# Dynamics of imbalanced quasi-one-dimensional binary Bose-Einstein condensate
in external potentials

K. K. Ismailov1,3, B. B. Baizakov1, F. Kh. Abdullaev1,2 and M. Salerno3 1
Physical-Technical Institute, Uzbek Academy of Sciences, 100084, Tashkent,
Uzbekistan
2 Department of Theoretical Physics, National University of Uzbekistan,
100174, Tashkent, Uzbekistan
3 Dipartimento di Fisica “E.R. Caianiello”, and INFN Gruppo Collegato di
Salerno,
Universitá di Salerno, Via Giovanni Paolo II, 84084 Fisciano, Salerno, Italy

###### Abstract

In the framework of coupled 1D Gross-Pitaevskii equations, we explore the
dynamics of a binary Bose-Einstein condensate where the intra-component
interaction is repulsive, while the inter-component one is attractive. The
existence regimes of stable self-trapped localized states in the form of
symbiotic solitons have been analyzed. Imbalanced mixtures, where the number
of atoms in one component exceeds the number of atoms in the other component,
are considered in parabolic potential and box-like trap. When all the intra-
species and inter-species interactions are repulsive, we numerically find a
new type of symbiotic solitons resembling dark-bright solitons. A variational
approach has been developed which allows us to find the stationary state of
the system and frequency of small amplitude dynamics near the equilibrium. It
is shown that the strength of inter-component coupling can be retrieved from
the frequency of the localized state’s vibrations.

###### pacs:

67.85.Hj, 03.75.Kk, 03.75.Lm

##  I Introduction

Two-component Bose-Einstein condensates (BEC) may show a variety of
interesting phenomena depending on the character and strength of intra-species
and inter-species forces kevrekidis2016 . Among all possible settings the case
of two self-repulsive BECs forming a bound state due to inter-species
attraction has been of special attention. In two- and three-dimensional free
space such a bound state of binary gas is unstable against collapse or
spreading malomed-book , though the stability provided by a two-dimensional
external periodic potential was reported ma2016 . By contrast, in a one-
dimensional free space stable localized state can emerge, which is also known
as a symbiotic soliton adhikari2005 ; perezgarcia2005 ; adhikari2008 . The
peculiarity of the symbiotic soliton is that its individual components cannot
exist separately from each other, therefore its dissociation into individual
solitons cannot occur. This fact lies in the core of their enhanced robustness
and unusual dynamical behavior during interactions with external potentials
javed2022  and mutual collisions. Although the properties of symbiotic
solitons have been reported in many publications, they were mainly focused on
balanced settings with all atoms bound in the symbiotic soliton. The influence
of excess atoms in imbalanced mixtures upon the static and dynamic properties
of symbiotic localized states remains less explored. It should be noted, that
previous work on self-trapped localized states in imbalanced ultra-cold atomic
mixtures mainly was concerned with quantum droplets tengstrand2022 ; flynn2022
, an issue that is beyond the scope of this paper.

Our objective in this work is to show that stable localized states similar to
symbiotic solitons can exist in imbalanced binary self-repulsive BECs when the
number of atoms in one component exceeds the number of atoms in the other
component. We address the situation when inter-component attraction slightly
overcomes intra-component repulsion so that the overall mean-field
nonlinearity is weakly attractive. A similar setting in a three-dimensional
case was employed to produce quantum droplets (see review articles luo2021 ;
guo2021 ; bottcher2021 ). Dynamics of symbiotic localized states will be
considered in the harmonic trap, rectangular potential well and toroidal
confinement for quantum gases.

This paper is organized as follows. In the next Section II, the system of
governing equations is introduced and a variational approach for the binary
BEC is developed. In Sec. III we consider the dynamics of symbiotic localized
states in imbalanced BEC mixtures, confined to a parabolic potential, box-like
and toroidal traps. Sec. IV summarizes the main findings of this work.

##  II The model and variational approach

The model is based on the following system of coupled quasi-one-dimensional
Gross-Pitaevskii equations (GPE)

$\displaystyle
iu_{t}+\frac{1}{2}u_{xx}-V_{1}(x)u+\left(g_{11}|u|^{2}+g_{12}|v|^{2}\right)u$
$\displaystyle=$ $\displaystyle 0,$ (1) $\displaystyle
iv_{t}+\frac{1}{2}v_{xx}-V_{2}(x)v+\left(g_{22}|v|^{2}+g_{12}|u|^{2}\right)v$
$\displaystyle=$ $\displaystyle 0,$ (2)

where $u(x,t),v(x,t)$ are the mean-field wave functions of the binary
condensate, $V_{1,2}$ are external trapping potentials for the components,
$g_{11},g_{22}$ are intra- and $g_{12}$ inter-component coupling coefficients.
The dimensionless quantities, entering these equations are scaled using the
frequency of the radial confinement $\omega_{\bot}$, atomic mass $m$ and
radial harmonic oscillator length $l_{\bot}=\sqrt{\hbar/m\omega_{\bot}}$ as
follows: time $t\rightarrow t\omega_{\bot}$, space $x\rightarrow x/l_{\bot}$,
wave function $\\{u,v\\}\rightarrow\sqrt{2|a_{s}|}\,\\{u,v\\}$. When
considering imbalanced settings with different number of atoms $N_{1,2}$ in
the components, we shall denote $u(x,t)$ and $v(x,t)$ as minority and majority
components, respectively.

In absence of external potentials, $V_{1,2}=0$, equal coefficients,
$g=g_{11}=g_{22}$, and equal number of atoms, $N=N_{1}=N_{2}$, Eqs.(1)-(2)
admit an exact quiescent solution ueda1990

$\displaystyle u(x,t)=v(x,t)=A\,{\rm
sech}\left[A\,\sqrt{g+g_{12}}\,x\right]\exp\left(i\Theta\right),\quad\Theta=(A^{2}/2)(g+g_{12})\,t,$
(3)

whose amplitude is linked to the norm $N=\int|u(x)|^{2}dx$ as
$A=(N/2)\sqrt{g+g_{12}}$. The overall coupling constant is assumed to be
positive $g+g_{12}>0$.

In presence of external potentials, analytic solutions are unavailable and one
has to recourse to approximate methods. For general settings stationary
solutions to Eqs.(1)-(2) are found in adhikari2008  by means of the
variational approach using Gaussian trial functions

$u(x,t)=A_{1}\exp\left[-\frac{x^{2}}{2a_{1}^{2}}+ib_{1}x^{2}+i\phi_{1}\right],\quad
v(x,t)=A_{2}\exp\left[-\frac{x^{2}}{2a_{2}^{2}}+ib_{2}x^{2}+i\phi_{2}\right],$
(4)

where time dependent variational parameters $A(t),a(t),b(t),\phi(t)$ define
the amplitude, width, chirp and phase of the components, respectively.
Standard transformations of the variational method anderson1983 ; malomed2002
with Lagrangian density corresponding to Eqs.(1)-(2)

${\cal
L}=\frac{i}{2}(uu^{\ast}_{t}-u^{\ast}u_{t}+vv^{\ast}_{t}-v^{\ast}v_{t})+\frac{1}{2}|u_{x}|^{2}+\frac{1}{2}|v_{x}|^{2}+V_{1}(x)|u|^{2}+V_{2}(x)|v|^{2}-\frac{g_{11}}{2}|u|^{4}-\frac{g_{22}}{2}|v|^{4}-g_{12}|u|^{2}|v|^{2},$
(5)

yield the effective Lagrangian $L=\int{\cal L}dx$ based on trial functions (4)
and harmonic traps $V_{1,2}=\beta_{1,2}\,x^{2}$

$\displaystyle L$ $\displaystyle=$
$\displaystyle\frac{N_{1}}{4a_{1}^{2}}+\frac{N_{2}}{4a_{2}^{2}}-\frac{g_{11}N_{1}^{2}}{2\sqrt{2\pi}a_{1}}-\frac{g_{22}N_{2}^{2}}{2\sqrt{2\pi}a_{2}}+N_{1}a_{1}^{2}b_{1}^{2}+N_{2}a_{2}^{2}b_{2}^{2}+\frac{1}{2}N_{1}a_{1}^{2}b_{1t}+\frac{1}{2}N_{2}a_{2}^{2}b_{2t}-$
(7)
$\displaystyle\frac{g_{12}N_{1}N_{2}}{\sqrt{\pi}(a_{1}^{2}+a_{2}^{2})^{1/2}}+\frac{1}{2}\beta_{1}N_{1}a_{1}^{2}+\frac{1}{2}\beta_{2}N_{2}a_{2}^{2}+N_{1}\phi_{1t}+N_{2}\phi_{2t}.$

The equations for the widths are derived from Euler-Lagrange equations
$d/dt(\partial L/\partial\dot{a}_{i})-\partial L/\partial a_{i}=0$, $i=1,2$.

$\displaystyle\ddot{a}_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{a_{1}^{3}}-\frac{g_{11}N_{1}}{\sqrt{2\pi}a_{1}^{2}}-\frac{2g_{12}N_{2}a_{1}}{\sqrt{\pi}(a_{1}^{2}+a_{2}^{2})^{3/2}}-2\,\beta_{1}a_{1},$
(8) $\displaystyle\ddot{a}_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{a_{2}^{3}}-\frac{g_{22}N_{2}}{\sqrt{2\pi}a_{2}^{2}}-\frac{2g_{12}N_{1}a_{2}}{\sqrt{\pi}(a_{1}^{2}+a_{2}^{2})^{3/2}}-2\,\beta_{2}a_{2},$
(9)

where the over-dot implies the time derivative,
$N_{1,2}=A_{1,2}^{2}a_{1,2}\sqrt{\pi}$ is the norm of the corresponding
component. The fixed point of this system allows us to find the stationary
waveform and explore its small amplitude dynamics.

Most convenient for analytical treatment is the above mentioned symmetric
setting $a=a_{1}=a_{2}$, $N=N_{1}=N_{2}$, $g=g_{11}=g_{22}$ with similar
external potentials $\beta=\beta_{1}=\beta_{2}$. Then the system Eqs.(8)-(9)
reduces to a single equation

$\ddot{a}=\frac{1}{a^{3}}-(g+g_{12})\frac{N}{\sqrt{2\pi}a^{2}}-2\beta a.$ (10)

The existence of stable bound state of two self-repulsive BECs ($g<0$) held
together solely by inter-component attraction ($g_{12}>0$), even in the
absence of trap ($\beta=0$), is evident from the shape of the potential

$U(a)=\frac{1}{2a^{2}}-(g+g_{12})\frac{N}{\sqrt{2\pi}a}+\beta a^{2},$ (11)

which corresponds to above equation of motion $\ddot{a}=-\partial U/\partial
a$, as illustrated in Fig. 1. In presence of harmonic potential
($\beta\not=0$) the width of the stationary localized state can be found from
solution of the algebraic equation

$a^{4}+p\,a+q=0,\quad p=\frac{(g+g_{12})N}{2\,\sqrt{2\pi}\beta},\quad
q=-\frac{1}{2\beta},$ (12)

whose real and positive roots arekorn-book

$a=-\frac{1}{2}\left(s^{1/2}-\sqrt{2ps^{-1/2}-s}\right),\quad
s=\frac{4q}{3r},\quad
r=\left(\frac{p^{2}}{2}+\frac{1}{2}\sqrt{p^{4}-\frac{256}{27}q^{3}}\right)^{1/3}.$
(13)

The corresponding chemical potential of the symbiotic soliton confined to a
parabolic potential can be found using a static variational approach

$\mu=\beta a^{2}-(g+g_{12})\frac{3N}{4\sqrt{2\pi}a}.$ (14)

The width of the trapped symbiotic soliton and its chemical potential
according to Eqs. (12)-(14) are shown in Fig. 2, which is drawn as follows.
First we calculate the width $a$ from Eq. (13) for given inter-component
coupling parameter $g_{12}$, then use it in Eq. (14). Symbols in these figures
correspond to numerical data, which are found from GPE as

$a_{1}(t)=\left(2\int_{-\infty}^{\infty}x^{2}|u(x,t)|^{2}dx/\int_{-\infty}^{\infty}|u(x,t)|^{2}dx\right)^{1/2},$
(15)

and similarly for $a_{2}(t)$. Numerical data for the chemical potential are
found from oscillation frequency of the stationary solution $Re(u(x,t))$. It
should be noted, that although the symbiotic soliton does’n exist for
$g_{12}\leq 1.01$, the localized state retains its gaussian shape owing to the
harmonic trap confinement of the BEC.

Figure 1: Left panel: A local minimum in the potential $U(a)$ signifies the
presence of a bound state of two self-repulsive BECs, held together only
thanks to the inter-species attraction in free space ($\beta=0$). When the
inter-component attraction is sufficiently weak the local minimum disappears
(black solid line), therefore the bound state doesn’t exist. Middle panel: The
shapes of stable symbiotic localized states in absence (red solid line) and in
presence (blue dashed line) of a parabolic potential for $g_{12}=1.02$. Right
panel: The frequency of internal vibrations of a symbiotic localized state as
a function of the inter-component coupling according to Eq. (16) (solid line)
and GPE numerical results (symbols). Parameter values: $N=N_{1}=N_{2}=100$,
$g=g_{11}=g_{22}=-1$.

Figure 2: Left panel: The localized state shrinks as the inter-component
coupling becomes stronger according to Eq. (12). Middle panel: The chemical
potential crosses zero around $g_{12}\simeq 1.01$ signaling the decay of the
symbiotic soliton at weaker inter-component attraction for $\beta=0.5$. This
corresponds to potential Eq. (11) being positive at its minimum. Right panel:
When the harmonic trap is present ($\beta\not=0$), the minimum in the
potential $U(a)$ always exists. Negative energy at minimum (blue dashed line)
corresponds to a bound state, while positive energy (green dashed line) is
evidence of the bound state decay. Parameter values are similar to previous
figure. Symbols correspond to numerical GPE simulations.

Imbalanced mixtures of binary BECs require an external potential for the
unbound excess atoms of the majority component to remain confined in space. In
fact, the bigger component can act as a trapping potential for the smaller
one. Therefore, trapping only the bigger component ($V_{1}=0,V_{2}\not=0$) is
sufficient for the existence of stable symbiotic localized states in an
external potential, as shown in Fig. 2.

##  III Numerical results

Small amplitude vibrations of a free ($\beta=0$) symbiotic soliton near the
equilibrium $a_{0}=\sqrt{2\pi}/[(g+g_{12})N]$ can be investigated by
representing $a(t)=a_{0}+\delta a(t)$ with $\delta a\ll a_{0}$. Expansion of
the right-hand side of Eq.(10) and keeping the terms up to the first order in
$\delta a$ yields the harmonic oscillator equation $\delta
a_{tt}+\omega^{2}\delta a=0$ with corresponding frequency:

$\omega=(g+g_{12})^{2}N^{2}/(2\pi).$ (16)

The Eq.(16) connects the frequency of the soliton vibrations, $\omega$, with
the coefficient of the inter-component coupling $g_{12}$. This relation is
verified by direct comparison with GPE numerical simulations, as shown in Fig.
1c. To induce vibrations on the soliton profiles a slight change (by one
percent) of the coefficient $g_{12}$ has been employed. At stronger inter-
component attraction ($g_{12}\simeq 1.1$) the oscillations become highly
anharmonic, that is why we show the GPE results up to $g_{12}=1.08$.

We remark that in the literature the properties of symbiotic localized states
are considered mainly for balanced settings, when the two components are self-
trapped in free space or stabilized by a periodic potential adhikari2005 ;
perezgarcia2005 ; adhikari2008 ; javed2022 . The imbalanced case, when the
number of atoms in one component is significantly greater than that in the
other component, remains less explored. In this connection, the motion of a
symbiotic localized state in the surrounding gas of unbound atoms of the
majority component represents particular interest.

The analytic approach for strongly imbalanced binary mixtures is complicated
by the presence of the external potential, which is necessary to prevent the
spreading of the unbound part of the majority component, and by unusual
waveforms of resulting symbiotic states, this making the choice of a suitable
ansatz for the variational method more difficult. Below we produce symbiotic
localized states in trapped binary self-repulsive BECs with attractive inter-
component interaction by numerical methods. In particular, external potentials
in the form of a soft-wall box

$V(x)=\frac{V_{0}}{2}\left[{\rm th}\left(\frac{x-h}{w}\right)-{\rm
th}\left(\frac{x+h}{w}\right)+2\right],$ (17)

where the parameters $V_{0},h,w$ characterize its strength, spatial extent and
width of the transition region, and in the form of a parabolic trap
$V(x)=\beta\,x^{2}$, will be considered. In both cases the external potential
is necessary only for the majority component, to prevent its excess atoms from
spreading. Note that the minority component, being immersed fully in the
majority one, doesn’t require any external potential. All of its atoms are
bound due to inter-species attraction, i.e. the majority component acts as a
confining medium for the minority one. A similar situation was shown to occur
also in the context of a BEC quantum dot salerno2005 .

The waveforms of typical symbiotic localized states with imbalanced atom
numbers in the two components ($N_{2}=4N_{1}$), and their dynamics are shown
Fig. 3. As it can be seen from the bottom panels of this figure, in both cases
the background medium (unbound part of the majority component) remains
unperturbed by a moving soliton with velocity below the Landau critical value
as expected for a superfluid (see below for the case of $v$ above the Landau
critical value). To induce the dynamics of the symbiotic localized state it is
sufficient to set in motion only the minority component by means of phase
imprinting $u\rightarrow u\exp{(ivx)}$.

a) b)

c) d)

Figure 3: Upper panels: Stable symbiotic localized states in binary self-
repulsive BECs with inter-species attraction, trapped in a box-like potential
(a) and harmonic trap (b). Lower panels: Density plots showing the motion of
the symbiotic localized state as a whole through the bath of unbound atomic
gas of the majority component (light blue region). In the box-potential the
symbiotic soliton moves along the straight line with constant velocity until
it reflects from the boundary (c). In the case of parabolic trap, the motion
is oscillatory with a constant period (d). The minority component is set in
motion with velocity $v=0.5$. Parameter values: $g_{11}=g_{22}=-1$,
$g_{12}=1.05$, $N_{1}=100$, $N_{2}=400$, $V_{0}=1000$, $h=4\pi$, $w=0.1$,
$\beta_{1}=0$, $\beta_{2}=0.2$.

Note that in our model, the symbiotic soliton is embedded in the background
gas of unbound atoms of the bigger component. When the trapping potential is
removed, the excess atoms of the imbalanced mixture move away from the center,
subsequently being absorbed by the domain boundaries, and pure symbiotic
soliton emerges, as illustrated in Fig. 4.

Figure 4: Evolution of the imbalanced self-repulsive binary BEC when the
harmonic trap is removed at $t=0$. Excess atoms of the majority component
(blue dashed line) move away from the center and get absorbed at domain
boundaries. A pure symbiotic soliton emerges at $t=30$.

###  III.1 All-repulsive interactions

Now we consider a special case of all-repulsive interactions, when the mixture
of two self-repulsive condensates ($g_{11}<0,g_{22}<0$) with inter-species
repulsion ($g_{12}<0$) is confined to a ring-type quasi-1D potential. In
numerical simulations such a setting is modeled via periodic boundary
conditions $u(0,t)=u(L,t),\ v(0,t)=v(L,t)$, with $L$ being the length of the
integration domain. It should be noted that ground states of binary
condensates with all-repulsive atomic interactions confined to 1D box
potentials was reported in Ref. parajuli2019 . In our setting the density
distortions caused by hard walls of the box potential parajuli2019  have been
avoided and the ring geometry enables continuous circulation of the symbiotic
soliton. In Fig. 5 the ground state wave profiles of a two-component BEC with
all-repulsive interactions is illustrated. As it can be seen, the stronger
cross-repulsion leads to a full depletion of the background component at the
origin, while a weaker cross-repulsion allows the non-zero amplitude of both
components within the overlap region. It should be emphasized that although
these localized modes resemble the well known dark-bright soliton, they cannot
exist in isolation and therefore belong to the class of symbiotic solitons.
Unlike previously reported symbiotic solitons existing due to inter-component
attraction, these localized modes exist only due to inter-component repulsion.
Similar solutions were also discussed in filatrella2014  for the cases of
immiscible binary BEC mixtures and Tonks-Girardeau gases. When a stable
symbiotic soliton in the all repulsive setting has been created, the dynamics
can be induced by setting in motion the minority component with some velocity.

Figure 5: Symbiotic soliton profiles of a binary condensate with all-repulsive
interactions confined to a ring-shaped quasi-1D trap. Numerical simulations of
the GPE (1)-(2) using the Pitaevskii damping procedure choi1998  are performed
with periodic boundary conditions. Starting wave profiles correspond to a flat
background for the majority component with amplitude $u(x,0)=5$ and a Gaussian
function for the minority component $v(x,0)=A\exp(-x^{2}/(2a^{2}))$ with
$A=6.71,a=1.25$. In the course of evolution with a phenomenological damping
term in the GPE a stable symbiotic soliton emerges. To improve numerical
convergence the absorbing boundary conditions has been employed for the
localized component $v(x,t)$ only. Other parameters are fixed as:
$g_{11}=g_{22}=-1$, $g_{12}=-1.05$ (left panel) and $g_{12}=-1.02$ (right
panel).

To illustrate superfluid property of the system we set in motion the minority
component with different velocities and observe emerging density modulations
on top of the majority component. The characteristic example is shown in Fig.
6. When the velocity is smaller than the speed of sound for the background
condensate $v<v_{s}=\sqrt{g_{3D}n/m}$, the excitations have not been produced,
while for the velocity exceeding the speed of sound ($v>v_{s}$) density
perturbations show up. These observations are in line with the Landau theory
of superfluidity pitaevskii-book .

Figure 6: Density plots for the majority component $|u(x,t)|^{2}$ according to
the numerical solution of the GPE (1)-(2). At sub-critical velocity ($v=0.1$)
the background condensate remains uniform (left panel) which is evidence of
its superfluid property. In contrast, at super-critical velocity ($v=1$)
notable density modulations emerge (right panel), signaling the disruption of
the superfluidity. Parameter values correspond to strength of cross-repulsion
($g_{12}=-1.05$) slightly exceeding that of the self-repulsion
($g_{11}=g_{22}=-1$).

###  III.2 Relevance to experimental conditions

The experimental realization of a binary BEC in a toroidal trap beattie2013
opened new possibilities in exploring the superfluidity in multi-component
systems. The condensate was created using 87Rb atoms in two different spin
states $|F=1,m_{F}=1\rangle$ and $|F=1,m_{F}=0\rangle$, each component
containing $N\sim 10^{4}$ atoms. The ring trap with transverse frequency
$\omega_{\bot}=2\pi\times 50$ Hz allowed tight confinement of the condensate
so that the dynamics along the ring can be considered in quasi-1D
approximation. If the average radius of the ring is $r\simeq 12\ \mu$m
chen2019  the length of the circular trap will be ${\cal L}=2\pi r\simeq 75\
\mu$m. The same-species and cross-species $s$-wave scattering lengths for 87Rb
slightly differ from each other egorov2013  and can be estimated in units of
Bohr radius ($a_{B}$) as $a_{11}=a_{22}\simeq-100\times a_{B}$ and
$a_{12}\simeq-98\times a_{B}$, these parameters being tunable by the Feshbach
resonance method chin2010 .

Now using the atomic mass of 87Ru $m=1.4\times 10^{-25}$ kg and the transverse
frequency $\omega_{\bot}=2\pi\times 50$ Hz $=$ 314 sec-1, we can estimate the
unit of length employed in numerical simulations
$l_{\bot}=\sqrt{\hbar/m\omega_{\bot}}\simeq 1.5\ \mu$m. The integration domain
length corresponds to $L=12\pi l_{\bot}\simeq 58\ \mu$m, while the volume
occupied by the condensate is $V=Ll_{\bot}^{2}\simeq 1.4\times 10^{-16}$m-3.
The atomic density is estimated as $n=N/V\simeq 7.3\times 10^{19}$m-3. The
coupling strength for the majority component
$g_{3D}=4\pi\hbar^{2}|a_{11}|/m\simeq 5.2\times 10^{-51}$ kg m5/sec2. Using
the above parameters we can estimate the speed of sound
$v_{s}=\sqrt{g_{3D}n/m}\simeq 1.6\times 10^{-3}$m/sec, which in dimensionless
units corresponds to $v_{s}/(\omega_{\bot}l_{\bot})\simeq 3$. The above
presented estimates show, that the parameter values used in our numerical
simulations qualitatively correspond to experimental conditions.

###  III.3 Details of numerical simulation

Numerical solution of the governing Eqs. (1)-(2) has been performed using the
method of fast Fourier transform agrawal-book  with 1024 Fourier modes
numrecipes  within the integration domain $x\in[-6\pi,6\pi]$ and with a time
step $\delta t=0.0005$. To find the ground state of imbalanced symbiotic
solitons in external potentials we employ the Pitaevskii phenomenological
damping procedure choi1998  and Nijhof’s method nijhof2000 , adapted for two-
component GPE. We employed absorbing boundary method to emulate the condition
of infinite integration domain. The accuracy of the numerical procedures was
controlled through conserved quantities, like the norm, energy, and momentum.

##  IV Conclusions

We have considered static and dynamic properties of symbiotic localized states
in imbalanced binary self-repulsive BECs, confined to quasi-one-dimensional
external potentials. These states can exist due to inter-component attraction.
The slow motion of the symbiotic localized state in the background gas of the
majority component occurs without exciting perturbations, which is evidence of
its superfluid nature. When the confining potential is removed, the imbalanced
self-trapped localized state smoothly transforms into pure symbiotic soliton.
In the course of evolution, the excess atoms of the majority component move
away from the origin and get absorbed at the domain boundaries. For binary
condensates with all-repulsive atomic interactions in a ring-type potential we
also have found symbiotic solitons which can exist due to inter-component
repulsion. The developed variational approach allowed us to reveal the
frequency of soliton’s vibrations as a function of the intra- and inter-
component coupling parameters. The analytic predictions are corroborated by
numerical GPE simulations.

## Acknowledgements

KKI acknowledges financial support from the Ministry of Innovative Development
of the Republic of Uzbekistan for a three months grant under the program
”Short Term Scientific Internship of Young Scientists in Foreign Scientific
Organizations”, contract No. 74 with the University of Salerno. KKI also
acknowledges the Physics Department ”E. R. Caianiello” for the hospitality
received and for the internship opportunity during which this work was
completed.

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