Exact wave solutions for Bose–Einstein condensates with time-dependentscattering length and spatiotemporal complicated potentialE. Kengne, A. Lakhssassi, R. Vaillancourt, and Wu-Ming Liu Citation: J. Math. Phys. 54, 051501 (2013); doi: 10.1063/1.4803458 View online: http://dx.doi.org/10.1063/1.4803458 View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v54/i5 Published by the American Institute of Physics. Additional information on J. Math. Phys.Journal Homepage: http://jmp.aip.org/ Journal Information: http://jmp.aip.org/about/about_the_journal Top downloads: http://jmp.aip.org/features/most_downloaded Information for Authors: http://jmp.aip.org/authors

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JOURNAL OF MATHEMATICAL PHYSICS 54, 051501 (2013)

Exact wave solutions for Bose–Einstein condensateswith time-dependent scattering length and spatiotemporalcomplicated potential

E. Kengne,1,2,3 A. Lakhssassi,2 R. Vaillancourt,3 and Wu-Ming Liu1

1National Laboratory for Condensed Matter Physics, Institute of Physics,Chinese Academy of Sciences, Beijing 100190, People’s Republic of China2Departement d’informatique et d’ingenierie, Universite du Quebec en Outaouais,101 St-Jean-Bosco, Succursale Hull, Gatineau(PQ) J8Y 3G5, Canada3Department of Mathematics and Statistics, Faculty of Science, University of Ottawa,585 King Edward Ave., Ottawa ON K1N 6N5, Canada

(Received 1 November 2012; accepted 16 April 2013; published online 7 May 2013)

We consider a cubic-quintic Gross–Pitaevskii equation which governs the dynamicsof Bose–Einstein condensate matter waves with time-dependent scattering length andspatiotemporal complex potential. By introducing phase-imprint parameters in thesystem, we present the integrable condition for the equation and obtain the exactanalytical solutions, which describe the propagation of a solitary wave. By applyingspecific time-modulated feeding/loss functional parameter, various types of magnetictrap strengths, and phase-imprint parameters, the dynamics of the solutions can becontrolled. Solitary wave solutions with breathing and snaking behaviors are reported.C© 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4803458]

I. INTRODUCTION

Since the realization of Bose–Einstein condensates (BEC) in dilute alkali-metal atomic vapours,1

the exploration of their dynamical properties through the studies of excitations in these quantummacroscopic media has been a field of very active research.2–16 Some years ago, experimental andtheoretical observations of dark and bright solitons in BECs trapped in optical lattices have drawnattention to various aspects of the dynamics of nonlinear matter waves, such as vortex dynamics,domain walls in binary BECs, soliton propagation, interference patterns, and modulational instability(MI).8, 17–25 The dynamics of the BEC solitary wave becomes one of its most important aspects; infact, the generation, dynamics, and management of a BEC solitary wave are important for a numberof BEC applications, like atomic interferometry,26 and different kinds of quantum phase transitions,27

as well as in the context of nonlinear physics, including nonlinear optics and hydrodynamics.At very low temperatures, the dynamics of a BEC is supposed to be fully condensed and is

described by the mean-field Gross–Pitaevskii (GP) equation, which is a time dependent nonlinearSchrodinger (NLS) equation with external trap potential; this GP equation takes the effects ofthe particle interactions through an effective meanfield into account and describes the condensatedynamics in a confined geometry.7, 28 As has been reported,29 the cubic nonlinear term in the GPequation that accounts for the two-body interatomic interactions generally appears as the dominantone. Hence, three-body interatomic interactions can be treated as a perturbation over the two-bodycase. In the simple but physically important case of a cigar-shaped trapping potential, the GPequation can be integrated out, resulting in the quasi-one-dimensional dimensionless cubic-quinticGross–Pitaevskii (C-QGP) equation,30, 31

i∂ψ

∂t= −1

2

∂2ψ

∂x2+ V ψ + g |ψ |2 ψ + χ0 |ψ |4 ψ, (1)

where the temporal and spatial coordinates t and x are measured in units 1/ω⊥ and a⊥, respectively.a⊥ = √

�/mω⊥ and a0 = √�/(mω0) are linear oscillator lengths in the transverse and cigar-axis

0022-2488/2013/54(5)/051501/14/$30.00 C©2013 AIP Publishing LLC54, 051501-1

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051501-2 Kengne et al. J. Math. Phys. 54, 051501 (2013)

directions, respectively; ω⊥ and ω0 are the corresponding harmonic-oscillator frequencies. Thenonlinearity parameters g = − 2as

3aBand χ0 = mg1 N 2

3π2�2a4⊥

represent, respectively, the two- and three-bodyinteratomic interactions coefficients, positive for repulsive interatomic interactions (or defocusingnonlinearities) and negative for attractive ones (focusing nonlinearities); here, as and aB are thes-wave scattering length and the Bohr radius, respectively, while g1 is an effective three-bodycontact interaction related to the GP equation, and N the total number of atoms. As discussed inRefs. 9, 32, and 33, g1 gives a mean-field contribution g1|ψ |6/3 to the energy density in a three-body system. This results in the correction of the GP equation with a term g1|ψ |4. In general,even though the three-body interaction parameter g1 could be a complex quantity, one can safelyneglect the imaginary part which represents the three-body loss when the density is not too highor the experimental period is not too long. Indeed, theoretical and experimental studies32 show,for 87Rb atoms, that Im[g1]/Re[g1]≈10− 3–10− 4. So the three-body recombination expressed bythe imaginary part of g1 is negligible. Usually the strength of the three-body interaction is verysmall when compared with strength of the two-body interaction, as pointed out by Gammal et al.34

Theoretical studies32 estimated that, for 87Rb atoms, g1 ≈ � × 10− 26 − � × 10− 27 cm2 s− 1,taking N ≈ 104–105 and a⊥ = 10− 6 m. The normalized macroscopic wave function ψ(x, t) isconnected to the original order parameter �(r, t) as follows:35, 36

�(r, t) = 1√2πaBa⊥

ψ

(x

a⊥ψ

, ω⊥t

)exp

(−iω⊥t − y2 + z2

2a2⊥

).

It is important to notice that, in the relevant experiments, a Feshbach resonance is utilized tomanipulate the sign of the s-wave scattering length from positive to negative;19, 20 in this situation,the s-wave scattering length is allowed to be a function of time t.37 On the other hand, in order tocontrol the dynamics of BEC in the trap, the trapping frequency in the elongated axis ω0 can also be afunction of time t.38 Therefore, the cubic nonlinearity parameter g and potential parameter V can betime- or spatiotemporal-dependent, and Eq. (1) can be used to describe the control and managementof BEC by properly choosing the two time- or spatiotemporal-dependent parameters.

In the present investigation, we consider a BEC with a time-dependent scattering length ina spatiotemporal-dependent parabolic background complex potential and gravitational field. For amore general investigation, the three terms of the external potential are taken to be time dependent:15

V (x, t) = k(t)x2 + λ(t)x + iγ (t). (2)

In potential (2),√|k(t)| = ω0/ω⊥ � 1 so that the strength of the magnetic trap k(t) expresses the

trapping frequency in the x direction; moreover, this parameter k(t) may be positive (confiningpotential) or negative (repulsive potential).39 It is important to point out that the quadratic term,k(t)x2, of potential (2) is the most physically relevant example of an external potential in the BECcase, giving the harmonic confinement of atoms by experimentally used magnetic traps. The linearterm of the potential λ(t)x may correspond to the gravitational field or some linear potentials.24 γ (t) isa small parameter related to the feeding (γ > 0) or loss (γ < 0) of atoms in the condensate resultingfrom the contact with the thermal cloud and three-body recombination.40 The GP equation (1)with potential (2) in which k(t) = γ = 0 may serve as a model to describe the dynamics ofthe condensate surface. In this case, λ is the force constant arising from linearizing the quadraticpotential energy near the surface and x is the coordinate normal to the surface such that the bulk ofthe condensate exists in the region x < 0.24

The experimental realization of solitary and periodic waves in BECs has stimulated intenseinterest in their properties, particularly their formation and propagation.16, 17, 19, 41–43 Due to the non-linearity arising from the interatomic interactions and due to the presence of an external potential inthe GP equation that describes the evolution of the solitary and periodic waves, these studies wereperformed either by solving the corresponding GP equation numerically or by using perturbativemethods. Interestingly enough, analytic methods have been used to find some analytical solitonicsolutions for the GP equation in one dimension with time- and space-dependent interatomic inter-actions and trapping potential strengths.11, 44, 45 It is clear that exact solutions allow for testing thevalidity of the GP equation at high densities and obtain the long-time evolution of the soliton where

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051501-3 Kengne et al. J. Math. Phys. 54, 051501 (2013)

numerical techniques may fail. They help to understand the formation, propagation, dynamics, andmanagement of a BEC solitary wave which are important for a number of BEC applications, likeatomic interferometry,26 and different kinds of quantum phase transitions,27 as well as in the contextof the nonlinear physics, including nonlinear optics and hydrodynamics. Motivated by the aboveworks and by the applications of exact solutions of GP equation in the theory of BECs, we aimin this paper, to explicitly present analytical solitary and periodic wave solutions for Eq. (1) withtime-dependent s-wave scattering length and spatiotemporal-dependent complex potential (2).

The rest of this paper is organized as follows. In Sec. II, we present the integrable conditionfor Eq. (1), and reduce Eq. (1) to a derivative cubic-quintic GP (DC-QGP) equation by engineeringthe imprinted phase and employing a modified lens-type transformation. In Sec. III, we apply asymmetry reduction method to the DC-QGP equation to obtain solitary wave-like solutions andJacobian elliptic function solutions of Eq. (1) with potential (2). Under the integrable condition, weinvestigate in Sec. IV the modulational instability and the dynamics of exact bright solitary wavesin one-dimensional BEC systems. Finally, we conclude by summarizing the main results in Sec. V.

II. DERIVATIVE CUBIC-QUINTIC GP EQUATION AND INTEGRABLE CONDITION

In order to derive the exact analytical expression which describes the propagation of solitarywaves and periodic elliptic function solutions, first we need to find the integrable condition forEq. (1) with potential (2). We begin by combining a modified lens-type transformation46 with aphase-imprint transformation as17, 18

ψ(x, t) = 1√(t)

�(X, T ) exp[η(t) + i f (t)

{x2 + σ (t)x

}+ iθ (x, t)], (3)

∂θ (x, t)

∂x= h(t) |ψ |2 + h(t), (4)

∂θ (x, t)

∂t= −i

h(t)

2

[ψ

∂ψ∗

∂x− ψ∗ ∂ψ

∂x

]+ h(t)h(t) |ψ |2 +

[h2(t) + ˜h(t)

]|ψ |4 , (5)

where θ (x, t) is the phase-imprint parameter, and (t), η(t), σ (t), f(t), h(t), h(t), ˜h(t), and T(t) areeight real valued functions of t, and X = x/(t). It is important to notice that the above transformationmakes sense only when h(t) is not the null function.

By demanding that

d f

dt+ 2 f 2 + k = 0, (6)

d

dt− 2 f = 0, (7)

dT

dt− 1

2(t)= 0, (8)

dσ

dt− σk + 2 f 2σ − λ

f= 0, (9)

dη

dt− γ = 0, (10)

h(t) + f (t)σ (t) = 0, (11)

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051501-4 Kengne et al. J. Math. Phys. 54, 051501 (2013)

Eq. (1) is converted into the perturbed DC-QGP equation with only time-dependent coefficientswithout external potential,

i∂�

∂T= −1

2

∂2�

∂ X2+ g0(t) |�|2 � + χ (t) |�|4 � − ig0d (t)

[�2 ∂�∗

∂ X+ |�|2 ∂�

∂ X

], (12)

where g0(t) = (g − fσh)exp [2η], χ (t) = 2−1(2χ0 + 2h + h2) exp [4η], and g0d(t) = hexp [2η].When working without the linear term of the external potential (2), i.e., when λ(t) ≡ 0, it is reasonableto use the trivial solution of Eq. (9), i.e., σ (t) ≡ 0. In general, the perturbed DC-QGP equation (12)with distributed coefficients is not integrable, and only a few solutions can be obtained under somerestrictions on its coefficients. In our analytical studies, we will restrict ourselves to the case ofconstant coefficients which certainly lead to a large class of solutions.

Henceforward, we work with

h(t) = α0 exp [−2η(t)] and ˜h(t) = 2α0 − α20

2exp [−4η(t)] − χ0, (13)

where α0 �= 0 and α0 are two arbitrary real numbers. With this choice of phase-imprint functionalparameters h(t) and ˜h(t), the coefficients of Eq. (12) become g0(t) = (gexp [2η] − α0fσ ), g0d(t)= α0, and χ (t) = α0. For the integrability of Eq. (12) with parameters (13), it is sufficient that g0(t)be constant. In other words, Eq. (12) with parameters (13) is integrable as soon as the nonlinearityparameter g(t) and the time-dependent coefficients of potential (2) satisfy the equation

(g exp [2η] − α0 f σ ) = g0 = const, (14)

g0 being any real constant. It is important to point out that f(t), σ (t), and η(t) depend only oncoefficients k(t), γ (t), and λ(t) of potential (2). Thus, the phase-imprint parameters h(t), h(t), and˜h(t) defined, respectively, by Eqs. (11) and (13), together with the integrability condition (14) dependonly on the functional parameters g(t), k(t), λ(t), and γ (t) of Eq. (1) with potential (2).

Under the integrability condition (14), Eq. (12) for �(X, T) becomes

i∂�

∂T= −1

2

∂2�

∂ X2+ g0 |�|2 � + α0 |�|4 � − iα0

[�2 ∂�∗

∂ X+ |�|2 ∂�

∂ X

], (15)

where g0 is an arbitrary real constant. It is important to point out that in our analysis, the usualintegrable cubic NLS equation cannot be obtained from Eq. (15). In fact, Eq. (15) will become theusual integrable cubic NLS equation if and only if the two last terms are absent, that is, if α0 = α0 = 0.It follows from Eq. (13) that the absence of the derivative term (last term of Eq. (15)) implies thath(t) = 0. Inserting this expression of h(t) in Eqs. (4) and (5) gives |ψ |4 = − 1

χ0

ddt

[xh(t) + h0(t)

],

where h0(t) is an arbitrary function of t. This proves the importance of the presence of the derivativeterm in Eq. (15) for our analytical analysis, and shows that without this term, the integrability ofEq. (15) will be destroyed. Thus, the virtue of the combined phase-imprint parametrization and themodified lens-type transformation (3)–(5) is that, with some algebraic manipulation, we not only findthe integrable condition (14) for Eq. (1) with potential (2), but also retrieve the well-known standardDC-QGP equation (15), also known as derivative cubic-quintic nonlinear Schrodinger (DC-QNLS)equation or derivative cubic-quintic Ginzburg-Landau (DC-QGL) equation. The complex Ginzburg–Landau equation (CGLE), probably the most celebrated nonlinear equation in physics, generallydescribes the dynamics of oscillating, spatially extended systems close to the onset of oscillations.The complex derivative Ginzburg-Landau (DCGL) equation arises as the envelope equation fora weakly subcritical bifurcation to counter-propagation waves, which is also of importance in thetheory of interaction behavior, including complete interpenetration as well as partial annihilation, forcollision between localized solutions corresponding to a single particle and to a two-particle state.47

Deissler and Brand47 showed numerically that the two derivative terms �2∂�*/∂X and |�|2∂�/∂Xcan significantly slow down the propagation speed of pulses and also cause nonsymmetry of pulses.

It is obvious that the problem of finding the time dependence of the parameters appearing inansatz (3) reduces to the solution of Riccati equation (6). With a special choice of the nonlinear

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051501-5 Kengne et al. J. Math. Phys. 54, 051501 (2013)

coefficient g(t), solution of Eq. (14), the system of Eqs. (6)–(10) has solution⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

k(t) = 14α2

1

[−1 + tanh

(1

2α1t)]

,

f (t) = 14α1

[−1 + tanh

(1

2α1t)]

,

η(t) = ∫ t0 γ (τ ) dτ,

(t) = 2−1(0)[1 + exp

(−α−11 t)]

,

σ (t) =(

exp (α1t) + exp α21−1α1

t) [

σ (0)2 + 2α1

∫ t0 λ(s) exp

(1α1

s − α1s)

ds],

g(t) = (g0

−1(t) + α0 f (t)σ (t))

exp[−2∫ t

0 γ (τ ) dτ],

T (t) = α12(0)

(1

exp(α−11 t)+1

+ ln[exp

(α−1

1 t)+ 1

]− 1+2 ln 22

),

(16)

where α1 is any real constant. For simplicity, we take T(0) = 0 and η(0) = 0. Requesting that T(t)→ + ∞ as t → + ∞ yields α1 > 0. It is obvious that k(t) given by Eq. (16) is negative; therefore,it corresponds to an attractive potential.

When k(t) ≡ k = const., the solutions of the system of Eqs. (6)–(10) are⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

f (t) = −√

k2 tan

[√2k (t + t∗)

],

(t) = (0)∣∣∣cos

√2k (t + t∗) cos−1

(t∗√2k

)∣∣∣ ,T (t) = (2k)−1/2 −2(0)

[cos2

(t∗√2k

)tan

√2k (t + t∗) − 2−1 sin

(2t∗√2k

)],

η(t) = ∫ t0 γ (s)ds,

σ (t) = tan−1[√

2k (t + t∗)] [√

2k

∫ t0 λ(s)ds + σ (0) tan

(t∗√2k

)],

(17)

where t∗ is any real number. This set of functional parameters is associated with a repulsivepotential. For simplicity, the initial conditions are T(0) = η(0) = 0. In this case of constant k,we may use the same expression for g(t) as in Eq. (16). Rescaling (17) signals singularities attn = (2n + 1)π/(2

√2k ) − t∗, where n is any integer. The singularity at tn is understandable; indeed,

in the present case, the “chirp” initial condition means the existence of a current at the initial momentof time. Due to the quadratic potential, this current periodically changes direction. The change ofthe current direction is accompanied by a phase singularity.48

Our last example corresponds to a complicated periodically time-modulated potential parameterk(t), given as

k(t)=−2A20+4A0 B0m2sn2 (μt, m)−2B2

0 m4sn4 (μt, m)+B0μm2sn (2μt, m)[1−m2sn4 (μt, m)

],

(18)where A0, B0, and μ are three arbitrary real constants with |A0| + |B0| > 0, 0 ≤ m ≤ 1, andsn(μt, m), the sine Jacobian elliptic function with modulus m. Depending on the values of A0 andB0, the potential functional parameter (18) can be positive or negative. As a particular solution ofRiccati equation (6) with potential parameter (18), we have

f (t) = A0 − B0m2sn2 (t, m) . (19)

With f(t) given by (19), one can, with more or less difficulty, solve Eqs. (7)–(9).For computational purpose, for both sets of functional parameters (16) and (17), we shall use

the periodically time-modulated gain/loss γ (t) = γ 0 + γ 1cos ω0t, where γ 0, ω0, and γ 1 are realconstants.49

Let us remark from Eqs. (16) and (17) that γ (t) obviously influences the number of atoms N in the

condensate. In fact, N = ∫R |ψ |2 dx = ∫

R |u|2 dx = exp(

2∫ t

0 γ (s)ds)

× ∫R |�(X, T )|2 d X which,

in the case of rescaling (16), grows exponentially if∫ t

0 γ (s)ds > 0 and decreases exponentially if∫ t0 γ (s)ds < 0.

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051501-6 Kengne et al. J. Math. Phys. 54, 051501 (2013)

III. SYMMETRY REDUCTION METHOD FOR SOLITARY WAVE-LIKE AND JACOBIANELLIPTIC FUNCTION SOLUTIONS UNDER THE INTEGRABLE CONDITION (14)

In this section, we apply the symmetry reduction method to find solitary wave-like and Jacobianelliptic function solutions of the GP equation (1) with potential (2) under the integrable condition(14). Here, we wish to find a solution on the non-vanishing continuous wave (cw) background�cw(X, T ) = ρcw exp (iϕcw), where ϕcw = q0 X − (

q20 + 2g0ρ

2cw + 2α0ρ

4cw

)T/2. We start with a

useful transformation.

A. Reduction of Eq. (15) to an ODE for Jacobian elliptic functions by a symmetryreduction method

In order to convert the DCGL equation (15) into an ordinary differential equation (ODE) forJacobian elliptic functions, we look for its traveling wave solutions

�(X, T ) = [ρcw + ρ(ξ )] exp

[iϕcw − iωT + i

∫ ξ

0θ (z′)dz′

], (20)

where ξ = μX − υT. Substitution of Eq. (17) into Eq. (15) yields

θ (ξ ) = υ − μq0

μ2− α0

μζ + C0

ζ, (21)

(dζ

dξ

)2

= αζ 4 + 4βζ 3 + 6γ ζ 2 + 4δζ + ε = R(ζ ), (22)

where {ζ (ξ ) = [ρcw + ρ(ξ )]2 , α = 4(α2

0 + 2α0)/(3μ2), β = g0/μ2, δ = δ0,

ε = −4C20 , γ = −2(3μ4)−1

[2μ2

(g0ρ

2cw + μα0C0 + ω

)+ (μq0 − υ)2],

(23)

C0 and δ0 being two arbitrary real constants. With the help of ansatz (20), we have converted theDCGL equation (15) to the ODE (22) for elliptic function solutions of the GP equation (1) withexternal potential (2).

The solutions to Eq. (22) with coefficients (23) are expressed in terms of the Weierstrass ellipticfunction ℘ as50–52

ζ (ξ ) = ζ0 +√

R(ζ0) d℘(ξ ;g2,g3)dξ

+ 12 R′(ζ0)[℘(ξ ; g2, g3) − 1

24 R′′(ζ0)] + 124 R(ζ0)R′′′(ζ0)

2[℘(ξ ; g2, g3) − 1

24 R′′(ζ0)]2 − 1

48 R(ζ0)R(I V )(ζ0), (24)

where ′ = ddζ

, ζ 0 is any real constant, g2 and g3 are invariants of the Weierstrass elliptic function℘(ξ ; g2, g3) related to the coefficients of the polynomial R(ζ ) by50

g2 = αε − 4βδ + 3γ 2, g3 = αγ ε + 2βγ δ − αδ2 − γ 3 − εβ2. (25)

It is obvious that if ζ 0 is a simple root of the polynomial R(ζ 0), then solution (24) will take thesimple form

ζ (ξ ) = ζ0 + R′(ζ0)

4[℘(ξ ; g2, g3) − 1

24 R′′(ζ0)] . (26)

In this paper, we are interested in positive (nonnegative, if C0 = 0) solitary wave-like solutions andbounded elliptic wave solutions of Eq. (22) with coefficients (23). Whittaker and Watson50 (also seeRef. 51) used the so-called discriminant of ℘(ξ ; g2, g3) defined in terms of the invariants g2 and g3

as

� = g32 − 27g2

3 (27)

to classify the behavior of the solution ζ (ξ ) of Eq. (22) with coefficients (23). The conditions

� �= 0 or � = 0, g2 > 0, g3 > 0 (28)

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051501-7 Kengne et al. J. Math. Phys. 54, 051501 (2013)

lead to periodic solutions, whereas the conditions

� = 0, g2 > 0, g3 ≤ 0 (29)

are associated with solitary wave-like solutions.

B. Solitary wave-like solutions

As we just pointed out, solitary wave-like solutions of Eq. (22) with coefficients (23) correspondto conditions (29). Under conditions (29), a solitary wave-like solution which is associated to anysimple root ζ 0 of the polynomial R(ζ ) is given by51

ζ (ξ ) = ζ0 + R′(ζ0)

4[e1 − 1

24 R′′(ζ0) + 3e1 cosh−2(√

3e1 ξ)] , (30)

where e1 = 3√−g3. Because C0 and δ0 are arbitrary constants, working under condition δε = 0 does

not impose any restriction on the DCGL equation (15). It is also obvious that a large set of solutionsof (22) can be found by choosing δ = ε = 0 in Eq. (22), thus simplifying the evaluation of theconditions of positivity of ζ (ξ ). In this special and interesting case, we have g2 = 3γ 2, g3 = −γ 3,� = 0, e1 = 3

√−g3 = γ > 0, and R(ζ ) = (αζ 2 + 4βζ + 6γ

)ζ 2. Therefore, the polynomial R(ζ )

admits two simple roots ζ0± = (2α)−1(−4β ±

√(4β)2 − 24αγ

), if 2β2 − 3αγ > 0. Inserting ζ 0 ±

into Eq. (30) yields

ζ±(ξ ) =γ[±4β + 2

√4β2 − 6αγ

] [3γ 2 + (

γ 2 − 1)

cosh2

(√3γ 3

2 ξ

)]2

[∓αγ 3 +

(±4β2 ∓ αγ (5 + γ )2 + 2β

√4β2 − 6αγ

)cosh2

(√3γ 3

2 ξ

)] . (31)

In order to write down the conditions for nonnegative solutions (31), we distinguish the two cases,γ = 1 and γ �= 1.

(1) If γ = 1 and 2β2 − 3αγ > 0, then for nonnegative solutions (31), it is necessary andsufficient that 4β3 − 3 (1 ± 6) αβ ± (

2β2 ∓ 9α)√

4β2 − 6α > 0 and either 18α ∓ 2β2

∓ β√

4β2 − 6α > 0 or 18α ∓ 2β2 ∓ β√

4β2 − 6α < 0 < −37α ± 4β2 ∓ 2β√

4β2 − 6α.(2) If 0 < γ �= 1 and 2β2 − 3αγ > 0, then for positive solutions (31), it is necessary and sufficient

that the following three conditions be satisfied simultaneously:

(i)(γ 2 − 1

) [8 (1 ± 1) β3 − 2

(6 ± (5 + γ )2)αβγ ± (

4 (1 ± 1) β2 ∓ αγ (5 + γ )2)√4β2 − 6αγ

]> 0;

(ii) either γ 2 − 1 > 0 or γ 2 − 1 < 0 < 1 − 4γ 2;(iii) either αγ (5 + γ )2 ∓ 4β2 ∓ 2β

√4β2 − 6αγ > 0 or αγ (5 + γ )2 ∓ 4β2

∓ 2β√

4β2 − 6αγ < 0 < −αγ (5 + γ )2 − αγ 3 ± 4β2 ± 2β√

4β2 − 6αγ .

It should be noted that the double root ζ 0 = − β/α of the polynomial R(ζ ) which correspondsto the case γ = β2/(3α) does not give any nonnegative solitary wavelike solutions.

In the following, we consider three examples to demonstrate the dynamics of solitary waves in1D BEC systems with different kinds of magnetic trap strength k(t) and periodically time-modulatedgain/loss γ (t) = γ 0 + γ 1cos ω0t. For these examples, we use α0 = 0, which corresponds to the caseof absent quintic term in Eq. (12).

First, we consider the time-independent strength of the magnetic trap k(t) given in system(16). According to this system, we have η(t) = γ0t + ω−1

0 γ1 sin ω0t . In Fig. 1(a), we plot η(t)corresponding to the feeding/loss functional parameter γ (t) = 0.1 + cos (1.1t) (red curve) and γ (t)= − 0.1 + cos (1.1t) (blue curve). The dynamics of a solitary wave-like solution is shown inFigs. 1(b) and 1(c) and in Figs. 2(a) and 2(b) with the above two feeding/loss functional parameters.The feeding/loss parameter γ (t) has an important effect on modulating the amplitude of the solutionψ(x, t) of the GP equation (1) with potential (2). It is seen that η(t) is periodically oscillating along the

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051501-8 Kengne et al. J. Math. Phys. 54, 051501 (2013)

FIG. 1. The dynamics of a solitary wave in a modulated complex spatiotemporal complicated potential and time-dependents-wave scattering length. These plots are generated with solution (31) and parameters γ = 1, g0 = − 1, μ = 1, α0 = 0.2, υ

= 0.5, (0) = 2, and two lens transformation parameters η(t) shown in (a). Plots (b) and (c) are, respectively, realized withγ (t) = 0.1 + cos (1.1t) (red curve in (a)) and γ (t) = − 0.1 + cos (1.1t) (blue curve in (a)).

t-axis with increasing or decreasing aspect. The corresponding ψ(x, t) will be an explicit breathingsolitary wave-like solution, and its density |ψ(x, t)|2 is depicted as a function of distance x and time t inFigs. 1(b) and 1(c) with γ (t) = 0.1 + cos (1.1t) and γ (t) = − 0.1 + cos (1.1t), respectively. For theseplots, we used solution (31) with parameters γ = 1, g0 = − 1, μ = 1, α0 = 0.2, υ = 0.5, (0) = 2.From Figs. 1(b) and 1(c) it can be seen that, with an increasing (decreasing) functional parameterη(t), the solitary wave has an increase (decrease) in the peak value and a broadening (compression)of its width. Via the parameter α0 (see expression (13) for h(t)), Fig. 2 shows the effect of thephase-imprint parameter h(t) on the dynamics of solitary waves in 1D BEC described by Eqs. (1)and (2). Here, plots (a) and (b) are generated at time t = 3 with periodically oscillating increasing anddecreasing η(t), respectively. It is seen from these plots that with both the increasing and decreasingfunctional parameter η(t), the solitary wave has increasing peak value and a broadening in widthwhen |α0| increases.

It is important to notice that the use of the periodically time-modulated gain/loss γ (t) = γ 0

+ γ 1cos ω0t corresponds to a temporal periodic modulation of the s-wave scattering length (see theexpression of g(t) in Eq. (16)).35, 36

Second,we consider the time-independent strength of the magnetic trap k which was used inthe creation of bright BEC solitons in the case of harmonic potential.20 In that experiment, ω = 2π i× 70 Hz and ω⊥ = 2π × 710 Hz, so k = 2κ2 (κ � 0.05). For this value of k, we use system (17) forlens transformation functional parameters. With the same parameters as in Fig. 1, the dynamics of aone-solitary wave in a modulated complex spatiotemporal complicated potential is shown in Fig. 3where plots (a) and (b) are associated with γ (t) = 0.1 + cos (1.1t) and γ (t) = − 0.1 + cos (1.1t),respectively. To avoid singularity and guarantee the variation of T(t) from zero to infinity, we focusour study on the case where t goes from zero to t0 = π /4κ − t∗. In Fig. 3, we use t∗ = 2. FromFigs. 3(a) and 3(b) it is seen that, with increasing (decreasing) functional parameter η(t) obtained

FIG. 2. Effect of the functional phase-imprint parameter h(t) (via the phase-imprint parameter α0) on the dynamics of solitarywaves in 1D BEC described by Eqs. (1) and (2). We use solution (31) and the same parameters as in Fig. 1 to realize plots(a) and (b). Here, we use (a) γ (t) = 0.1 + cos (1.1t) and (b) γ (t) = − 0.1 + cos (1.1t).

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051501-9 Kengne et al. J. Math. Phys. 54, 051501 (2013)

FIG. 3. The dynamics of a solitary wave in a modulated complex spatiotemporal complicated potential and time-dependents-wave scattering length associated with Eq. (31) in the case of a time-independent strength of the magnetic trap with thesame parameters as in Fig. 1 and k = 0.005. (a) γ (t) = 0.1 + cos (1.1t); (b) γ (t) = − 0.1 + cos (1.1t).

with γ (t) = 0.1 + cos (1.1t) (with γ (t) = − 0.1 + cos (1.1t)), the solitary wave in Fig. 3(a) (inFig. 3(b)) has an increase (decrease) in the peak value and a broadening (compression) in its width.Plots of Fig. 3 show that the solitary waves have a breathing behavior.

Finally, we consider the temporal periodic modulation of strength of the magnetic trap k(t) (18)given in terms of Jacobian elliptic function with parameters A0 = − 0.02, B0 = 0.1, m = 0.98,μ = 2. For this third example, we use solution (31) with parameters g0 = μ = 1, γ = 2, (0) = 1,υ = 0.75, and α0 = 0.495 to demonstrate the dynamics of a bright one-solitary wave and the effectof the phase-imprint parameter h(t) on the dynamics of solitary waves in Figs. 4 and 5, respectively.Plots of Fig. 4 are generated with constant feeding/loss parameter (a) γ (t) = 0.1 + 0.098 cos 1.1t,(b) γ (t) = − 0.1 + 0.098 cos 1.1t, and (c) γ (t) = − 0.31 + 0.098 cos 1.1t. From the plots of thisfigure, it is seen that: (i) the solitary wave has a compression in its width when propagating; (ii) inthe presence of feeding of atoms in the condensate, the solitary wave propagates with an increase inthe peak value; the same situation also happens in the presence of loss of atoms in the condensate,when the loss parameter is very small [see plot (b) which has been obtained with γ (t) = − 0.001];(iii) when the loss of atoms in the condensate is more significant, the solitary wave propagates, as itis seen from plot (c), with a decrease in the peak value; (iv) Fig. 4 shows a solitary wave propagatingon an oscillating trajectory and the amplitude of the wave density also oscillates; the oscillatoryaspect of the wave is due to the temporal periodic modulation of both the loss/feeding parameterγ (t) and the strength of the magnetic trap k(t).

In order to evidence the effect of the functional phase-imprint parameter h(t) on the dynamicsof solitary waves, we have plotted in Fig. 5 the density |ψ(x, t)|2 at a given time as a functionof the spatial variable x and the phase-imprint parameter α0. Here, we use the same parametervalues for k(t) and solution (31) as in Fig. 4. A constant feeding/loss parameter γ has been used:

FIG. 4. The dynamics of a bright solitary wave in a spatiotemporal complex potential with a temporal periodic modulationof the strength of the magnetic trap k(t) given by Eq. (18) with parameters A0 = − 0.02, B0 = 0.1, m = 0.98, μ = 2.(a) γ (t) = 0.1 + 0.098 cos 1.1t, (b) γ (t) = − 0.1 + 0.098 cos 1.1t, and (c) γ (t) = − 0.31 + 0.098 cos 1.1t. These plotsare associated with the density |ψ(x, t)|2 obtained from solution (31) and with parameters g0 = μ = 1, γ = 2, (0) = 1,υ = 0.75, and α0 = 0.495.

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051501-10 Kengne et al. J. Math. Phys. 54, 051501 (2013)

FIG. 5. Effect of the functional phase-imprint parameter h(t) on the dynamics of solitary waves in the case of a temporalperiodic modulation of the strength of the magnetic trap k(t) given by Eq. (18) with the same parameters values as inFig. 4. (a) γ (t) = 0.1 + 0.098 cos 1.1t, (b) γ (t) = − 0.31 + 0.098cos 1.1t. The two plots are realized with density |ψ(x, t)|2obtained from solution (31) at time t = 5 and with parameters g0 = μ = 1, γ = 2, (0) = 1, υ = 0.75.

(a) γ (t) = 0.1 + 0.098 cos 1.1t and (b) γ (t) = − 0.31 + 0.098 cos 1.1t. The two plots show thebehavior of the solitary wave at time t = 5. It is seen from these plots that for both (a) feeding and (b)loss of atoms in the condensate, the peak value of the solitary wave obtained from solution (31) withγ �= 1 decreases and its width is compressed as |α0| increases. Thus, one can use the phase-imprintparameter α0 to readjust the peak value and the width of the solitary wave.

C. Periodic Jacobian elliptic function solutions

In this section, we work under conditions (28) leading to periodic Jacobian elliptic functionsolutions of Eq. (22). For periodic solution, we shall use the Weierstrass elliptic function50–52

℘(ξ ; g2, g3) = e3 + e1 − e3

sn2(√

e1 − e3 ξ, m) , (32)

where e1 ≥ e2 ≥ e3 are the roots of the equation

4s3 − g2s − g3 = 0. (33)

For simplicity, we restrict ourselves to solutions associated with simple roots of the polynomial R(ζ ).In the case of the simple root ζ 0 of R(ζ ), Eqs. (26) and (32) give the periodic solution

ζ (ξ ) = 2 (e1 − e3) ζ0 + [αζ 3

0 + 4βζ 20 + (2e3 + 5γ ) ζ0 + 2δ

]sn2(√

e1 − e3 ξ, m)

2 (e1 − e3) + [2e3 − γ − αζ 2

0 − 2βζ0]

sn2(√

e1 − e3 ξ, m) , (34)

with m = e2−e3e1−e3

. Solution (34) is subjected to the non-negativity condition. One can easily verify

that e1 = 3g3+g2√

3g2

4g2and e2 = e3 = − 3g3

2g2are the only solutions of Eq. (33), if � = 0. We then have

m = 0 and Eq. (34) is just a trigonometric solution. Thus, periodic Jacobian elliptic function solutionscan be obtained only with � �= 0; moreover, we work only with � > 0. Indeed, only positive �

guarantees real solutions of Eq. (33).53 The simplest example of solution (34) is obtained withε = 0 and δ �= 0, and corresponds to the simple root ζ 0 = 0. This solution reads

ζ (ξ ) = 2δsn2(√

e1 − e3 ξ, m)

2 (e1 − e3) + [2e3 − γ ] sn2(√

e1 − e3 ξ, m) , m = e2 − e3

e1 − e3. (35)

For a positive solution (35), it is necessary and sufficient that either δ [2e3 − γ ] > 0 and(e1 − e3) (2e3 − γ ) > 0 or δ [2e3 − γ ] < 0 and (2e1 − γ )/(2e3 − γ ) < 0.

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051501-11 Kengne et al. J. Math. Phys. 54, 051501 (2013)

IV. MODULATIONAL INSTABILITY AND DYNAMICS OF EXACT BRIGHT SOLITARY WAVESOLUTION UNDER THE INTEGRABLE CONDITION

We start this section with the modulational instability of the DC-QGP equation (15). We lookfor perturbed cw solutions of the form

�(X, T ) = (ρcw + εb) exp

[i

{q0 X − q2

0 + 2g0ρ2cw + 2α0ρ

4cw

2T

}], (36)

analyzing the O(ε) terms:

b(X, T ) = b1 exp [i (Q X − �T )] + b∗2 exp

[−i(Q X − �∗T

)]. (37)

The dispersion relation connecting the wave number Q and frequency � of the perturbation is(� + Q

(q0 − ρ2

cwα0))2 = Q2

[1

4Q2 + ρ2

cw

(g0 + ρ2

cw

(α2

0 + 2α0))]

. (38)

Dispersion relation (38) implies that the instability region for the DC-QGP equation (15) appearsfor perturbation wave numbers Q2 < −4ρ2

cw

[g0 + ρ2

cw

(α2

0 + 2α0)]

, and, in particular, only forg0 + ρ2

cw

(α2

0 + 2α0)

< 0. This last inequality means that either g0 < 0 or α20 + 2α0 < 0.

We now use the ODE for Jacobian elliptic functions, (22) and (23), to find under the integrablecondition (14) the exact bright solitary wave solution of the GP equation (1) with complex poten-tial (2). Seeking the bright solitary wave solution of Eq. (22) with coefficients (23) in the formζ (ξ ) = A + Bcosh − 1(Cξ ) and going back to the initial variable x and t, we find

|ψ(x, t)|2 =−3μ2

√6(β2 − γ α

)exp [2η(t)]

4(α20 + 2α0)(t)

×⎡⎣ g0

μ2√

6(β2 − γ α

) ± 1

cosh(

3 |μ|√

γ α−β2

2(α20+2α0)

(μ x

(t) − υ∫ t

0ds

2(s) ds))⎤⎦ , (39)

if

β2 − γ α > 0, α20 + 2α0 < 0. (40)

For a positive solution (39), it is necessary and sufficient that

g0

μ2√

6(β2 − γ α

) > 1 for sign “ − ” and g0 ≥ 0 for sign “ + ”. (41)

Condition α20 + 2α0 < 0 is just the necessary condition of modulational instability. Because g0

> 0 is nonnegative as one can see from Eq. (41), bright solitary wave propagates in both regionsof modulational stability (when g0 + ρ2

cw

(α2

0 + 2α0)

> 0) and of modulational instability (wheng0 + ρ2

cw

(α2

0 + 2α0)

< 0). For solution (39), we shall distinguish two cases, namely, g0 = 0 and g0

�= 0.

(i) Case g0 = 0. Going back to Eq. (14), it is seen that g0 = 0 just means that the nonlinearityparameter is taken to be g(t) = α0f(t)σ (t)exp [ − 2η(t)]. In this case of g0 = 0, (39) becomes

|ψ(x, t)|2 =−3μ2

√6(β2 − γ α

)exp

[2∫ t

0 γ (s)ds + 2γ (0)]

4(α20 + 2α0)(t) cosh

(3 |μ| μ

√γ α−β2

2(α20+2α0)

(1

(t) x − υμ

∫ t0

ds2(s) ds

)) , (42)

where (t) is a positive solution of Eq. (7); for simplicity, we take T(0) = 0 for the solutionof Eq. (8). The right-hand side of Eq. (42) is just the amplitude of a bright one-solitary wavesolution for the GP equation (1) with potential (2). From density (42) we conclude that: (i) the

amplitude of a bright solitary wave is proportional to −1(t) exp[2∫ t

0 γ (s)ds + 2γ (0)], while

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051501-12 Kengne et al. J. Math. Phys. 54, 051501 (2013)

FIG. 6. The density of bright solitary waves given by Eq. (39) for |ψ + (x, t)|2 with the temporal periodic modulation ofthe strength of the magnetic trap k(t) given by Eq. (18) with the parameters A0 = 0.1, B0 = 0.2, m = 0.98, and μ = 2. Theparameters chosen to generate these plots are (0) = 1, γ (0) = 0, g0 = 1, ρcw = 1.1, q0 = 0.1, ω = − 4, υ = 0.75, μ = 1,C0 = 1, γ (t) = γ 0 + 0.098 cos (1.1t): (a) γ 0 = 0.1, α0 = 0.1, α0 = −1; (b) γ 0 = 0.1, α0 = 0.1, α0 = −0.1; (c) γ 0 = − 0.1,α0 = 0.1, α0 = −1; (d) γ 0 = − 0.1, α0 = 0.1, α0 = −0.1.

its width is proportional to (t), so the total number of BEC atoms will be∫ +∞−∞ |ψ(x, t)|2 dx

= π2

√− 3

α20+2α0

exp[2∫ t

0 γ (s)ds + 2γ (0)]. Therefore, with increasing (decreasing) parameter

(t), the solitary wave has a decrease (increase) in peak value and a compression (broadening)in width. Thus, Eq. (42) can be used to describe the broadening (compression) of a brightsolitary wave when (t) decreases (increases) with time; (ii) the centre of the bright solitarywave is κ(t) = υμ−1(t)

∫ t0

ds2(s) ds; by using system (6)–(11), one easily shows that the centre

κ of a bright solitary wave satisfies the following equation:

d2κ

dt2+ 2k(t)κ = 0. (43)

Equation (43) means that the centre of mass of the macroscopic wave packet behaves like aclassical particle, and allows one to manipulate the motion of bright solitary waves in BECsystems by controlling the quadratic part of the external complex potential.

(ii) Case g0 �= 0. In this case, Eq. (14) gives g(t) = −1(t) [g0 + α0(t) f (t)σ (t)]

exp[−2∫ t

0 γ (s)ds − 2γ (0)]

and Eq. (39) defines two densities, |ψ − (x, t)|2 and |ψ + (x, t)|2. As

in the case of g0 = 0, the velocity for the corresponding solitary waves still satisfies Eq. (43), but

the amplitude of such solitary waves is not proportional to −1(t) exp[2∫ t

0 γ (s)ds + 2γ (0)],

which shows in Fig. 6 the density |ψ + (x, t)|2 of bright solitary waves. To realize this figure,we use the temporal periodic modulation of the strength of the magnetic trap k(t) given byEq. (18) with parameters A0 = 0.1, B0 = 0.2, m = 0.98, and μ = 2. The top panels [plots(a) and (b)] and the bottom panels [plots (c) and (d)] are generated with the periodic mod-ulation feeding parameter γ (t) = 0.1 + 0.098 cos (1.1t) and the modulation loss parameterγ (t) = − 0.1 + 0.098 cos (1.1t), respectively. The left panels show the evolution of brightsolitary waves in the modulational instability region g0 + ρ2

cw

(α2

0 + 2α0)

< 0, while those ofthe right panels show the evolution of bright solitary waves in the modulational stability regiong0 + ρ2

cw

(α2

0 + 2α0)

> 0. From these plots we can see that the density of the solitary wave has

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051501-13 Kengne et al. J. Math. Phys. 54, 051501 (2013)

strong variations along its propagation, and thus shows a snaking behavior. The variations aregetting more and more significant as the wave is propagating, and the amplitude of the densityalso oscillates. In the case of feeding (loss) of atoms in the condensate, the solitary wavespropagate with increasing (decreasing) peak value.

Remark: It is important to notice that the amplitude of the density is more significant when thesolitary wave propagates in the modulational stability region g0 + ρ2

cw

(α2

0 + 2α0)

> 0; this is easilyseen from the right panels of Fig. 6. This does not contradict the propagation of a bright solitarywave in the modulational stability region; this is just due to the fact that the factor −(3/4)μ2(α2

0

+ 2α0)−1√

6(β2 − γ α

)of the wave amplitude is more significant when g0 + ρ2

cw

(α2

0 + 2α0)

> 0.

V. CONCLUSION

In conclusion, we have studied the cubic-quintic GP equation which describes the dynamics ofthe BEC matter waves with time-dependent s-wave scattering length and time-dependent complexcomplicated potential. By combining a modified lens-type transformation with a phase-imprinttransformation, we reduced the one-dimensional cubic-quintic GP equation (1) with the externalpotential (2) to the derivative cubic-quintic Ginzburg-Landau equation (15) under the integrablecondition (14), and the solutions of the cubic-quintic GP equation are thus constructed via thoseof the DC-QGL equation. With the help of the F-expansion method, we gave explicit analyticalperiodic and solitary wave solutions. Under the integrable condition (14), we derived the exactanalytical expression for modulational instability and a bright one-solitary wave embedded in a cwbackground. By applying different time-modulations to the functional feeding/loss parameter andby using various types of time-independent strengths of the magnetic trap k(t), we showed that thesolitary waves can exhibit breathing and snaking behaviors. Furthermore, we have investigated theeffect of the phase-imprint parameter, responsible for the integrability condition, on the dynamics ofsolitary waves and found that this phase-imprint parameter can be used to manage the peak valuesand the width of the solitary wave. Different solitary wave solutions found in this work imply thatcontrol of the strength k(t) of the magnetic trap, the functional feeding/loss parameter γ (t), and thefunctional phase-imprint h(t) allows us to manipulate the motion of solitary waves in BEC systems.

ACKNOWLEDGMENTS

This work was supported by the Chinese Academy of Sciences Visiting Professorship forSenior International Scientists, the NKBRSFC under Grant Nos. 2011CB921502, 2012CB821305,2009CB930701, and 2010CB922904, the NSFC under Grant Nos. 10934010 and 60978019, and theNSFC-RGC under Grant Nos. 11061160490 and 1386-N-HKU74810.

The first author, E.K., dedicates this work to his father-in-law, Papa SADO Jean Le Prince.

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