, and Th. Busch,  Coherent impurity transport in an attractive binary Bose-Einstein condensate,  New J. Phys. 21, 053019 (2019).
  * (20) M. I. Shaukat, E. V. Castro, and H. Tercas,  Slow sound in matter-wave dark soliton gases,  Phys. Rev. B 99, 205408 (2019).
  * (21) E. G. M. van Kempen, S. J. J. M. F. Kokkelmans, D. J. Heinzen, and B. J. Verhaar,  Interisotope Determination of Ultracold Rubidium Interactions from Three High-Precision Experiments,  Phys. Rev. Lett. 88, 093201 (2002).
  * (22) A. Widera, O. Mandel, M. Greiner, S. Kreim, T. W. H$\ddot{a}$nsch, and I. Bloch,  Entanglement Interferometry for Precision Measurement of Atomic Scattering Properties,  Phys. Rev. Lett. 92, 160406 (2004).
  * (23) Th. Busch and J. R. Anglin,  Dark-Bright Solitons in Inhomogeneous Bose-Einstein Condensates,  Phys. Rev. Lett. 87, 010401 (2001).
  * (24) L. M. Aycock, H. M. Hurst, D. K. Efimkin, D. Genkina, H.-I Lu, V. M. Galitski, and I. B. Spielman,  Brownian motion of solitons in a Bose-Einstein condensate,  Proc. Natl. Acad. Sci. 114, 2503 (2017).
  * (25) Y. S. Kivshar, and W. Królikowski,  Lagrangian approach for dark solitons,  Opt. Commun. 114, 353 (1995).
  * (26) L.-C. Zhao, W. Wang, Q. Tang, Z.-Y. Yang, W.-L. Yang, and J. Liu,  Spin soliton with a negative-positive mass transition,  Phys. Rev. A 101, 043621 (2020).
  * (27) L.-Z. Meng, S.-W. Guan, and L.-C. Zhao,  Negative mass effects of a spin soliton in Bose-Einstein condensates,  Phys. Rev. A 105, 013303 (2022).
  * (28) A. Radouani,  Soliton and phonon production by an oscillating obstacle in a quasi-one-dimensional trapped repulsive Bose-Einstein condensate,  Phys. Rev. A 70, 013602 (2004).
  * (29) S. Burger, L. D. Carr, P. Öhberg, K. Sengstock, and A. Sanpera,  Generation and interaction of solitons in Bose-Einstein condensates,  Phys. Rev. A 65, 043611 (2002).
  * (30) P. Engels, and C. Atherton,  Stationary and Nonstationary Fluid Flow of a Bose-Einstein Condensate Through a Penetrable Barrier,  Phys. Rev. Lett. 99, 160405 (2007).
  * (31) N. G. Parker, N. P. Proukakis, M. Leadbeater, and C. S. Adams,  Deformation of dark solitons in inhomogeneous Bose-Einstein condensates,  J. Phys. B: At., Mol. Opt. Phys. 36, 2891 (2003).
  * (32) D. E. Pelinovsky, D. J. Frantzeskakis, and P. G. Kevrekidis,  Oscillations of dark solitons in trapped Bose-Einstein condensates,  Phys. Rev. E 72, 016615 (2005).
  * (33) A. Radouani,  Gray solitons and phonons in repulsive Bose-Einstein condensates confined in one-dimensional nonharmonic traps,  Phys. Rev. A 68, 043620 (2003).
  * (34) V. V. Konotop, V. M. Pérez-García, Y.-F. Tang, and L. Vázquez,  Interaction of a dark soliton with a localized impurity,  Phys. Lett. A 236, 314 (1997).
  * (35) V. Hakim,  Nonlinear Schrödinger flow past an obstacle in one dimension,  Phys. Rev. E 55, 2835 (1997).
  * (36) C. Raman, M. Köhl, R. Onofrio, D. S. Durfee, C. E. Kuklewicz, Z. Hadzibabic, and W. Ketterle,  Evidence for a Critical Velocity in a Bose-Einstein Condensed Gas,  Phys. Rev. Lett. 83, 2502 (1999).
  * (37) K. Bongs, S. Burger, S. Dettmer, D. Hellweg, J. Arlt, W. Ertmer, and K. Sengstock,  Coherent manipulation and guiding of Bose-Einstein condensates by optical dipole potentials,  C. R. Acad. Sci. Paris 2, 671 (2001).
  * (38) N. Pavloff,  Breakdown of superfluidity of an atom laser past an obstacle,  Phys. Rev. A 66, 013610 (2002).
  * (39) G. Theocharis, D. J. Frantzeskakis, P. G. Kevrekidis, R. Carretero-Gonz$\acute{a}$lez, and B. A. Malomed,  Dark soliton dynamics in spatially inhomogeneous media: Application to Bose-Einstein condensates,  Math. Comput. Simul. 69, 537 (2005).
  * (40) G. Theocharis, D. J. Frantzeskakis, R. Carretero-Gonz$\acute{a}$lez, P. G. Kevrekidis, and B. A. Malomed,  Controlling the motion of dark solitons by means of periodic potentials: Application to Bose-Einstein condensates in optical lattices,  Phys. Rev. E 71, 017602 (2005).
  * (41) H. Lyu, Y. Zhang, and T. Busch, Detection of roton and photon excitations in a spin-orbit-coupled Bose-Einstein condensate with a moving barrier,  Phys. Rev. A 106,013302 (2022).
  * (42) A. R. Fritsch, M. Lu, G. H. Reid, A. M. Piñeiro, and I. B. Spielman,  Creating solitons with controllable and near-zero velocity in Bose-Einstein condensates,  Phys. Rev. A 101, 053629 (2020).
  * (43) G. Huang, M. G. Velarde, and V. A. Makarov,  Dark solitons and their head-on collisions in Bose-Einstein condensates,  Phys. Rev. A 64, 013617 (2001).
  * (44) S. Stellmer, C. Becker, P. Soltan-Panahi, E.-M. Richter, S. Dörscher, M. Baumert, J. Kronjäger, K. Bongs, and K. Sengstock,  Collisions of Dark Solitons in Elongated Bose-Einstein Condensates,  Phys. Rev. Lett. 101, 120406 (2008).
  * (45) A. Weller, J. P. Ronzheimer, C. Gross, J. Esteve, and M. K. Oberthaler,  Experimental Observation of Oscillating and Interacting Matter Wave Dark Solitons,  Phys. Rev. Lett. 101, 130401 (2008).
  * (46) J. J. Chang and P. Engels,  Formation of Dispersive Shock Waves by Merging and Splitting Bose-Einstein Condensates,  Phys. Rev. Lett. 101, 170404 (2008).
  * (47) G. Theocharis, A. Weller, J. P. Ronzheimer, C. Gross, M. K. Oberthaler, P. G. Kevrekidis, and D. J. Frantzeskakis,  Multiple atomic dark solitons in cigar-shaped Bose-Einstein condensates,  Phys. Rev. A 81, 063604 (2010).
  * (48) V. E. Zakharov, and A. B. Shabat,  Interaction between solitons in a stable medium,  Sov. Phys. JETP 37, 823 (1973).