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Matter wave solitons in coupled system with external potentials
Physics Letters A 375 (2011) 3017–3020
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Matter wave solitons in coupled system with external potentials Li-Chen Zhao a,b,∗ , Shao-Ling He c a b c
Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China Science and Technology Computation Physics Laboratory, Institute of Applied Physics and Computational Mathematics, Beijing 100088, China Department of Mathematics, China University of Mining and Technology, Beijing 100083, China
a r t i c l e
i n f o
Article history: Received 9 April 2011 Received in revised form 3 June 2011 Accepted 15 June 2011 Available online 21 June 2011 Communicated by A.R. Bishop Keywords: Coupled GP-equation Lax-pair Bright soliton
a b s t r a c t We present Lax-pair corresponding to the coupled Gross–Pitaevskii equation (CGPE) which governs the evolution of the macroscopic wave function of two components Bose–Einstein condensates trapped in time-dependent harmonic potential. Kinds of soliton solutions can be derived from the Lax-pair through Darboux transformation conveniently. Furthermore, soliton management in two-component Bose–Einstein condensate would be realized base on that the shape and motion of soliton in both components are investigated analytically. Moreover, it is found that there is a transformation existed between the nonautonomous coupled system and Manakov model. © 2011 Elsevier B.V. All rights reserved.
1. Introduction It is well known that the evolution of Bose–Einstein Condensate (BEC) at nearly absolute zero temperature can be described well by Gross–Pitaevskii equation [1–4]. Scientists usually study on the dynamics of BEC through solving the nonlinear equation numerically [5] or analytically [4]. Between these studies, the study on soliton is a very exiting field due to its particle-like properties. The dynamics of soliton in single-component BEC have been researched widely [6–10]. However, the multi-components BEC have been realized experimentally in recent years [11–13]. The evolution of the n-components condensate will be described by the so called coupled Gross–Pitaevskii equations (CGPE), particularly,
∂ 2 ψi ∂ψi + + R j (t )|ψ j |2 ψi + V (x, t )ψi = 0, ∂t ∂ x2 n
i
(1)
j =1
where i = 1, 2, . . . , n and ψi denotes the wave function of i-th component in BEC. R j (t ) is nonlinearity management parameter which describes the variation of scattering length and can be controlled well by Feshbach resonance. V (x, t ) stands for the trapping potential and other potentials. Without considering the effects of trap potentials, some studies have been done in [14–16]. In fact, the external potentials cannot be ignored in many circumstances. For example, the condensates are usually trapped in harmonic potential [1,4,17]. However, the studies on soliton in these systems with external potentials are still scant theoretically
*
Correspondence to: Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China. E-mail address: [email protected] (L.-C. Zhao). 0375-9601/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2011.06.034
[17]. It is well known that the nonlinear evolution equations in integrable systems all have their corresponding Lax-pair expressions. The form of Lax-pair possesses profound geometrical and physical significance. From the Lax-pair, one can generate analytical soliton solutions conveniently through related Darboux transformation. Therefore, it is meaningful to derive the Lax-pair of the CGPE to study these nonlinear systems. In this Letter, we present Lax-pair and the related Darboux transformation corresponding to the two components CGPE. For simplicity, we calculate one family of soliton solution from trivial seeds. Additionally, based on expressions which describe the evolution of soliton’s properties, and the compatibility condition, soliton management is able to be realized in explicit ways. These results would stimulate studies on soliton management. Additionally, it is found that there is a transformation existed which can be used to get soliton solutions of nonautonomous system from Manakov model conveniently. 2. The model and its Lax-pair We start by considering a general two-component BEC in the time-dependent harmonic potential M (t )x2 . Additionally, for atoms in the nK mK temperature regime, the effect of the Earth’s gravitational field is by no means negligible especially in the case of magnetic trapping. To describe the effects of gravitational field or other linear potentials, we can introduce an arbitrary timedependent linear potential f (t )x to study the influence of them conveniently. Considering the atoms transformed between condensates and thermal cloud, it is suitable to add complex potentials in CGPE to describe their effects on the condensate, marked by iG j (x, t ). Then, the CGPE is of the form
3018
L.-C. Zhao, S.-L. He / Physics Letters A 375 (2011) 3017–3020
∂ 2ψ ∂ψ + 2 + V 1 (x, t )ψ + R 11 |ψ|2 + R 12 |φ|2 ψ = 0, ∂t ∂x ∂φ ∂ 2 φ + 2 + V 2 (x, t )φ + R 21 |ψ|2 + R 22 |φ|2 φ = 0, i ∂t ∂x i
(2) (3)
where V j (x, t ) = M (t )x2 + f (t )x + iG j (x, t ) ( j = 1, 2). It is obvious that the coupled system is a generalized nonautonomous coupled system [4]. When f (t ) = 0 and G j (t ) = 0, vector soliton in the system has been studied through a special transformation [18]. When f (t ) = 0, M (t ) = constant, and G j (t ) = 0, dynamics of soliton has been researched through the similarity transformations method [19]. In this Letter, we will solve the generalized CGPE analytically through the Lax-pair method. To solve the coupled nonlinear equations, we assume the solutions of them have the form as
ψ(x, t ) = p (x, t ) exp iC (t )x2 − 2C (t ) + G 1 (t ) dt , φ(x, t ) = q(x, t ) exp iC (t )x2 − 2C (t ) + G 2 (t ) dt .
(4) (5)
It is well known that all integrable systems must have some certain integrate conditions, or else it is impossible to get soliton solution analytically. It is found that when the nonlinear parameters are chosen as
R 11 = R 21 = 2σ exp
R 22 = R 12 = 2σ exp and M (t ) = 4C 2 (t ) −
4C (t ) + 2G 2 (t ) dt ,
dC (t ) , dt
(6)
ipt + p xx + i4C (t )xp x + 2σ | p |2 + |q|2 p
−2
dC (t ) dt
2
x p + f (t )xp = 0,
(7)
iqt + q xx + i4C (t )xq x + 2σ | p |2 + |q|2 q
−2
dC (t ) dt
(8)
Then the corresponding Lax-pair of Eqs. (7) and (8) can be presented as
⎛
⎞ ⎛ ⎞ Φ1 Φ1 ∂ x ⎝ Φ2 ⎠ = U ⎝ Φ2 ⎠ , Φ3 Φ3 ⎛ ⎞ ⎛ ⎞ Φ1 Φ1 ∂t ⎝ Φ2 ⎠ = V ⎝ Φ2 ⎠ , Φ3 Φ3 ⎛
−i 23 λ − i 23 ζ x ⎝ U= σ p¯ −σ q¯ ⎛
(9)
A
V = U λ + ⎝ −i σ p¯x i σ q¯x
−p i λ + 3i ζ x 3 0
−ip x B i σ q¯ p
q 0
⎞ ⎠,
(10)
i λ + 3i ζ x 3
⎞
iq x i σ q p¯ ⎠ . C
¯ u¯ 2i (λ − λ) 4σ + |u |2 + | v |2
¯ ) v¯ 2i (λ − λ
4σ + |u |2 + | v |2
, (12)
,
3 2 where u = Φ Φ1 and v = Φ1 . For example, we get soliton solution from zero seed solution. After solving the Lax-pair to get the solution of Φ1 , Φ2 , Φ3 , the solution of Eqs. (7) and (8) can be presented by the transformation Eq. (12). Then we can present the solution of Eqs. (2) and (3) from Eqs. (4) and (5) directly
ψ(x, t ) =
4β B 21 exp [θ1 (x, t )]
(11)
,
(13)
,
(14)
4σ + ( B 221 + B 231 ) exp [ϕ (x, t )] 4β B 31 exp [θ2 (x, t )] 4σ + ( B 221 + B 231 ) exp [ϕ (x, t )]
where
θ1 (x, t ) = −iC (t )x − i λ¯ x + − i λ¯ 2 dt ,
2
2C (t ) − G 1 (t ) dt
θ2 (x, t ) = −iC (t )x2 − i λ¯ x + 2C (t ) − G 2 (t ) dt − i λ¯ 2 dt , ϕ (x, t ) = F (t ) dt − 2β xe 4C (t ) dt , and
F (t ) = −4β
Hereafter, the overbar denotes the complex conjugate. The expressions of A , B, and C are given as following
1 2 A = −i ζt x2 + i f (t )x + i σ | p |2 + i σ |q|2 , 3 3 1 2 1 B = i ζt x − i f (t )x − i σ | p |2 , 6 3 1 2 1 C = i ζt x − i f (t )x − i σ |q|2 , 6 3
p1 = p0 −
φ(x, t ) =
x2 q + f (t )xq = 0.
where
From the Lax-pair presented above, one can perform the following Darboux transformation to get soliton solution of Eqs. (7) and (8)
Φ
one can simplify Eqs. (2) and (3) as
3. The soliton solution and its dynamics
q1 = q0 −
4C (t ) + 2G 1 (t ) dt ,
t
where ζ = 4C (t ) and λ = [α − f (t )e 0 −4C (τ ) dτ dt + i β]e 4C (t ) dt . Between these expressions, the parameter σ can be chosen as a constant number. For BEC system, σ < 0 corresponds to repulsive and σ > 0 for attractive atomic interaction. From this Lax-pair, bright and dark soliton can be derived conveniently. However, we have introduced some certain relations between nonlinear coefficients and trap potential, to get make the model integrable. The dC (t ) relations Eq. (6) and M (t ) = 4C 2 (t ) − dt , could be seen as integrable conditions for our model. For experimental viewpoint, we emphasize that the nonlinear term R 12 should be equal to R 21 for BEC system, which leads to G 1 (t ) = G 2 (t ) and R 11 = R 12 = R 21 = R 22 . But for optical pulse propagating in left and right-handed (normal) nonlinear mediums, the relations can be satisfied without any more constraints [20].
t
α−
f
τ e
τ 0
−4C (τ ) dτ
dτ
e
t 0
8C (τ ) dτ
.
0
Between these expressions, the parameter α , β , B 21 and B 31 are all real numbers which relate with the initial condition of soliton, such as initial coordinate, initial velocity, and initial shape. Between the above solution, when C (t ) = 0, f (t ) = 0, and G j (t ) = 0, the generalized model will become the well-known Manakov model which can be studied analytically [21]. This is a generic solution which can be used to study on properties of bright solitons in BEC trapped in many kinds of potentials and many other systems. For example, this integrable system is able to be used to study on the multi-mode soliton propagation in nonlinear fibers
L.-C. Zhao, S.-L. He / Physics Letters A 375 (2011) 3017–3020
[14–16]. When σ > 0, it corresponds to focusing behavior and anomalous dispersion fibers, and σ < 0 corresponds to defocussing behavior and normal dispersion fibers. The solutions we present are bright–bright soliton solutions. However, it should be pointed out that the dark soliton solutions cannot be derived from a trivial seed solution through Darboux transformation. To get the dark–dark and bright–dark soliton solutions, one should perform Darboux transformation from nontrivial seed solution. For incoherent solitons which means there is no interference between such two components, the total density distribution |U (x, t )|2 = |ψ|2 + |φ|2 . We show the evolution of total density distribution with incoherent solitons in Fig. 1. It is shown that the peak can be well controlled by complex potentials. From the coefficients given in figure, one can see that the oscillation of soliton’s peak comes from the periodic complex potential. This can be proved through investigating the properties of total soliton analytically. From the soliton solution of each BEC component, we can calculate nonautonomous soliton’s peak, width and the motion of its wave center by assuming that the maximum value of density correspond to the wave center, and the half-value width is the width of soliton [22]. The expressions which describe properties of soliton in both components BEC can be calculated. We find solitons’ width and the trajectories evolve in the same way in the two components. The evolution of them can be given as following (with C (t ) is a real function): the evolution of width is
Fig. 1. (Color online.) The dynamics of total density distribution with incoherent solitons in the parabolic potential under time-dependent interaction between atoms of condensates. The explicit coefficients are C (t ) = −0.005, f (t ) = 0, α = 0.02, β = 4, B 21 = 4, B 31 = 5, σ = 0.2, G 1 (t ) = 2 cos(3t ) and G 2 (t ) = 0.05.
3019
√
W (t ) =
ln(3 + 2 2)
β
exp
−4C (t ) dt ,
and the motion of its wave center is F (t ) dt − ln 4σ
xc (t ) =
B 221 + B 231
2β
exp
(15)
−4C (t ) dt .
(16)
However, the evolution of soliton’ s peak evolve in different way. For ψ(x, t ) component, soliton’s peak, marked by |ψ|2max , is presented as
|ψ|2max
=
β 2 B 221 ( B 221 + B 231 )σ
exp
4C (t ) − 2G 1 (t ) dt ,
(17)
and the soliton in φ(x, t ) component is
|φ|2max =
β 2 B 231 ( B 221 + B 231 )σ
exp
4C (t ) − 2G 2 (t ) dt .
(18)
From the explicit expressions which describe the main properties of solitons, it is convenient to study the effects of each physical operation on the nonautonomous solitons. It is obvious that the time-dependent linear potential does not affect the shape of soliton, and complex potentials do not have any effects on the motion of soliton. When the explicit operations are chosen, the corresponding soliton solution can be given directly. For example, when the parameter C (t ) is chosen, the evolution of soliton’s width will be determined exactly through Eq. (15). To control the motion of soliton, one can choose the formation of the linear potential f (t )x for the soliton with certain initial condition. The peak of soliton can be determined through choosing the form of C (t ) and G j (t ) from Eqs. (17) and (18). Especially, when G j (t ) = 2C (t ), the peak of soliton will keep unchanged. We emphasize that the experimental management of the trap potential and Feshbach resonance should dC (t ) be derived from M (t ) = 4C 2 (t ) − dt and Eq. (6), which can seen as the integrate condition for our system. This provides many possibilities to control the evolution of soliton exactly. To get more stable soliton in both components, one can choose the parameter C (t ) smaller to make soliton’s width evolve more slowly. The complex potential G j (t ) can be chosen as 2C (t ) to keep the peak of soliton unchanged. As an example, C (t ) is chosen to be −0.0005, and G j (t ) = −0.001, we show the evolution of soliton of two component under this condition in Fig. 2. It is shown that the evolution of solitons are close to the standard soliton. From the integrable condition, we can know how to manage the trap potential and Feshbach resonance technology to get soliton with the properties.
Fig. 2. (a) The evolution of soliton in the first component ψ ; (b) The evolution of soliton in the second component φ . It is shown that the solitons are similar to the standard soliton. The explicit coefficients are C (t ) = −0.0005, f (t ) = 0, α = 0, β = 4, B 21 = 4, B 31 = 5, σ = 0.2, G 1 (t ) = −0.001 and G 2 (t ) = −0.001.
3020
L.-C. Zhao, S.-L. He / Physics Letters A 375 (2011) 3017–3020
4. A transformation into the Manakov system X.G. He et al. have shown that there is a transformation between nonautonomous system and the standard nonlinear Schrödinger equations [23]. Furthermore, we find that there is a transformation exist between the above coupled integrable system and the well-known Manakov model. Assuming the solution of Eqs. (2) and (3) have the following forms
ψ = Q 1 r (x, t ), s(t ) exp ia(x, t ) + h(t ) − G 1 (t ) dt , φ = Q 2 r (x, t ), s(t ) exp ia(x, t ) + h(t ) − G 2 (t ) dt ,
with
a(x, t ) = −
h(t ) 2
x2 + h1 (t )x + h2 (t ),
2h(t ) dt − 2
r (x, t ) = x exp
s(t ) =
exp
h1 (t ) exp
Acknowledgements
2h(t ) dt dt ,
This work is supported by the National Fundamental Research Program of China (Contract Nos. 2007CB814800, 2011CB921503), the National Science Foundation of China (Contract Nos. 10725521, 91021021, 11075020, 11078001).
4h(t ) dt dt ,
R 11 = R 21 = 2σ exp 2h(t ) + 2G 1 (t ) dt , R 22 = R 12 = 2σ exp 2h(t ) + 2G 2 (t ) dt ,
References
where h(t ), h1 (t ) and h2 (t ) satisfy the following relations
M (t ) = − f (t ) = dh2 (t ) dt
1 dh(t )
2 dt dh1 (t ) dt
+ h(t )2 ,
− 2h1 (t )h(t ),
= −h1 (t )2 ,
one can transform the coupled Eqs. (2) and (3) to be the Manakov model
i Q 1 , T + Q 1 , X X + 2 σ | Q 1 |2 + | Q 2 |2 Q 1 = 0 , i Q 2 , T + Q 2 , X X + 2 σ | Q 1 |2 + | Q 2 |2 Q 2 = 0 .
bitrary time-dependent linear potential and complex potential. As an example, the soliton solutions are derived from trivial seeds. From the soliton solutions, we calculate expressions of soliton’s width and peak to describe the evolution of their shape. The motion of soliton are investigated analytically too. It is shown that the linear potential just affect soliton’s motion without changing its shape, and the complex potentials just have effects on the peak of soliton without any influence on the motion of solitons. Therefore, for certain evolution of soliton, the explicit ways to adjust the parameters are able to be deduced from the expressions and compatibility condition. We believe that these results have great potentiality in soliton application. Moreover, we present a transformation between the coupled nonautonomous system and the well-known Manakov model, which could be used to derive soliton solutions conveniently too.
(19)
Between the above equations, the footnote T or X means partial derivation of T or X , and T , X represent s(t ), r (x, t ) respectively. Therefore, it is convenient to derive soliton solutions of the nonautonomous system from the ones of Manakov model too. 5. Conclusion We present the Lax-pair and Darboux transformation of the two components CGPE in a time-dependent harmonic trap with an ar-
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Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Matter wave solitons in coupled system with external potentials Li-Chen Zhao a,b,∗ , Shao-Ling He c a b c
Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China Science and Technology Computation Physics Laboratory, Institute of Applied Physics and Computational Mathematics, Beijing 100088, China Department of Mathematics, China University of Mining and Technology, Beijing 100083, China
a r t i c l e
i n f o
Article history: Received 9 April 2011 Received in revised form 3 June 2011 Accepted 15 June 2011 Available online 21 June 2011 Communicated by A.R. Bishop Keywords: Coupled GP-equation Lax-pair Bright soliton
a b s t r a c t We present Lax-pair corresponding to the coupled Gross–Pitaevskii equation (CGPE) which governs the evolution of the macroscopic wave function of two components Bose–Einstein condensates trapped in time-dependent harmonic potential. Kinds of soliton solutions can be derived from the Lax-pair through Darboux transformation conveniently. Furthermore, soliton management in two-component Bose–Einstein condensate would be realized base on that the shape and motion of soliton in both components are investigated analytically. Moreover, it is found that there is a transformation existed between the nonautonomous coupled system and Manakov model. © 2011 Elsevier B.V. All rights reserved.
1. Introduction It is well known that the evolution of Bose–Einstein Condensate (BEC) at nearly absolute zero temperature can be described well by Gross–Pitaevskii equation [1–4]. Scientists usually study on the dynamics of BEC through solving the nonlinear equation numerically [5] or analytically [4]. Between these studies, the study on soliton is a very exiting field due to its particle-like properties. The dynamics of soliton in single-component BEC have been researched widely [6–10]. However, the multi-components BEC have been realized experimentally in recent years [11–13]. The evolution of the n-components condensate will be described by the so called coupled Gross–Pitaevskii equations (CGPE), particularly,
∂ 2 ψi ∂ψi + + R j (t )|ψ j |2 ψi + V (x, t )ψi = 0, ∂t ∂ x2 n
i
(1)
j =1
where i = 1, 2, . . . , n and ψi denotes the wave function of i-th component in BEC. R j (t ) is nonlinearity management parameter which describes the variation of scattering length and can be controlled well by Feshbach resonance. V (x, t ) stands for the trapping potential and other potentials. Without considering the effects of trap potentials, some studies have been done in [14–16]. In fact, the external potentials cannot be ignored in many circumstances. For example, the condensates are usually trapped in harmonic potential [1,4,17]. However, the studies on soliton in these systems with external potentials are still scant theoretically
*
Correspondence to: Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China. E-mail address: [email protected] (L.-C. Zhao). 0375-9601/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2011.06.034
[17]. It is well known that the nonlinear evolution equations in integrable systems all have their corresponding Lax-pair expressions. The form of Lax-pair possesses profound geometrical and physical significance. From the Lax-pair, one can generate analytical soliton solutions conveniently through related Darboux transformation. Therefore, it is meaningful to derive the Lax-pair of the CGPE to study these nonlinear systems. In this Letter, we present Lax-pair and the related Darboux transformation corresponding to the two components CGPE. For simplicity, we calculate one family of soliton solution from trivial seeds. Additionally, based on expressions which describe the evolution of soliton’s properties, and the compatibility condition, soliton management is able to be realized in explicit ways. These results would stimulate studies on soliton management. Additionally, it is found that there is a transformation existed which can be used to get soliton solutions of nonautonomous system from Manakov model conveniently. 2. The model and its Lax-pair We start by considering a general two-component BEC in the time-dependent harmonic potential M (t )x2 . Additionally, for atoms in the nK mK temperature regime, the effect of the Earth’s gravitational field is by no means negligible especially in the case of magnetic trapping. To describe the effects of gravitational field or other linear potentials, we can introduce an arbitrary timedependent linear potential f (t )x to study the influence of them conveniently. Considering the atoms transformed between condensates and thermal cloud, it is suitable to add complex potentials in CGPE to describe their effects on the condensate, marked by iG j (x, t ). Then, the CGPE is of the form
3018
L.-C. Zhao, S.-L. He / Physics Letters A 375 (2011) 3017–3020
∂ 2ψ ∂ψ + 2 + V 1 (x, t )ψ + R 11 |ψ|2 + R 12 |φ|2 ψ = 0, ∂t ∂x ∂φ ∂ 2 φ + 2 + V 2 (x, t )φ + R 21 |ψ|2 + R 22 |φ|2 φ = 0, i ∂t ∂x i
(2) (3)
where V j (x, t ) = M (t )x2 + f (t )x + iG j (x, t ) ( j = 1, 2). It is obvious that the coupled system is a generalized nonautonomous coupled system [4]. When f (t ) = 0 and G j (t ) = 0, vector soliton in the system has been studied through a special transformation [18]. When f (t ) = 0, M (t ) = constant, and G j (t ) = 0, dynamics of soliton has been researched through the similarity transformations method [19]. In this Letter, we will solve the generalized CGPE analytically through the Lax-pair method. To solve the coupled nonlinear equations, we assume the solutions of them have the form as
ψ(x, t ) = p (x, t ) exp iC (t )x2 − 2C (t ) + G 1 (t ) dt , φ(x, t ) = q(x, t ) exp iC (t )x2 − 2C (t ) + G 2 (t ) dt .
(4) (5)
It is well known that all integrable systems must have some certain integrate conditions, or else it is impossible to get soliton solution analytically. It is found that when the nonlinear parameters are chosen as
R 11 = R 21 = 2σ exp
R 22 = R 12 = 2σ exp and M (t ) = 4C 2 (t ) −
4C (t ) + 2G 2 (t ) dt ,
dC (t ) , dt
(6)
ipt + p xx + i4C (t )xp x + 2σ | p |2 + |q|2 p
−2
dC (t ) dt
2
x p + f (t )xp = 0,
(7)
iqt + q xx + i4C (t )xq x + 2σ | p |2 + |q|2 q
−2
dC (t ) dt
(8)
Then the corresponding Lax-pair of Eqs. (7) and (8) can be presented as
⎛
⎞ ⎛ ⎞ Φ1 Φ1 ∂ x ⎝ Φ2 ⎠ = U ⎝ Φ2 ⎠ , Φ3 Φ3 ⎛ ⎞ ⎛ ⎞ Φ1 Φ1 ∂t ⎝ Φ2 ⎠ = V ⎝ Φ2 ⎠ , Φ3 Φ3 ⎛
−i 23 λ − i 23 ζ x ⎝ U= σ p¯ −σ q¯ ⎛
(9)
A
V = U λ + ⎝ −i σ p¯x i σ q¯x
−p i λ + 3i ζ x 3 0
−ip x B i σ q¯ p
q 0
⎞ ⎠,
(10)
i λ + 3i ζ x 3
⎞
iq x i σ q p¯ ⎠ . C
¯ u¯ 2i (λ − λ) 4σ + |u |2 + | v |2
¯ ) v¯ 2i (λ − λ
4σ + |u |2 + | v |2
, (12)
,
3 2 where u = Φ Φ1 and v = Φ1 . For example, we get soliton solution from zero seed solution. After solving the Lax-pair to get the solution of Φ1 , Φ2 , Φ3 , the solution of Eqs. (7) and (8) can be presented by the transformation Eq. (12). Then we can present the solution of Eqs. (2) and (3) from Eqs. (4) and (5) directly
ψ(x, t ) =
4β B 21 exp [θ1 (x, t )]
(11)
,
(13)
,
(14)
4σ + ( B 221 + B 231 ) exp [ϕ (x, t )] 4β B 31 exp [θ2 (x, t )] 4σ + ( B 221 + B 231 ) exp [ϕ (x, t )]
where
θ1 (x, t ) = −iC (t )x − i λ¯ x + − i λ¯ 2 dt ,
2
2C (t ) − G 1 (t ) dt
θ2 (x, t ) = −iC (t )x2 − i λ¯ x + 2C (t ) − G 2 (t ) dt − i λ¯ 2 dt , ϕ (x, t ) = F (t ) dt − 2β xe 4C (t ) dt , and
F (t ) = −4β
Hereafter, the overbar denotes the complex conjugate. The expressions of A , B, and C are given as following
1 2 A = −i ζt x2 + i f (t )x + i σ | p |2 + i σ |q|2 , 3 3 1 2 1 B = i ζt x − i f (t )x − i σ | p |2 , 6 3 1 2 1 C = i ζt x − i f (t )x − i σ |q|2 , 6 3
p1 = p0 −
φ(x, t ) =
x2 q + f (t )xq = 0.
where
From the Lax-pair presented above, one can perform the following Darboux transformation to get soliton solution of Eqs. (7) and (8)
Φ
one can simplify Eqs. (2) and (3) as
3. The soliton solution and its dynamics
q1 = q0 −
4C (t ) + 2G 1 (t ) dt ,
t
where ζ = 4C (t ) and λ = [α − f (t )e 0 −4C (τ ) dτ dt + i β]e 4C (t ) dt . Between these expressions, the parameter σ can be chosen as a constant number. For BEC system, σ < 0 corresponds to repulsive and σ > 0 for attractive atomic interaction. From this Lax-pair, bright and dark soliton can be derived conveniently. However, we have introduced some certain relations between nonlinear coefficients and trap potential, to get make the model integrable. The dC (t ) relations Eq. (6) and M (t ) = 4C 2 (t ) − dt , could be seen as integrable conditions for our model. For experimental viewpoint, we emphasize that the nonlinear term R 12 should be equal to R 21 for BEC system, which leads to G 1 (t ) = G 2 (t ) and R 11 = R 12 = R 21 = R 22 . But for optical pulse propagating in left and right-handed (normal) nonlinear mediums, the relations can be satisfied without any more constraints [20].
t
α−
f
τ e
τ 0
−4C (τ ) dτ
dτ
e
t 0
8C (τ ) dτ
.
0
Between these expressions, the parameter α , β , B 21 and B 31 are all real numbers which relate with the initial condition of soliton, such as initial coordinate, initial velocity, and initial shape. Between the above solution, when C (t ) = 0, f (t ) = 0, and G j (t ) = 0, the generalized model will become the well-known Manakov model which can be studied analytically [21]. This is a generic solution which can be used to study on properties of bright solitons in BEC trapped in many kinds of potentials and many other systems. For example, this integrable system is able to be used to study on the multi-mode soliton propagation in nonlinear fibers
L.-C. Zhao, S.-L. He / Physics Letters A 375 (2011) 3017–3020
[14–16]. When σ > 0, it corresponds to focusing behavior and anomalous dispersion fibers, and σ < 0 corresponds to defocussing behavior and normal dispersion fibers. The solutions we present are bright–bright soliton solutions. However, it should be pointed out that the dark soliton solutions cannot be derived from a trivial seed solution through Darboux transformation. To get the dark–dark and bright–dark soliton solutions, one should perform Darboux transformation from nontrivial seed solution. For incoherent solitons which means there is no interference between such two components, the total density distribution |U (x, t )|2 = |ψ|2 + |φ|2 . We show the evolution of total density distribution with incoherent solitons in Fig. 1. It is shown that the peak can be well controlled by complex potentials. From the coefficients given in figure, one can see that the oscillation of soliton’s peak comes from the periodic complex potential. This can be proved through investigating the properties of total soliton analytically. From the soliton solution of each BEC component, we can calculate nonautonomous soliton’s peak, width and the motion of its wave center by assuming that the maximum value of density correspond to the wave center, and the half-value width is the width of soliton [22]. The expressions which describe properties of soliton in both components BEC can be calculated. We find solitons’ width and the trajectories evolve in the same way in the two components. The evolution of them can be given as following (with C (t ) is a real function): the evolution of width is
Fig. 1. (Color online.) The dynamics of total density distribution with incoherent solitons in the parabolic potential under time-dependent interaction between atoms of condensates. The explicit coefficients are C (t ) = −0.005, f (t ) = 0, α = 0.02, β = 4, B 21 = 4, B 31 = 5, σ = 0.2, G 1 (t ) = 2 cos(3t ) and G 2 (t ) = 0.05.
3019
√
W (t ) =
ln(3 + 2 2)
β
exp
−4C (t ) dt ,
and the motion of its wave center is F (t ) dt − ln 4σ
xc (t ) =
B 221 + B 231
2β
exp
(15)
−4C (t ) dt .
(16)
However, the evolution of soliton’ s peak evolve in different way. For ψ(x, t ) component, soliton’s peak, marked by |ψ|2max , is presented as
|ψ|2max
=
β 2 B 221 ( B 221 + B 231 )σ
exp
4C (t ) − 2G 1 (t ) dt ,
(17)
and the soliton in φ(x, t ) component is
|φ|2max =
β 2 B 231 ( B 221 + B 231 )σ
exp
4C (t ) − 2G 2 (t ) dt .
(18)
From the explicit expressions which describe the main properties of solitons, it is convenient to study the effects of each physical operation on the nonautonomous solitons. It is obvious that the time-dependent linear potential does not affect the shape of soliton, and complex potentials do not have any effects on the motion of soliton. When the explicit operations are chosen, the corresponding soliton solution can be given directly. For example, when the parameter C (t ) is chosen, the evolution of soliton’s width will be determined exactly through Eq. (15). To control the motion of soliton, one can choose the formation of the linear potential f (t )x for the soliton with certain initial condition. The peak of soliton can be determined through choosing the form of C (t ) and G j (t ) from Eqs. (17) and (18). Especially, when G j (t ) = 2C (t ), the peak of soliton will keep unchanged. We emphasize that the experimental management of the trap potential and Feshbach resonance should dC (t ) be derived from M (t ) = 4C 2 (t ) − dt and Eq. (6), which can seen as the integrate condition for our system. This provides many possibilities to control the evolution of soliton exactly. To get more stable soliton in both components, one can choose the parameter C (t ) smaller to make soliton’s width evolve more slowly. The complex potential G j (t ) can be chosen as 2C (t ) to keep the peak of soliton unchanged. As an example, C (t ) is chosen to be −0.0005, and G j (t ) = −0.001, we show the evolution of soliton of two component under this condition in Fig. 2. It is shown that the evolution of solitons are close to the standard soliton. From the integrable condition, we can know how to manage the trap potential and Feshbach resonance technology to get soliton with the properties.
Fig. 2. (a) The evolution of soliton in the first component ψ ; (b) The evolution of soliton in the second component φ . It is shown that the solitons are similar to the standard soliton. The explicit coefficients are C (t ) = −0.0005, f (t ) = 0, α = 0, β = 4, B 21 = 4, B 31 = 5, σ = 0.2, G 1 (t ) = −0.001 and G 2 (t ) = −0.001.
3020
L.-C. Zhao, S.-L. He / Physics Letters A 375 (2011) 3017–3020
4. A transformation into the Manakov system X.G. He et al. have shown that there is a transformation between nonautonomous system and the standard nonlinear Schrödinger equations [23]. Furthermore, we find that there is a transformation exist between the above coupled integrable system and the well-known Manakov model. Assuming the solution of Eqs. (2) and (3) have the following forms
ψ = Q 1 r (x, t ), s(t ) exp ia(x, t ) + h(t ) − G 1 (t ) dt , φ = Q 2 r (x, t ), s(t ) exp ia(x, t ) + h(t ) − G 2 (t ) dt ,
with
a(x, t ) = −
h(t ) 2
x2 + h1 (t )x + h2 (t ),
2h(t ) dt − 2
r (x, t ) = x exp
s(t ) =
exp
h1 (t ) exp
Acknowledgements
2h(t ) dt dt ,
This work is supported by the National Fundamental Research Program of China (Contract Nos. 2007CB814800, 2011CB921503), the National Science Foundation of China (Contract Nos. 10725521, 91021021, 11075020, 11078001).
4h(t ) dt dt ,
R 11 = R 21 = 2σ exp 2h(t ) + 2G 1 (t ) dt , R 22 = R 12 = 2σ exp 2h(t ) + 2G 2 (t ) dt ,
References
where h(t ), h1 (t ) and h2 (t ) satisfy the following relations
M (t ) = − f (t ) = dh2 (t ) dt
1 dh(t )
2 dt dh1 (t ) dt
+ h(t )2 ,
− 2h1 (t )h(t ),
= −h1 (t )2 ,
one can transform the coupled Eqs. (2) and (3) to be the Manakov model
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bitrary time-dependent linear potential and complex potential. As an example, the soliton solutions are derived from trivial seeds. From the soliton solutions, we calculate expressions of soliton’s width and peak to describe the evolution of their shape. The motion of soliton are investigated analytically too. It is shown that the linear potential just affect soliton’s motion without changing its shape, and the complex potentials just have effects on the peak of soliton without any influence on the motion of solitons. Therefore, for certain evolution of soliton, the explicit ways to adjust the parameters are able to be deduced from the expressions and compatibility condition. We believe that these results have great potentiality in soliton application. Moreover, we present a transformation between the coupled nonautonomous system and the well-known Manakov model, which could be used to derive soliton solutions conveniently too.
(19)
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