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Dynamics and matter-wave solitons in bose-einstein condensates with two- and three-body interactions.

1. Introduction

Bose-Einstein condensation was first predicted by Einstein and Indian physicist Bose in 1924-1925. It is an exotic quantum phenomenon that was observed in dilute atomic gases for the first time in 1995 [1-3]. The "condensate" here is a state of matter of a dilute gas of bosons at temperatures close to 0 kelvins, which is different from the "condensate" in day life. One of the interesting dynamical features in the context of Bose-Einstein condensate (BEC) is the formation of matter wave solitons such as bright solitons [4, 5], dark solitons [5], vortex solitons [6], and gap solitons [7], which have been experimentally achieved before. Recent experimental techniques for managing nonlinearity have attracted considerable attention. For example, nonlinearity management arises in atomic physics for the Feshbach resonance [4, 8] of the scattering length of interatomic interactions in BECs, where the interaction strength can be characterized by a single parameter, the s-wave scattering length [a.sub.s]. Across a Feshbach resonance the length [a.sub.s] can in principle be varied from -ot to +[infinity], where [a.sub.s] <0 ([a.sub.s] > 0) corresponds to effectively attractive (repulsive) interactions. Thus in this situation, one can deal with the governing equations with the nonlinearity coefficients being functions of time or space [4, 9-12].

In the mean-field theory, a BEC system can be well described by the Gross-Pitaevskii (GP) equation [13-15], whose coefficient in front of the cubic term comes from the interatomic interaction. Under certain condition, the GP equation can be converted into the classical nonlinear Schrodinger equation (NLS). It is known that at low densities the three-body interactions can be neglected and the swave two-body interactions achieve a dominant position. However, the three-body interactions play a key role in BEC at high densities. Similarly, a BEC with two- and three-body interactions can be described by the GP equation with cubicquintic nonlinearity, also called variable coefficient cubicquintic nonlinear Schrodinger (CQNLS) equation [16-18]:

i [partial derivative][psi]/[partial derivative]t + 1/2 [[partial derivative].sup.2]/[partial derivative][x.sup.2] + V(x, t)[psi] + g (t) [[absolute value of ([psi])].sup.2][psi] + G (t) [[absolute value of ([psi])].sup.4] [psi] = 0, (1)

where [psi] is matter-wave function, V(x, t) is external potential, g(t) is two-body interaction coefficient, and G(t) is the three-body interaction coefficient. Function g(t) is positive (negative) for attractive (repulsive) condensates and the same as function G(t). Here functions g(t), G(t), and V(x,t) are experimentally controlled.

In this paper, we investigate the matter-wave soliton solutions and dynamics in BEC with two- and three-body interactions trapped by harmonic potential. The paper is organized as follows. In Section 2, the exact matter-wave soliton solutions of the variable coefficient cubic-quintic nonlinear Schrodinger equation are obtained by using similarity transformation. In Section 3, the density distributions and dynamics of the matter-wave solitons are investigated by analyzing their figures. We summarize our results in the conclusions.

2. Exact Matter-Wave Soliton Solutions of CQNLS Equation (1)

2.1. Similarity Transformation. In what follows, we solve the variable coefficient cubic-quintic nonlinear Schrodinger equation (1) by means of similarity transformation. Through analyzing the properties of the exact solutions, new density distributions and dynamics of BEC with two- and three-body interactions are found.

Assume the variable coefficient CQNLS equation 1) can be transformed into the following CQNLS equation:

i [partial derivative]U/[partial derivative]T + [[partial derivative].sup.2]/[partial derivative][X.sup.2] + [[sigma].sup.2] [[absolute value of (U)].sup.2] U + [[sigma].sub.3] [[absolute value of (U)].sup.4] U = 0, (2)

where [[sigma].sub.2], [[sigma].sub.3] are constants.

By the principle of similarity transformation, suppose the variable coefficient CQNLS equation (1) has the following form of exact solution:

[psi] = [rho]U(T, X) [e.sup.[phi]], (3)

[[absolute value of ([psi])].sup.2] = [[rho].sup.2] [[absolute value of (U(T,X))].sup.2], (4)

where [rho] is amplitude, [phi] is phase, and [rho] and [phi] are functions t and x; function U(T, X) solves the CQNLS equation (2), T is a function of time t, and X is a function of t and x.

Making use of the symbolic computation software Maple, substituting the similarity transformation (3) and (4) into the variable coefficient CQNLS equation (1), and letting U(T, X) satisfy the CQNLS equation (2), we obtain a set of partial differential equations (PDEs) of functions T, X, p, [phi], V, g, and G as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)

From the first equation in (5), that is, ([partial derivative]X/[partial derivative]x) - 2([partial derivative]T/[partial derivative]t) = 0, it is found that function X is linear in variable x. Furthermore, from ([[partial derivative].sup.2]X/[partial derivative][x.sup.2])[rho] + 2([partial derivative][rho]/[partial derivative]x)([partial derivative]X/[partial derivative]x) = 0 we have ([partial derivative][rho]/[partial derivative]x)([partial derivative]X/[partial derivative]x) = 0. Because [partial derivative]X/[partial derivative]x = 0, we have [partial derivative]p/[partial derivative]x = 0, and then [rho] is only a function of time t. So we can write functions X and [rho] as

X = [[tau].sub.1]x + [[tau].sub.2], (6)

[rho] = [rho](t), (7)

where [[tau].sub.1] and [[tau].sub.2] are functions of time t.

Inserting (6) and (7) into the fourth and fifth equations in (5), we have the expressions of g and G as

g = [[sigma].sup.2]([partial derivative]T/[partial derivative]t)/[[rho].sup.2], G = [[sigma].sub.3]([partial derivative]T/[partial derivative]t)/[[rho].sup.4]. (8)

Because [rho] is only a function of time t; from the last equation in (5) we have 2([partial derivative][rho]/[partial derivative]t) + ([[partial derivative].sup.2][phi]/ [partial derivative][x.sup.2])[rho] = 0, so function [phi] is a quadratic function of variable x. Thus we can assume function [phi] as

[phi] = [[eta].sub.1][x.sup.2] + [[eta].sub.2]x + [[eta].sub.3], (9)

where [[eta].sub.1], [[eta].sub.2], and [[eta].sub.3] are all functions of time t.

Finally, the external potential V(x,t) is usually harmonic potential in real experiments, so we let

V(X,t) = [[omega].sup.2][x.sup.2]/2, (10)

where [omega] is the frequency of harmonic potential, which is a function of time t.

Substituting (6)--(10) into (5) and simplifying, we have

[[tau].sub.1.sup.2] -2 [partial derivative]T/[partial derivative]t = 0, (11)

2[[tau].sub.1][[eta].sub.1]x + [[tau].sub.1][[eta].sub.2] + [partial derivative][[tau].sub.1]/[partial derivative]t x + [partial derivative][[tau].sup.2]/[partial derivative]t = 0, (12)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (13)

[partial derivative][rho]/[partial derivative]t + [[eta].sub.1][rho] = 0, (14)

which become the following ordinary differential equations (ODEs) of functions [[tau].sub.1], [[tau].sub.2], [[eta].sub.1], [[eta].sub.2], [[eta].sub.2], T, and [rho] by further calculating

[[tau].sub.1.sup.2] - 2[partial derivative]T/[partial derivative]t = 0, (15)

[[tau].sub.1][[eta].sub.2] + [partial derivative][[tau].sub.2]/[partial derivative]t = 0, 2[[tau].sub.1][[eta].sub.1] + [partial derivative][[tau].sub.1]/[partial derivative]t =0, (16)

2 [partial derivative][[eta].sub.2]/[partial derivative]t + 4[[eta].sub.1][[eta].sub.2] = 0, [[omega].sup.2] - 2 [partial derivative][[eta].sub.1]/[partial derivative]t - 4[[eta].sub.1.sup.2] = 0, (17)

2 [partial derivative][rho]/[partial derivative]t + 2[[eta].sub.1][rho] = 0, -2 [partial derivative][[eta].sub.3]/[partial derivative]t - [[eta].sub.2.sup.2] = 0. (18)

The exact solutions of the ODEs in (15)--(18) are

[[eta].sub.1] = [[eta].sub.1], (19)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (20)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (21)

T = 1/2 [integral] [[tau].sub.1.sup.2] dt + [C.sub.4], (22)

[[eta].sub.3] = -1/2 [integral] [[eta].sub.2.sup.2] dt + [C.sub.3], (23)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (24)

[omega] = [+ or -] [square root of (2([partial derivative][[eta].sub.1]/[partial derivative]t + 2[[eta].sub.1.sup.2]))], (25)

[[tau].sub.2] = - [integral] [[tau].sub.1][[eta].sub.2]dt + [C.sub.1], (26)

where we assume the frequency of the harmonic potential to be positive; that is, [omega] = [square root of (2(([partial derivative][[eta].sub.1]/[partial derivative]t) + 2[[eta].sub.1.sub.2]))], and [C.sub.1],..., [C.sup.6] are constants.

Now we discuss the frequency of the harmonic potential to by six cases.

Case 1. Let [omega] = [[OMEGA].sub.0] (constant); that is, the frequency of the harmonic potential is invariant. Then we can get function [[eta].sub.1] from [omega] = [square root of (2(([partial derivative][[eta].sub.1]/[partial derivative]t) + 2[[eta].sub.1.sup.2]))]. So other variables can be determined by (19)-(26). In current case, the function [[eta].sub.1] is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (27)

where [[OMEGA].sub.0] is a constant.

Case 2. Let function [[eta].sub.1] be linear in time t; that is, [[eta].sub.1] = [[OMEGA].sub.1]t with [[OMEGA].sub.1] a constant, and then we have

[omega] = [square root of (2 [[OMEGA].sub.1](1 + 2[[OMEGA].sub.1][t.sup.2])]. (28)

Case 3. Let function [[eta].sub.1] be an exponential function of time t; that is, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (29)

Case 4. Let function [[eta].sub.1] be a hyperbolic function of time t; that is, [[eta].sub.1] = cosh([[OMEGA].sub.3]t), and then we have

[omega] = [square root of 2 (sinh ([[OMEGA].sub.3]t) + 4 cosh [([[OMEGA].sup.3]t).sup.2])], (30)

where sinh denotes hyperbolic sine function and cosh denotes hyperbolic cosine function.

Case 5. Let function [[eta].sub.1] be another exponential function of time t; that is, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (31)

Case 6. Finally, let function [[eta].sub.1] be [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (32)

and here and above [[OMEGA].sub.1], [[OMEGA].sub.2],..., [[OMEGA].sub.5] are nonzero constants.

2.2. Exact Solutions of the CQNLS Equation (2). Up to now, we have derived the coefficients in the similarity transformation. The next thing is to find the exact solutions of the CQNLS equation (2) to formulate the wave function U(T, X). In what follows, we list three types of exact solutions of the CQNLS equation (2).

Type 1. Consider

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (33)

where cn is Jacobi elliptic cosine function [19, 20] with module k (0 [less than or equal to] k [less than or equal to] 1), and [l.sub.0] and [d.sub.2] are nonzero constants and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (34)

where the constant K satisfies

K = 1/6(- 24[[sigma].sub.3][k.sup.2] + 12[[sigma].sub.3]

+6 [[square root of (9[[sigma].sub.22][d.sub.2.sup.2] + 24[[sigma].sub.2][[sigma].sub.3][d.sub.2] + (l6[k.sup.4] - 16[k.sup.2] + 16) [[sigma].sub.3.sup.2]))].sup.2]. (35)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = (36)

where [l.sub.0] and [d.sub.2] are nonzero constants and Q = [square root of ([(4 [[sigma].sub.3] + 3[[sigma].sub.2][d.sub.2]).sup.2])].

Type 3. Consider

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (37)

where [l.sub.0] and [d.sub.2] are nonzero constants.

2.3. Exact Matter-Wave Soliton Solutions of (1). Three types of exact solutions [U.sub.1](T, X), [U.sub.2](T, X), [U.sub.3] (T, X) of the CQNLS equation (2) have been obtained above. In order to achievethe exact matter-wave soliton solutions of the variable coefficient CQNLS equation (1), we only need to combine the solutions [U.sub.1](T, X), [U.sub.2](T, X), [U.sub.3](T, X) with the similarity transformation (3). Thus the exact matter-wave soliton solution of the variable coefficient CQNLS equation is

[psi](x,t) = [rho]U(T,X)[e.sup.i[phi]], (38)

where X, [phi], and [rho] are given by (6), (9), and (24) and functions [[tau].sub.1], [[tau].sub.2], [[eta].sub.1], [[eta].sub.2], [[eta].sub.3], and T are given by (19)-(26). Here the value of function [[eta].sub.1] is the key to the solution. We have listed six cases of choices of function [[eta].sub.1] above. But we only choose [[eta].sub.1] = [[OMEGA].sub.1]t in the following calculation. So from (19)-(26), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (39)

where erf is error function (also named Gaussian error function).

To sum up, when the external potential is time-dependent harmonic potential V(x,t) = 2[[OMEGA].sub.1](1 + 2[[OMEGA].sub.1][t.sup.2])[x.sup.2], the coefficients of two- and three-body interactions are g(t) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and we derive three families of exact matter-wave soliton solutions of the variable coefficient CQNLS equation (1) as follows.

Family 1. Consider

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (40)

Family 2. Consider

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (41)

Family 3. Consider

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]], (42)

where the functions T,X are given by (39) and Q = [square root of ([(4 [[sigma].sub.3] + 3[[sigma].sub.2][d.sup.2]).sup.2])].

Remark 1. By considering the rest five cases of frequency [omega] and function [[eta].sub.1], we can also obtain other exact matter-wave soliton solutions of the variable coefficient CQNLS equation (1) under different types of two- and three-body interactions.

3. Density Distributions and Dynamics of the Matter-Wave Solitons

In this Section, we investigate the density distributions and dynamics of the matter-wave soliton solutions (40)-(42) by analyzing their figures. It is noted that the frequency of the harmonic potential is [omega] = [square root of (2[[OMEGA].sub.1](1 + 2[[OMEGA].sub.1][t.sup.2]))], and the coefficients of two- and three-body interactions are [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which are monotone nonincreasing function of time t.

3.1. Density Distribution and Dynamics of Solution [[psi].sub.1] (x, t). For the matter-wave soliton solution [[psi].sub.1](x, t) in (40), we choose the parameters as follows: [C.sub.1] = [C.sub.2] = ... = [C.sub.6] =1, [[OMEGA].sub.1] = 0.01, [[sigma].sub.2] = -0.5, [[sigma].sub.3] = -1, [d.sub.2] = 2, [l.sub.0] = 1, and k = 0.99. Here the time range is [0,15], time size is 0.01, the space range is [-20,20], and space size is 0.05.

The density distributions of the matter-wave soliton solution [[psi].sub.1](x, t) are shown in Figure 1. Here the coefficient of two-body interaction is g(t) < 0 and that of three-body interaction is G(t) < 0, which denotes that both two-body and three-body interactions are repulsive. According to the periodical property of the Jacobi elliptic functions, the density distribution [[absolute value of ([[psi].sub.1](x,t))].sup.2] is periodic. We can choose elliptic modulus k from 0 < k < 1, and we let k = 0.99 here. It is observed from Figure 1 that there are five peaks in the density distribution; that is, it has five periods in space. Under the effects of time-dependent harmonic potential and the repulsive two- and three-body interactions, the density distribution of wave function [[psi].sub.1](x,t) decreases with time. We can also get various density distributions by choosing k from 0 < k < 1.

3.2. Density Distribution and Dynamics of Solution [[psi].sub.2](x, t). We now study the matter-wave soliton solution [[psi].sub.2](x, t) in (41). The parameters here are [C.sub.1] = [C.sub.2] = ... = [C.sub.6] = 1, [[OMEGA].sub.1] = 0.01, [[sigma].sub.2] = 1/3, [[sigma].sub.3] = -0.5, [d.sub.2] = -1, and [l.sub.0] = 1. The time range is [0, 10], time size is 0.05, the space range is [-200, 200], and space size is 0.1.

The density distributions of the matter-wave soliton solution [[psi].sub.2](x,t) are shown in Figure 2. Here the coefficient of two-body interaction is g(t) > 0, and that of threebody interaction is G(t) < 0, which denotes that the twobody interaction is attractive and the three-body interaction is repulsive. This is a matter-wave bright soliton and is a localized nonlinear wave. It is seen from the color bar that as time goes on the density distribution of wave function, [[psi].sub.2](x,t) diminishes. Thus we find that attractive two-body interaction and repulsive three-body interaction do not support stable matter-wave bright soliton, which is consistent with the real experiments.

3.3. Density Distribution and Dynamics of Solution [[psi].sub.3](x,t). Finally, we analyze the matter-wave soliton solution [[psi].sub.3](x, t) in (42). The parameters here are [C.sub.1] = [C.sub.2] = ... = [C.sub.6] = 1, [[OMEGA].sub.1] = 0.01, [[sigma].sub.2] = -2, [[sigma].sub.3] = -1, [d.sub.2] = 5, and [l.sub.0] = 1. The time range is [0,10], time size is 0.02, the space range is [-10,10], and space size is 0.01.

The density distributions of the matter-wave soliton solution [[psi].sub.3](x, t) are shown in Figure 3. Here the coefficient of two-body interaction is g(t) < 0, and that of three-body interaction is G(t) < 0, which denotes that both two-body interaction and three-body interactions are repulsive. This is a matter-wave dark soliton and is also a localized nonlinear wave. It is also observed that the density distribution of wave function [[psi].sub.3](x, t) also diminishes with time.

4. Conclusions

In summary, we have studied the matter-wave solitons and dynamics of Bose-Einstein condensates with time-dependent two- and three-body interactions in time-dependent external potential. We find that when the nonlinear coefficients [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the cubic-quintic nonlinear Schrodinger equation supports three families of exact solutions. Moreover, six possible frequencies of harmonic potential are given. Finally, in the case of the harmonic potential V(x, t) = 2[[OMEGA].sub.1](1 + 2[[omega].sub.1][t.sup.2])[x.sup.2], we examine the density distributions and dynamics of the matter-wave soliton solutions by analyzing their plots. It is found that under the effect of time-dependent two- and three-body interactions along with time-dependent harmonic potential, the density distributions of the matter-wave solitons diminish with time. This is consistent with the real Bose-Einstein condensate experiments. There are indeed many papers [16-18, 21-28] studying the exact solutions of the nonlinear Schrodinger equations by similarity transformations, but they discuss either the matter-wave solitons of Bose-Einstein condensate with two-body interactions or the matterwave solitons of Bose-Einstein condensate with spatially inhomogeneous interactions. To our knowledge, the three families of exact solutions (40)-(42) for the cubic-quintic nonlinear Schrodinger are proposed for the first time.

http://dx.doi.org/10.1155/2014/307135

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the Talent Youth Program of Beijing Municipal Commission of Education (Grant no. YETP0984) and the Science and Technology Project of Beijing Municipal Commission of Education (Grant no. KM201311232021).

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Jing Chen, (1) Jie Yang, (2) and Lu Zhang (2)

(1) School of Statistics and Mathematics, Central University of Finance and Economics, Beijing 100081, China

(2) School of Science, Beijing Information Science and Technology University, Beijing 100192, China

Correspondence should be addressed to Jing Chen; chenjingcufe@163.com

Received 8 March 2014; Revised 15 June 2014; Accepted 24 June 2014; Published 5 August 2014

Academic Editor: Victor V. Moshchalkov
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