Exact Solutions for Equations of Bose-Fermi Mixtures in One-Dimensional Optical Lattice
We present two new families of stationary solutions for equations of Bose-Fermi mixtures with an elliptic function potential with modulus k. We also discuss particular cases when the quasiperiodic solutions become periodic ones. In the limit of a sinusoidal potential (k → 0) our solutions model a qu...
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irk-123456789-1473632019-02-15T01:23:00Z Exact Solutions for Equations of Bose-Fermi Mixtures in One-Dimensional Optical Lattice Kostov, N.A. Gerdjikov, V.S. Valchev, T.I. We present two new families of stationary solutions for equations of Bose-Fermi mixtures with an elliptic function potential with modulus k. We also discuss particular cases when the quasiperiodic solutions become periodic ones. In the limit of a sinusoidal potential (k → 0) our solutions model a quasi-one dimensional quantum degenerate Bose-Fermi mixture trapped in optical lattice. In the limit k → 1 the solutions are expressed by hyperbolic function solutions (vector solitons). Thus we are able to obtain in an unified way quasi-periodic and periodic waves, and solitons. The precise conditions for existence of every class of solutions are derived. There are indications that such waves and localized objects may be observed in experiments with cold quantum degenerate gases. 2007 Article Exact Solutions for Equations of Bose-Fermi Mixtures in One-Dimensional Optical Lattice / N.A. Kostov, V.S. Gerdjikov, T.I. Valchev // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 27 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 37K20; 35Q51; 74J30; 78A60 http://dspace.nbuv.gov.ua/handle/123456789/147363 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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We present two new families of stationary solutions for equations of Bose-Fermi mixtures with an elliptic function potential with modulus k. We also discuss particular cases when the quasiperiodic solutions become periodic ones. In the limit of a sinusoidal potential (k → 0) our solutions model a quasi-one dimensional quantum degenerate Bose-Fermi mixture trapped in optical lattice. In the limit k → 1 the solutions are expressed by hyperbolic function solutions (vector solitons). Thus we are able to obtain in an unified way quasi-periodic and periodic waves, and solitons. The precise conditions for existence of every class of solutions are derived. There are indications that such waves and localized objects may be observed in experiments with cold quantum degenerate gases. |
| format |
Article |
| author |
Kostov, N.A. Gerdjikov, V.S. Valchev, T.I. |
| spellingShingle |
Kostov, N.A. Gerdjikov, V.S. Valchev, T.I. Exact Solutions for Equations of Bose-Fermi Mixtures in One-Dimensional Optical Lattice Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Kostov, N.A. Gerdjikov, V.S. Valchev, T.I. |
| author_sort |
Kostov, N.A. |
| title |
Exact Solutions for Equations of Bose-Fermi Mixtures in One-Dimensional Optical Lattice |
| title_short |
Exact Solutions for Equations of Bose-Fermi Mixtures in One-Dimensional Optical Lattice |
| title_full |
Exact Solutions for Equations of Bose-Fermi Mixtures in One-Dimensional Optical Lattice |
| title_fullStr |
Exact Solutions for Equations of Bose-Fermi Mixtures in One-Dimensional Optical Lattice |
| title_full_unstemmed |
Exact Solutions for Equations of Bose-Fermi Mixtures in One-Dimensional Optical Lattice |
| title_sort |
exact solutions for equations of bose-fermi mixtures in one-dimensional optical lattice |
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Інститут математики НАН України |
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2007 |
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http://dspace.nbuv.gov.ua/handle/123456789/147363 |
| citation_txt |
Exact Solutions for Equations of Bose-Fermi Mixtures in One-Dimensional Optical Lattice / N.A. Kostov, V.S. Gerdjikov, T.I. Valchev // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 27 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| work_keys_str_mv |
AT kostovna exactsolutionsforequationsofbosefermimixturesinonedimensionalopticallattice AT gerdjikovvs exactsolutionsforequationsofbosefermimixturesinonedimensionalopticallattice AT valchevti exactsolutionsforequationsofbosefermimixturesinonedimensionalopticallattice |
| first_indexed |
2025-07-11T01:55:04Z |
| last_indexed |
2025-07-11T01:55:04Z |
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1837313752156864512 |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 3 (2007), 071, 14 pages
Exact Solutions for Equations of Bose–Fermi Mixtures
in One-Dimensional Optical Lattice
Nikolay A. KOSTOV †, Vladimir S. GERDJIKOV ‡ and Tihomir I. VALCHEV ‡
† Institute of Electronics, Bulgarian Academy of Sciences,
72 Tsarigradsko chaussee, 1784 Sofia, Bulgaria
E-mail: nakostov@inrne.bas.bg
‡ Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences,
72 Tsarigradsko chaussee, 1784 Sofia, Bulgaria
E-mail: gerjikov@inrne.bas.bg, valtchev@inrne.bas.bg
Received March 30, 2007, in final form May 17, 2007; Published online May 30, 2007
Original article is available at http://www.emis.de/journals/SIGMA/2007/071/
Abstract. We present two new families of stationary solutions for equations of Bose–Fermi
mixtures with an elliptic function potential with modulus k. We also discuss particular
cases when the quasiperiodic solutions become periodic ones. In the limit of a sinusoidal
potential (k → 0) our solutions model a quasi-one dimensional quantum degenerate Bose–
Fermi mixture trapped in optical lattice. In the limit k → 1 the solutions are expressed
by hyperbolic function solutions (vector solitons). Thus we are able to obtain in an unified
way quasi-periodic and periodic waves, and solitons. The precise conditions for existence
of every class of solutions are derived. There are indications that such waves and localized
objects may be observed in experiments with cold quantum degenerate gases.
Key words: Bose–Fermi mixtures; one dimensional optical lattice
2000 Mathematics Subject Classification: 37K20; 35Q51; 74J30; 78A60
1 Introduction
Over the last decade, the field of cold degenerate gases has been one of the most active areas
in physics. The discovery of Bose–Einstein Condensates (BEC) in 1995 (see e.g. [1, 2]) greatly
stimulated research of ultracold dilute Boson-Fermion mixtures. This interest is driven by the
desire to understand strongly interacting and strongly correlated systems, with applications in
solid-state physics, nuclear physics, astrophysics, quantum computing, and nanotechnologies.
An important property of Bose–Fermi mixtures wherein the fermion component is dominant
is that the mixture tends to exhibit essentially three-dimensional character even in a strongly
elongated trap. During the last decade, great progress has been achieved in the experimental
realization of Bose–Fermi mixtures [3, 4], in particular Bose–Fermi mixtures in one-dimensional
lattices. Optical lattices provide a powerful tool to manipulate matter waves, in particular
solitons. The Pauli exclusion principle results in the extension of the fermion cloud in the
transverse direction over distances comparable to the longitudinal dimension of the excitations.
It has been shown recently, however, that the quasi-one-dimensional situation can nevertheless
be realized in a Bose–Fermi mixture due to strong localization of the bosonic component [5, 6].
With account of the effectiveness of the optical lattice in managing systems of cold atoms, their
effect on the dynamics of Bose–Fermi mixtures is of obvious interest. Some of the aspects of
this problem have already been explored within the framework of the mean-field approximation.
In particular, the dynamics of the Bose–Fermi mixtures were explored from the point of view
of designing quantum dots [8]. The localized states of Bose–Fermi mixtures with attractive
(repulsive) Bose–Fermi interactions are viewed as a matter-wave realization of quantum dots
mailto:nakostov@inrne.bas.bg
mailto:gerjikov@inrne.bas.bg
mailto:valtchev@inrne.bas.bg
http://www.emis.de/journals/SIGMA/2007/071/
2 N.A. Kostov, V.S. Gerdjikov and T.I. Valchev
and antidots. The case of Bose–Fermi mixtures in optical lattices is investigated in detail
and the existence of gap solitons is shown. In particular, in [8] it is obtained that the gap
solitons can trap a number of fermionic bound-state levels inside both for repulsive and attractive
boson-boson interactions. The time-dependent dynamical mean-field-hydrodynamic model to
study the formation of fermionic bright solitons in a trapped degenerate Fermi gas mixed with
a Bose–Einstein condensate in a quasi-one-dimensional cigar-shaped geometry is proposed in [9].
Similar model is used to study mixing-demixing in a degenerate fermion-fermion mixture in [10].
Modulational instability, solitons and periodic waves in a model of quantum degenerate boson-
fermion mixtures are obtained in [11].
Our aim is to derive two new classes of quasi-periodic exact solutions of the time dependent
mean field equations of Bose–Fermi mixture in one-dimensional lattice. We also study some
limiting cases of these solutions. The paper is organized as follows. In Section 2 we give the
basic equations. Section 3 is devoted to derivation of the first class quasi-periodic solutions
with non-trivial phases. A system of Nf + 1 equations, which reduce quasi-periodic solutions to
periodic are derived. In Section 4 we present second class (type B) nontrivial phase solutions.
In Section 5 we obtain 14 classes of elliptic solutions. Section 6 is devoted to two special limits,
to hyperbolic and trigonometric functions. In Section 7 preliminary results about the linear
stability of solutions are given. Section 8 summarizes the main conclusions of the paper.
2 Basic equations
At mean field approximation we consider the following Nf + 1 coupled equations [7, 8, 12, 11]
i~
∂Ψb
∂t
+
1
2mB
∂2Ψb
∂x2
− VΨb − gBB|Ψb|2Ψb − gBFρfΨb = 0, (2.1)
i~
∂Ψf
j
∂t
+
1
2mF
∂2Ψf
j
∂x2
− VΨf
j − gBF|Ψb|2Ψf
j = 0, (2.2)
where ρf =
Nf∑
i=1
|Ψf
i |2 and
gBB =
2aBB
as
, gBF =
2aBF
asα
, α =
mB
mF
, as =
√
~
mBω⊥
,
aBB and aBF are the scattering lengths for s-wave collisions for boson-boson and boson-fermion
interactions, respectively. In recent experiments [13, 14] the quantum degenerate mixtures of 40K
and 87Rb are studied where mB = 87mp , mB = 40mp and ω⊥ = 215 Hz. Equations (2.1), (2.2)
have been studied numerically in [7]. The formation of localized structures containing bosons
and fermions has been reported in the particular case in which the interspecies scattering length
aBF is negative, which is the case of the 40K-87Rb mixture. An appropriate class of periodic
potentials to model the quasi-1D confinement produced by a standing light wave is given by [15]
V = V0sn 2(αx, k),
where sn (αx, k) denotes the Jacobian elliptic sine function with elliptic modulus 0 ≤ k ≤ 1.
Experimental realization of two-component Bose–Einstein condensates have stimulated con-
siderable attention in general [16] and in particular in the quasi-1D regime [17, 18] when the
Gross–Pitaevskii equations for two interacting Bose–Einstein condensates reduce to coupled
nonlinear Schrödinger (CNLS) equations with an external potential. In specific cases the two
component CNLS equations can be reduced to the Manakov system [19] with an external po-
tential.
Exact Solutions for Equations of Bose–Fermi Mixtures 3
Important role in analyzing these effects was played by the elliptic and periodic solutions
of the above-mentioned equations. Such solutions for the one-component nonlinear Schrödinger
equation are well known, see [20] and the numerous references therein. Elliptic solutions for the
CNLS and Manakov system were derived in [21, 22, 23].
In the presence of external elliptic potential explicit stationary solutions for NLS were derived
in [15, 24, 25]. These results were generalized to the n-component CNLS in [18]. For 2-component
CNLS explicit stationary solutions are derived in [26].
3 Stationary solutions with non-trivial phases
We restrict our attention to stationary solutions of these CNLS
Ψb(x, t) = q0(x) exp
(
−iω0
~
t+ iΘ0(x) + iκ0
)
, (3.1)
Ψf
j (x, t) = qj(x) exp
(
−iωj
~
t+ iΘj(x) + iκ0,j
)
, (3.2)
where j = 1, . . . , Nf , κ0, κ0,j , are constant phases, qj and Θ0, Θj(x) are real-valued functions
connected by the relation
Θ0(x) = C0
∫ x
0
dx′
q20(x′)
, Θj(x) = Cj
∫ x
0
dx′
q2j (x′)
, (3.3)
C0,Cj , j = 1, . . . , Nf being constants of integration. Substituting the ansatz (3.1), (3.2) in
equations (2.1) and separating the real and imaginary part we get
1
2mB
q30q0xx − gBBq
6
0 − V q40 − gBF
Nf∑
i=1
q2i
q40 + ω0q
4
0 =
1
2mB
C2
0, (3.4)
1
2mF
q3j qjxx − gBFq
2
0q
4
j − V q4j + ωjq
4
j =
1
2mF
C2
j .
We seek solutions for q20 and q2j , j = 1, . . . , Nf as a quadratic function of sn (αx, k):
q20 = A0sn 2(αx, k) +B0, q2j = Ajsn 2(αx, k) +Bj . (3.5)
Inserting (3.5) in (3.4) and equating the coefficients of equal powers of sn (αx, k) results in the
following relations among the solution parameters ωj , Cj , Aj and Bj and the characteristic of
the optical lattice V0, α and k:
A0 =
α2k2 −mFV0
mFgBF
,
Nf∑
j=1
Aj =
α2k2
gBF
(
1
mB
− gBB
mFgBF
)
− V0
gBF
(
1− gBB
gBF
)
, (3.6)
ω0 =
α2(k2 + 1)
2mB
+ gBBB0 + gBF
Nf∑
i=1
Bi +
α2k2
2mB
B0
A0
,
ωj =
α2(k2 + 1)
2mF
+ gBFB0 +
α2k2
2mF
Bj
Aj
, (3.7)
C2
0 =
α2B0
A0
(A0 +B0)(A0 +B0k
2), C2
j =
α2Bj
Aj
(Aj +Bj)(Aj +Bjk
2), (3.8)
where j = 1, . . . , Nf . Next for convenience we introduce
B0 = −β0A0, Bj = −βjAj , j = 1, . . . , Nf ,
4 N.A. Kostov, V.S. Gerdjikov and T.I. Valchev
Table 1. W = gBFmFWB/(mBWF).
1 β0 ≤ 0 βj ≤ 0 A0 ≥ 0 Aj ≥ 0 gBF ≷ 0 gBB ≶ W V0 ≶ α2k2/mF
2 β0 ≤ 0 1 ≤ βj ≤ 1/k2 A0 ≥ 0 Aj ≤ 0 gBF ≷ 0 gBB ≷ W V0 ≶ α2k2/mF
3 1 ≤ β0 ≤ 1/k2 βj ≤ 0 A0 ≤ 0 Aj ≥ 0 gBF ≷ 0 gBB ≷ W V0 ≷ α2k2/mF
4 1 ≤ β0 ≤ 1/k2 1 ≤ βj ≤ 1/k2 A0 ≤ 0 Aj ≤ 0 gBF ≷ 0 gBB ≶ W V0 ≷ α2k2/mF
then
C2
0 = α2A2
0β0(β0 − 1)(1− β0k
2), C2
j = α2A2
jβj(βj − 1)(1− βjk
2), j = 1, . . . , Nf .
In order for our results (3.5) to be consistent with the parametrization (3.1)–(3.3) we must ensure
that both q0(x) and Θ0(x) are real-valued, and also qj(x) and Θj(x) are real-valued; this means
that C2
0 ≥ 0 and q20(x) ≥ 0 and also C2
j ≥ 0 and q2j (x) ≥ 0 (see Table 1, WB = (α2k2 −mBV0),
WF = (α2k2−mFV0)). An elementary analysis shows that with l = 0, . . . , Nf one of the following
conditions must hold
a) Al ≥ 0, βl ≤ 0, b) Al ≤ 0, 1 ≤ βl ≤
1
k2
.
Although our main interest is to analyze periodic solutions, note that the solutions Ψb, Ψf
j
in (2.1), (2.2) are not always periodic in x. Indeed, let us first calculate explicitly Θ0(x) and
Θj(x) by using the well known formula, see e.g. [27]:∫ x
0
du
℘(αu)− ℘(αv)
=
1
℘′(αv)
[
2xζ(αv) +
1
α
ln
σ(αu− αv)
σ(αu+ αv)
]
,
where ℘, ζ, σ are standard Weierstrass functions.
In the case a) we replace v by iv0 and v by ivj , set sn 2(iαv0; k) = β0 < 0, sn 2(iαvj ; k) =
βj < 0 and
e1 =
1
3
(2− k2), e2 =
1
3
(2k2 − 1), e3 = −1
3
(1 + k2),
and rewrite the l.h.s in terms of Jacobi elliptic functions:∫ x
0
du sn 2(iαv; k)sn 2(αu; k)
sn 2(iαv; k)− sn 2(αu; k)
= −β0x− β2
0
∫ x
0
du
sn 2(αu, k)− β0
,
and for j = 1, . . . , Nf we have∫ x
0
du sn 2(iαv; k)sn 2(αu; k)
sn 2(iαv; k)− sn 2(αu; k)
= −βjx− β2
j
∫ x
0
du
sn 2(αu, k)− βj
.
Skipping the details we find the explicit form of
Θ0(x) = C0
∫ x
0
du
A0(sn 2(αu; k)− β0
= −τ0x+
i
2
ln
σ(αx+ iαv0)
σ(αx− iαv0)
,
τ0 = iαζ(iαv0) +
α
β0
√
−β0(1− β0)(1− k2β0).
and for Θj(x), j = 1, . . . , Nf we have
Θj(x) = Cj
∫ x
0
du
Aj(sn 2(αu; k)− βj)
= −τjx+
i
2
ln
σ(αx+ iαvj)
σ(αx− iαvj)
, (3.9)
Exact Solutions for Equations of Bose–Fermi Mixtures 5
τj = iαζ(iαvj) +
α
βj
√
−βj(1− βj)(1− k2βj).
These formulae provide an explicit expression for the solutions Ψb, Ψf
j with nontrivial phases;
note that for real values of v0 Θ0(x), vj Θj(x) are also real. Now we can find the conditions
under which Qj(x, t) are periodic. Indeed, from (3.9) we can calculate the quantities T0, Tj
satisfying:
Θ0(x+ T0)−Θ0(x) = 2πp0, Θj(x+ Tj)−Θj(x) = 2πpj , j = 1, . . . , Nf .
Then Ψb, Ψf
j will be periodic in x with periods T0 = 2m0ω/α, Tj = 2mjω/α if there exist pairs
of integers m0, p0, and mj , pj , such that:
m0
p0
= −π [αv0ζ(ω) + ωτ0/α]−1 ,
mj
pj
= −π [αvjζ(ω) + ωτj/α]−1 , j = 1, . . . , Nf .
where ω (and ω′) are the half-periods of the Weierstrass functions.
4 Type B nontrivial phase solutions
For the first time solutions of this type were derived in [15, 24, 25] for the case of nonlin-
ear Schrödinger equation and in [18] for the n-component CNLSE. For Bose–Fermi mixtures
solutions of this type are possible
• when we have two lattices VB and VF,
• when mB = mF.
We seek the solutions in one of the following forms:
q20 = A0sn (αx, k) +B0, q2j = Ajsn (αx, k) +Bj , (4.1)
q20 = A0cn (αx, k) +B0, q21 = Ajcn (αx, k) +Bj , (4.2)
q20 = A0dn (αx, k) +B0, q21 = Ajdn (αx, k) +Bj , j = 1, . . . , Nf . (4.3)
In the first case (4.1) we have
VB =
3α2k2
8mB
, VF =
3α2k2
8mF
A0 = − α2k2
4mFgBF
Bj
Aj
,
B1
A1
= · · · =
BNf
ANF
,
∑
j
Aj = − α2k2
4mBgBF
B0
A0
− A0gBB
gBF
,
ω0 =
α2(k2 + 1)
8mB
+ gBBB0 + gBFB1 −
α2k2
8mB
B2
0
A2
0
,
ωj =
α2(k2 + 1)
8mF
+ gBFB0 −
α2k2
8mF
B2
j
A2
j
,
C2
0 =
α2
4A2
0
(B2
0 −A2
0)(A
2
0 −B2
0k
2), C2
j =
α2
4A2
j
(B2
j −A2
j )(A
2
j −B2
j k
2).
We remark that due to relations B1
A1
= · · · =
BNf
ANF
we have that all qj of the fermion fields are
proportional to q1.
6 N.A. Kostov, V.S. Gerdjikov and T.I. Valchev
5 Examples of elliptic solutions
Using the general solution equations (3.6)–(3.8) we have the following special cases: (these
solutions are possible only when we have some restrictions on gBB, gBF, and V0 see the Table 1)
Example 1. Suppose that B0 = Bj = 0. Therefore we have
q0(x) =
√
A0sn (αx, k), qj =
√
Ajsn (αx, k), (5.1)
A0 =
α2k2 −mFV0
mFgBF
,
∑
j
Aj =
α2k2
gBF
(
1
mB
− gBB
mF gFB
)
− V0
gBF
(
1− gBB
gBF
)
. (5.2)
For the frequencies ω0 and ωj we have
ω0 =
α2(1 + k2)
2mB
, ωj =
α2(1 + k2)
2mF
.
as well as C0 = Cj = 0.
Example 2. Let B0 = −A0 and Bj = −Aj hold true. Then we have
q0(x) =
√
−A0cn (αx, k), qj(x) =
√
−Ajcn (αx, k). (5.3)
The coefficients A0 and Aj have the same form as (5.2). The frequencies ω0 and ωj now look
as follows
ω0 =
α2(1− 2k2)
2mB
+ V0, ωj =
α2(1− 2k2)
2mF
+ V0.
The constants C0 and Cj are equal to zero again.
Example 3. B0 = −A0/k
2 and Bj = −Aj/k
2. In this case we obtain
q0(x) =
√
−A0
k
dn (αx, k), qj(x) =
√
−Aj
k
dn (αx, k),
ω0 =
α2(k2 − 2)
2mB
+
V0
k2
, ωj =
α2(k2 − 2)
2mF
+
V0
k2
. (5.4)
As before C0 = Cj = 0.
Example 4. B0 = 0 and Bj = −Aj . The result reads
q0(x) =
√
A0sn (αx, k), qj(x) =
√
−Ajcn (αx, k),
ω0 =
α2(1− k2)
2mB
+ V0 +A0gBB, ωj =
α2
2mF
. (5.5)
By analogy with the previous examples the constants A0, Aj , C0 and Cj are given by formu-
lae (5.2) and C0, Cj are all zero.
Example 5. B0 = 0 and Bj = −Aj/k
2. Thus one gets
q0(x) =
√
A0sn (αx, k), qj(x) =
√
−Aj
k
dn (αx, k),
ω0 =
α2(k2 − 1)
2mB
+
V0
k2
+
A0gBB
k2
, ωj =
α2k2
2mF
. (5.6)
Exact Solutions for Equations of Bose–Fermi Mixtures 7
Table 2. Trivial phase solutions in the generic case. We use the quantity W = gBFmFWB/(mBWF).
1 q0 =
√
A0sn (αx, k) gBF ≷ 0 gBB ≶ W V0 ≶ α2k2/mF
qj =
√
Ajsn (αx, k)
2 q0 =
√
−A0cn (αx, k) gBF ≷ 0 gBB ≶ W V0 ≷ α2k2/mF
qj =
√
−Ajcn (αx, k)
3 q0 =
√
−A0dn (αx, k)/k gBF ≷ 0 gBB ≶ W V0 ≷ α2k2/mF
qj =
√
−Ajdn (αx, k)/k
4 q0 =
√
A0sn (αx, k) gBF ≷ 0 gBB ≷ W V0 ≶ α2k2/mF
qj =
√
−Ajcn (αx, k)
5 q0 =
√
A0sn (αx, k) gBF ≷ 0 gBB ≷ W V0 ≶ α2k2/mF
qj =
√
−Ajdn (αx, k)/k
6 q0 =
√
−A0cn (αx, k) gBF ≷ 0 gBB ≷ W V0 ≷ α2k2/mF
qj =
√
Ajsn (αx, k)
7 q0 =
√
−A0cn (αx, k) gBF ≷ 0 gBB ≶ W V0 ≷ α2k2/mF
qj =
√
−Ajdn (αx, k)/k
8 q0 =
√
−A0dn (αx, k)/k gBF ≷ 0 gBB ≷ W V0 ≷ α2k2/mF
qj =
√
Ajsn (αx, k)
9 q0 =
√
−A0dn (αx, k)/k gBF ≷ 0 gBB ≶ W V0 ≷ α2k2/mF
qj =
√
−Ajcn (αx, k)
Example 6. Let B0 = −A0 and Bj = 0. Hence we have
q0(x) =
√
−A0cn (αx, k), qj(x) =
√
Ajsn (αx, k),
ω0 =
α2
2mB
− gBBA0, ωj =
α2(1− k2)
2mF
+ V0.
Example 7. Let B0 = −A0 and Bj = −Aj/k
2. We obtain
q0(x) =
√
−A0cn (αx, k), qj(x) =
√
−Aj
k
dn (αx, k),
ω0 =
V0
k2
− α2
2mB
+
1− k2
k2
A0gBB, ωj = V0 −
α2k2
2mF
.
Example 8. Suppose B0 = −A0/k
2 and Bj = 0. Then
q0(x) =
√
−A0
k
dn (αx, k), qj(x) =
√
Ajsn (αx, k),
ω0 =
α2k2
2mB
− gBBA0
k2
, ωj =
α2(k2 − 1)
2mF
+
V0
k2
.
Example 9. Let B0 = −A0/k
2 and Bj = −Aj . Thus
q0(x) =
√
−A0
k
dn (αx, k), qj(x) =
√
−Ajcn (αx, k),
ω0 = V0 −
α2k2
2mB
+
k2 − 1
k2
gBBA0, ωj =
V0
k2
− α2
2mF
.
All these cases when V0 = 0 and j = 2 are derived for the first time in [11].
8 N.A. Kostov, V.S. Gerdjikov and T.I. Valchev
5.1 Mixed trivial phase solution
Example 10. When B0 = 0, B1 = 0, B2 = −A2, Bj = −Aj/k
2, j = 3, . . . , Nf the solutions
obtain the form
q0 =
√
A0sn (αx, k), q1 =
√
A1sn (αx, k),
q2 =
√
−A2cn (αx, k), qj =
√
−Ajdn (αx, k)/k, j = 3, . . . , Nf .
Using equations (3.6)–(3.8) we have
A0 =
α2k2 − V0mF
mF gBF
,
Nf∑
j=1
Aj = α2k2
(
1
mBgBF
− gBB
mF g2
BF
)
− V0
(
1
gBF
− gBB
g2
BF
)
,
ω0 =
α2(k2 − 1)
2mB
+
gBF
k2
(
A1 + (1− k2)A2
)
+
gBBA0
k2
+
V0
k2
,
ω1 =
α2(1 + k2)
2mF
, ω2 =
1
2mF
α2, ωj =
α2k2
2mF
, j = 3, . . . , NF .
Example 11. Let B0 = B1 = 0 and Bj = −Aj where j = 2, . . . , Nf . Therefore the solutions
read
q0(x) =
√
A0sn (αx, k), q1(x) =
√
A1sn (αx, k), qj(x) =
√
−Ajcn (αx, k).
Then we obtain for frequencies the following results
ω0 =
α2(1− k2)
2mB
+ V0 + gBBA0 + gBFA1, ω1 =
α2(1 + k2)
2mF
, ωj =
α2
2mF
.
Example 12. Suppose B0 = −A0, B1 = 0, B2 = −A2 and Bj = −Aj/k
2 where j = 3, . . . , Nf .
The solutions have the form
q0(x) =
√
−A0cn (αx, k), q1(x) =
√
A1sn (αx, k),
q2(x) =
√
−A2cn (αx, k), qj(x) =
√
−Ajdn (αx, k)/k.
The frequencies are
ω0 =
V0
k2
− α2
2mB
+
1− k2
k2
(gBBA0 + gBFA2) +
gBF
k2
A1, ω1 = V0 +
α2(1− k2)
2mF
,
ω2 = V0 +
α2(1− 2k2)
2mF
, ωj = V0 −
α2k2
2mF
.
Example 13. Let B0 = −A0, B1 = −A1 and Bj = −Aj/k
2 for j = 2, . . . , Nf . Then
q0(x) =
√
−A0cn (αx, k), q1(x) =
√
−A1cn (αx, k), qj(x) =
√
−Ajdn (αx, k)/k,
ω0 =
V0
k2
− α2
2mB
+
1− k2
k2
(gBBA0 + gBFA1) , ω1 = V0 +
α2(1− 2k2)
2mF
,
ωj = V0 −
α2k2
2mF
.
Example 14. Let B0 = −A0/k
2, B1 = −A1 and Bj = −Aj/k
2 for j = 2, . . . , Nf . Hence
q0(x) =
√
−A0dn (αx, k)/k, q1(x) =
√
−A1cn (αx, k), qj(x) =
√
−Ajdn (αx, k)/k,
ω0 =
α2(k2 − 2)
2mB
+
V0
k2
+
1− k2
k2
(gBBA0 + gBFA1),
ω1 =
V0
k2
− α2
2mF
, ωj =
V0
k2
+
α2(k2 − 2)
2mF
.
Certainly these examples do not exhaust all possible combinations of solutions and it is easy
to extend this list.
Exact Solutions for Equations of Bose–Fermi Mixtures 9
6 Vector soliton solutions
6.1 Vector bright-bright soliton solutions
When k → 1, sn (αx, 1) = tanh(αx) and B0 = −A0, Bj = −Aj we obtain that the solutions
read
q0 =
√
−A0
1
cosh(αx)
, qj =
√
−Aj
1
cosh(αx)
,
where A0 ≤ 0 as well as Aj ≤ 0. Using equations (3.6)–(3.8) we have
A0 =
α2 − V0mF
mFgBF
, V = V0 tanh2(αx),
Nf∑
j=1
Aj =
α2
gBF
(
1
mB
− gBB
mFgBF
)
− V0
gBF
(
1− gBB
gBF
)
,
ω0 = V0 −
1
2mB
α2, ωj = V0 −
1
2mF
α2.
As a consequence of the restrictions on A0 and Aj one can get the following unequalities
gBF > 0, V0 ≥
α2
mF
, gBB ≤
(α2 −mBV0)mF
(α2 −mFV0)mB
gBF,
gBF < 0, V0 ≤
α2
mF
, gBB ≥
(α2 −mBV0)mF
(α2 −mFV0)mB
gBF.
Vector bright soliton solution when V0 = 0 is derived for the first time in [11].
6.2 Vector dark-dark soliton solutions
When k → 1 and B0 = Bj = 0 are satisfied the solutions read
q0(x) =
√
A0 tanh(αx), qj(x) =
√
Aj tanh(αx).
The natural restrictions A0 ≥ 0 and Aj ≥ 0 lead to
gBF > 0, gBB ≤
(α2 −mBV0)mF
(α2 −mFV0)mB
gBF, V0 ≤ α2/mF,
gBF < 0, gBB ≥
(α2 −mBV0)mF
(α2 −mFV0)mB
gBF, V0 ≥ α2/mF,
A0 =
α2 −mFV0
mFgBF
,
∑
j
Aj =
α2
gBF
(
1
mB
− gBB
mF gFB
)
− V0
gBF
(
1− gBB
gBF
)
. (6.1)
For the frequencies ω0 and ωj and the constants C0 and Cj we have
ω0 =
α2
mB
, ωj =
α2
mF
, C0 = Cj = 0. (6.2)
10 N.A. Kostov, V.S. Gerdjikov and T.I. Valchev
6.3 Vector bright-dark soliton solutions
When k → 1, B0 = −A0 and Bj = 0, we have
q0(x) =
√
−A0
cosh(αx)
, qj(x) =
√
Aj tanh(αx),
ω0 =
α2
2mB
− gBBA0, ωj = V0, C0 = Cj = 0.
The parameters A0 and Aj are given by (6.1). In this case we have the following restrictions
gBF > 0, gBB ≥
(α2 −mBV0)mF
(α2 −mFV0)mB
gBF, V0 ≥ α2/mF,
gBF < 0, gBB ≤
(α2 −mBV0)mF
(α2 −mFV0)mB
gBF, V0 ≤ α2/mF.
6.4 Vector dark-bright soliton solutions
When k → 1 and provided that B0 = 0 and Bj = −Aj the result is
q0(x) =
√
A0 tanh(αx), qj(x) =
√
−Aj
cosh(αx)
, ω0 = V0 +A0gBB, ωj =
α2
2mF
.
By analogy with the previous examples the constants A0, Aj , C0 and Cj are given by formu-
lae (6.1) and (6.2) respectively. The restrictions now are
gBF > 0, gBB ≥
(α2 −mBV0)mF
(α2 −mFV0)mB
gBF, V0 ≤ α2/mF,
gBF < 0, gBB ≤
(α2 −mBV0)mF
(α2 −mFV0)mB
gBF, V0 ≥ α2/mF.
6.5 Vector dark-dark-bright soliton solutions
Let B0 = B1 = 0 and Bj = −Aj where j = 2, . . . , Nf . Therefore the solutions read
q0(x) =
√
A0 tanh(αx), q1(x) =
√
A1 tanh(αx), qj(x) =
√
−Aj sech(αx).
Then we obtain for frequencies the following results
ω0 = V0 + gBBA0 + gBFA1, ω1 =
α2
mF
, ωj =
α2
2mF
.
These examples are by no means exhaustive.
6.6 Nontrivial phase, trigonometric limit
In this section we consider a trap potential of the form Vtrap = V0 cos(2αx), as a model for an
optical lattice. Our potential V is similar and differs only with additive constant. When k → 0,
sn (αx, 0) = sin(αx)
q20 = A0 sin2(αx) +B0, q2j = Aj sin2(αx) +Bj , (6.3)
V = V0 sin2(αx) =
1
2
(V0 − V0 cos(2αx)), (6.4)
Exact Solutions for Equations of Bose–Fermi Mixtures 11
Table 3. W = gBFmFWB/(mBWF).
1 β0 ≤ 0 βj ≤ 0 A0 ≥ 0 Aj ≥ 0 gBF ≷ 0 gBB ≶ gBF V0 ≶ 0
2 β0 ≤ 0 βj ≥ 1 A0 ≥ 0 Aj ≤ 0 gBF ≷ 0 gBB ≷ gBF V0 ≶ 0
3 β0 ≥ 1 βj ≤ 0 A0 ≤ 0 Aj ≥ 0 gBF ≷ 0 gBB ≷ gBF V0 ≷ 0
4 β0 ≥ 1 βj ≥ 1 A0 ≤ 0 Aj ≤ 0 gBF ≷ 0 gBB ≶ gBF V0 ≷ 0
Using equations (3.6)–(3.8) again we obtain the following result when (see Table 3)
A0 = − V0
gBF
,
Nf∑
j=1
Aj = − V0
gBF
(
1− gBB
gBF
)
,
ω0 =
1
2mB
α2 +B0gBB + gBF
Nf∑
i=1
Bi, ωj =
1
2mF
α2 + gBFB0,
C2
0 = α2B0(A0 +B0), C2
j = α2Bj(Aj +Bj),
where
Θ0(x) = arctan
(√
A0 +B0
B0
tan(αx)
)
, Θj(x) = arctan
(√
Aj +Bj
Bj
tan(αx)
)
.
This solution is the most important from the physical point of view [8].
7 Linear stability, preliminary results
To analyze linear stability of our initial system of equations we seek solutions in the form
ψ0(x, t) = (q0(x) + εφ0(x, t)) exp
(
− iω0
~
t+ iΘ0(x) + iκ0
)
,
ψj(x, t) = (q1(x) + εφj(x, t)) exp
(
− iωj
~
t+ iΘ1(x) + iκ1
)
.
and obtain the following linearized equations
~
Φ0
Φ1
...
ΦNf
,t
=
Λ0 U1 U2 . . . UNf
V1 Λ1 0 . . . 0
V2 0 Λ2 . . . 0
...
...
...
. . .
...
VNf
0 0 . . . ΛNf
Φ0
Φ1
...
ΦNf
, Φ0 =
(
φR
0
φI
0,
)
, Φj =
(
φR
j
φI
j
)
,
where
Λ0 =
(
S0 L0,−
L0,+ S0
)
, Uj =
(
0 0
U0,j 0
)
, Λj =
(
Sj Lj,−
Lj,+ Sj
)
, Vj =
(
0 0
U1,j 0
)
,
S0 = − C0
mBq0
∂x
(
1
q0
)
, L0,− = − 1
2mB
(
∂2
xx −
C2
0
q40
)
+ V + gBBq
2
0 + gBFq
2
1 − ω0,
U0,j = −2gBFq
2
0, L0,+ =
1
2mB
(
∂2
xx −
C2
0
q40
)
− V − 3gBBq
2
0 − gBFq
2
1 + ω0,
Sj = − Cj
mFqj
∂x
(
1
qj
)
, Lj,− = − 1
2mF
(
∂2
xx −
C2
j
q40
)
+ V + gBFq
2
0 − ωj ,
12 N.A. Kostov, V.S. Gerdjikov and T.I. Valchev
U1,j = −2gBFq0qj , Lj,+ =
1
2mF
(
∂2
xx −
C2
j
q40
)
− V − gBFq
2
0 + ωj , j = 1, . . . , Nf .
The analysis of the latter matrix system is a difficult problem and only numerical simulations
are possible. Recently a great progress was achieved for analysis of linear stability of periodic
solutions of type (3.1), (3.2) (see e.g. [15, 24, 25, 18, 26] and references therein). Nevertheless
the stability analysis is known only for solutions of type (5.1)–(5.6) and solutions with nontrivial
phase of type (6.3) and (6.4). Linear analysis of soliton solutions is well developed, but it is out
scope of the present paper.
Finally we discuss three special cases:
Case I. Let B0 = Bj = 0 then for j = 1, . . . , Nf and q0 =
√
A0sn (αx, k), qj =
√
Ajsn (αx, k)
we have the following linearized equations:
~φR
0,t = − 1
2mB
∂2
xxφ
I
0 +
V0 + gBBA0 + gBF
∑
j
Aj
sn 2(αx, k)φI
0 − ω0φ
I
0,
~φI
0,t =
1
2mB
∂2
xxφ
R
0 −
V0 + 3gBBA0 + gBF
∑
j
Aj
sn 2(αx, k)φR
0
+ ω0φ
R
0 − 2gBFA0sn 2(αx, k)
∑
j
φR
j ,
~φR
j,t = − 1
2mF
∂2
xxφ
I
j + (V0 + gBFA0) sn 2(αx, k)φI
j − ωjφ
I
j ,
~φI
j,t =
1
2mF
∂2
xxφ
R
j − (V0 + gBFA0) sn 2(αx, k)φR
j + ωjφ
R − 2gBF
√
A0Ajsn 2(αx, k)φR
0 .
Case II. Let B0 = −A0, Bj = −Aj then for q0 =
√
−A0cn (αx, k), qj =
√
−Ajcn (αx, k) we
obtain the following linearized equations:
~φR
0,t = − 1
2mB
∂2
xxφ
I
0 +
V0 + gBBA0 + gBF
∑
j
Aj
sn 2(αx, k)φI
0
−
gBBA0 + gBF
∑
j
Aj + ω0
φI
0,
~φI
0,t =
1
2mB
∂2
xxφ
R
0 +
3gBBA0 + gBF
∑
j
Aj + ω0
φR
0
−
V0 + 3gBBA0 + gBF
∑
j
Aj
sn 2(αx, k)φR
0 + 2gBFA0
(
1− sn 2(αx, k)
)∑
j
φR
j ,
~φR
j,t = − 1
2mF
∂2
xxφ
I
j + (V0 + gBFA0)sn 2(αx, k)φI
j − (gBFA0 + ωj)φI
j ,
~φI
j,t =
1
2mF
∂2
xxφ
R
j − (V0 + gBFA0)sn 2(αx, k)φR
j + (gBFA0 + ωj)φR
j
− 2gBF
√
A0Aj
(
1− sn 2(αx, k)
)
φR
0 , j = 1, . . . , Nf .
Case III. Let B0 = −A0/k
2, Bj = −Aj/k
2 therefore the solutions are
q0 =
√
−A0dn (αx, k)/k, qj =
√
−Ajdn (αx, k)/k,
Exact Solutions for Equations of Bose–Fermi Mixtures 13
and we obtain the following linearized equations
~φR
0,t = − 1
2mB
∂2
xxφ
I
0 +
V0 + gBBA0 + gBF
∑
j
Aj
sn 2(αx, k)φI
0
−
gBBA0 + gBF
∑
j
Aj + k2ω0
φI
0
k2
,
~φI
0,t =
1
2mB
∂2
xxφ
R
0 +
3gBBA0 + gBF
∑
j
Aj + k2ω0
φR
0
k2
,
−
V0 + 3gBBA0 + gBF
∑
j
Aj
sn 2(αx, k)φR
0 +
2gBFA0(1− k2sn 2(α, k))
k2
∑
j
φR
j ,
~φR
j,t = − 1
2mF
∂2
xxφ
I
j + (V0 + gBFA0) sn 2(αx, k)φI
j −
gBFA0 + k2ωj
k2
φI
j ,
~φI
j,t =
1
2mF
∂2
xxφ
R
j − (V0 + gBFA0) sn 2(αx, k)φR
j +
gBFA0 + k2ωj
k2
φR
j
−
2gBF
√
A0Aj
(
1− k2sn 2(α, k)
)
φR
0
k2
, j = 1, . . . , Nf .
These cases are by no means exhaustive.
8 Conclusions
In conclusion, we have considered the mean field model for boson-fermion mixtures in optical
lattice. Classes of quasi-periodic, periodic, elliptic solutions, and solitons have been analyzed in
detail. These solutions can be used as initial states which can generate localized matter waves
(solitons) through the modulational instability mechanism. This important problem is under
consideration.
Acknowledgements
The present work is supported by the National Science Foundation of Bulgaria, contract No
F-1410.
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http://arxiv.org/abs/cond-mat/0404320
http://arxiv.org/abs/cond-mat/0503097
http://arxiv.org/abs/cond-mat/0509257
http://arxiv.org/abs/cond-mat/0607119
http://arxiv.org/abs/nlin.SI/0512020
http://arxiv.org/abs/cond-mat/0205015
http://arxiv.org/abs/cond-mat/0405419
http://arxiv.org/abs/cond-mat/0007174
http://arxiv.org/abs/cond-mat/0007091
http://arxiv.org/abs/cond-mat/0012354
http://arxiv.org/abs/cond-mat/0007117
http://arxiv.org/abs/cond-mat/0010099
1 Introduction
2 Basic equations
3 Stationary solutions with non-trivial phases
4 Type B nontrivial phase solutions
5 Examples of elliptic solutions
5.1 Mixed trivial phase solution
6 Vector soliton solutions
6.1 Vector bright-bright soliton solutions
6.2 Vector dark-dark soliton solutions
6.3 Vector bright-dark soliton solutions
6.4 Vector dark-bright soliton solutions
6.5 Vector dark-dark-bright soliton solutions
6.6 Nontrivial phase, trigonometric limit
7 Linear stability, preliminary results
8 Conclusions
References
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