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Compensation Process and Generation of ChirpedFemtosecond Solitons and Double-kink Solitons inBose-Einstein Condensates with Time-dependentAtomic Scattering Length in a Time-varyingComplex PotentialEmmanuel Kengne ( [email protected] )
Zhejiang University https://orcid.org/0000-0002-1197-6194Ahmed Lakhssassi
University of Quebec at Outaouais Alexandre-Tache Library: Universite du Quebec en OutaouaisBibliotheque Alexandre-Tache
Research Article
Keywords: Bose-Einstein condensate, Gross-Pitaevskii equation, Compensation process, Chirpedfemtosecond solitons, Double-kink solitons
Posted Date: March 12th, 2021
DOI: https://doi.org/10.21203/rs.3.rs-282232/v1
License: This work is licensed under a Creative Commons Attribution 4.0 International License. Read Full License
Compensation process and generation of chirped femtosecond solitons and
double-kink solitons in Bose-Einstein condensates with time-dependent atomic
scattering length in a time-varying complex potential
Emmanuel Kengne1∗ and Ahmed Lakhssassi21 School of Physics and Electronic Information Engineering,Zhejiang Normal University, Jinhua 321004, China and
2 Departement d’informatique et d’ingenierie, Universite du Quebec en Outaouais. 101 St-Jean-Bosco,Succursale Hull, Gatineau(PQ) J8Y 3G5, Canada
(Dated: February 26, 2021)
We consider the one-dimensional (1D) cubic-quintic Gross–Pitaevskii (GP) equation, which gov-erns the dynamics of Bose–Einstein condensate (BEC) matter waves with time-varying scatteringlength and loss/gain of atoms in a harmonic trapping potential. We derive the integrability con-ditions and the compensation condition for the 1D GP equation and obtain, with the help of acubic-quintic nonlinear Schrodinger (NLS) equation with self-steepening and self-frequency shift,exact analytical solitonlike solutions with the corresponding frequency chirp which describe the dy-namics of femtosecond solitons and double-kink solitons propagating on a vanishing background.Our investigation shows that under the compensation condition, the matter wave solitons maintaina constant amplitude, the amplitude of the frequency chirp depends on the scattering length, whilethe motion of both the matter wave solitons and the corresponding chirp depend on the externaltrapping potential. More interesting, the frequency chirps are localized and their feature dependson the sign of the self-steepening parameter. Our study also shows that our exact solutions canbe used to describe the compression of matter wave solitons when the absolute value of the s-wavescattering length increases with time.
Keywords: Bose-Einstein condensate; Gross-Pitaevskii equation; Compensation process;Chirped femtosecond solitons; Double-kink solitons
PACS numbers: 03.75.Kk, 03.75.Lm, 42.65.Sf
I. INTRODUCTION
First realized experimentally in 1995 for rubidium [1], lithium [2, 3] and sodium [4], Bose–Einstein condensates is asignificant, rapidly growing research area at the forefront of contemporary physics. Its attraction lies in its ability todisplay phenomena well-known from other fields, such as condensed-matter physics, the nonlinear physics, includingnonlinear optics and hydrodynamics, in a clear and unambiguous manner, allowing accurate and powerful theory tobe applied to “real-life” experimentally relevant situations. BECs provide unique opportunities for exploring quantumphenomena on a macroscopic scale. Bose–Einstein condensates can be thought of a macroscopic number of particlessharing the same wave function, and the atoms have similar coherence properties to a laser. The development ofthe atom laser promises state-of-the-art measurement devices, but more fundamentally, BECs allow us to model thequantum fields that underlie nearly all of modern physics in systems where we have unparalleled experimental control.Predicting the behavior of these systems is a major theoretical challenge, and designing improved atomic sources is acritical requirement for future experiments.At ultralow temperatures, the properties of weakly interacting bosonic gases trapped in a potential V is usually
described by the time-dependent, nonlinear, mean-field Gross–Pitaevskii equation for the order parameter Ψ(r, t) atposition r and time t [5–11]
i~∂Ψ(r, t)
∂t=
[− ~
2
2m∇2 + V (r, t) + g0 |Ψ(r, t)|2 + χ0 |Ψ(r, t)|4
]Ψ(r, t). (1)
Here, g0 = 4~2as(t)/m and χ0 are the strengths of time-dependent two-body and three-body interatomic interactions,respectively, ~ is the Planck constant, m is the atomic mass of the single bosonic atom and as(t) is the time-dependents-wave scattering length which can be tuned to any desired value by using the Feshbach resonance technique. In this
∗Corresponding author’s Email: [email protected]
2
work, we consider a cigar-shaped harmonic trapping potential with the elongated axis in the x−direction as used inexperiment [12], having the form V (r, t) = m
2
[ω2x2 + ω2
⊥
(y2 + z2
)]+ iγ in which ω << ω⊥ so that the variation
of the profile of the order parameter can be expected to be in the elongated direction (here, γ is a small parameteradded to take into account the loss or the gain of atoms in the condensate). When compared with the strength ofthe two-body interaction, the strength of the three-body interaction is usually very small as pointed out by Gammalet al. [10]. Accordingly we consider for our study that χ0 = χ0g0, where 0 ≤ χ0 < 1 is a real parameter. Therefore,the three-body interactions can be, as the two-body interactions, controlled by the tuning of s-wave scattering length[10]. By introducing the transformation
Ψ(r, t) =1√
2πaBa⊥exp
[−iω⊥t−
y2 + z2
2a2⊥
] (x
a⊥, ω⊥t
),
we obtain that in the physically important case of the cigar-shaped BECs, the GP Eq. (1) is reduced into the one-dimensional (1D) distributed cubic-quintic nonlinear Schrodinger (NLS) equation with an external harmonic trappingpotential
i∂ψ
∂t+
1
2
∂2ψ
∂x2+ g |ψ|2 ψ + χ0g |ψ|4 ψ −
(αx2 + iγ
)= 0, (2)
where aB is the Bohr radius. In Eq. (2), t and x are the temporal and spatial coordinates measured in 1/ω⊥
and a⊥ =√~/ (mω⊥), respectively. g = −2as/(3aB) is the nonlinearity parameter which is negative (positive) for
repulsive (attractive) interatomic interactions. α = ±ω2/(2ω2⊥) is the strength of the magnetic trap and is either
positive (confining potential) or negative (repulsive potential). Because |α| << 1, parameter α expresses the trappingfrequency in the x− direction [14]. γ is a small parameter related to the feeding (γ > 0) or loss (γ < 0) of atomsin the condensate resulting from the contact with the thermal cloud and three-body recombination [15, 16]. In thisstudy, the s-wave scattering length is allowed to be a function of time t [17] so that parameter g of the nonlinearityand strength α of the magnetic trap are be time-dependent; also, parameter γ of the gain/loss of atoms is assumedto be time-dependent. Therefore, Eq. (2) can be used to describe the control and management of BECs by properlychoosing these three time-dependent parameters g, α, and γ and the time-independent parameter χ0.The purpose of this paper is to study via the GP Eq. (2) the generation of chirped femtosecond solitons and
double-kink solitons in BECs with time-dependent atomic scattering length in a time-varying complex potential.Due to their wide applications in many different areas of physics and engineering such as ultrahigh-bit-rate opticalcommunication systems, ultrafast physical processes, infrared time-resolved spectroscopy, as well as optical samplingsystems, ultrashort (femtosecond) pulses have been extensively studied over the past several years [18–21]. It has beenestablished that physical systems whose dynamics can be described by the NLS equation with higher-order terms suchas third-order dispersion, self-steepening, and self-frequency shift may support the propagation of ultrashort pulses[22, 23]. Although the effect of third-order dispersion is significant for femtosecond pulses when the group velocitygroup (GVD) is close to zero, Vyas et al. [24] showed that it can be neglected for pulses whose width is of the order of100 fs or more, having power of the order of 1 W and GVD far away from zero. However, the effects of self-steepeningand self-frequency shift terms are still dominant and cannot be ignored. Alka et al. [25] demonstrated that thecompeting cubic-quintic nonlinearity induces propagating chirped solitonlike dark (bright) solitons and double-kinksolitons in the nonlinear Schrodinger equation with self-steepening and self-frequency shift. Extensive research workhas been carried out on chirped solitons and pulses because of their application in pulse compression or amplificationand thus they are particularly useful in the design of fiber-optic amplifiers, optical pulse compressors, BECs, andsolitary-wave-based communications links [26–30].In this paper, we present a thorough analysis of the dynamics of chirped femtosecond solitons and double-kink
solitons of BECs with time-varying atomic scattering length whose wave function is described by the GP Eq. (2).Our study is greatly facilitated by the so-called phase-imprint method which consists of reducing the GP Eq. (2) intoa higher-order nonlinear Schrodinger (HO-NLS) equation with self-steepening and self-frequency shift terms for whichwe can directly construct the exact solutions. Here, we derive families of chirped solitonlike solutions for the HO-NLSequation, by adopting a nonlinear chirping ansatz which differs from that used by many authors [25, 29]. With thehelp of these chirped solitonlike solutions of the HO-NLS equation, we build the chirped solitonlike solutions of the GPEq. (2) and he corresponding frequency chirps, which are the use to investigate the dynamics of chirped femtosecondsolitons and double-kink solitons in BECs described by Eq. (2). The rest of this paper is organized as follows: Themain transformation leading to the integrability conditions of Eq. (2) and the compensation condition is presentedin Section II and the HO-NLS equation whose chirped solitonlike solutions lead to chirped femtosecond soliton anddouble-kink soliton solutions of the GP Eq. (2) is derived and integrated. With the use of exact chirped solitonlikesolutions of the HO-NLS equation, we investigate in Section III the dynamics of chirped femtosecond solitons anddouble-kink solitons in the BECs under consideration. The main results are summarized in Section IV.
3
II. HIGHER-ORDER NONLINEAR SCHRODINGER EQUATION FOR THE GP EQ. (2)
A. Integrability and compensation conditions for the GP Eq. (2)
In order to obtain the integrable conditions of Eq. (2) and to reduce the GP Eq. (2) into a HO-NLS equation, weperform a lens-type transformation of the form
ψ(x, t) = φ(X,T ) exp[i(γx2 + θ
)](3)
in which T is a real function of time t, X =√
gg0x for any real constant g0 having the same sign as g, and θ = θ(X,T )
is a real function of variables X and T . We then demand that
dT
dt=
g
g0, (4a)
1
g
dg
dt+ 4γ = 0, (4b)
dγ
dt+ 2γ2 + α = 0, (4c)
∂θ
∂X= α0 |φ|2 , (4d)
∂θ
∂T= i
(3
2α0 − β0
)(φ∂φ∗
∂X+ φ∗
∂φ
∂X
)+
[g0χ0 −
1
2β20 +
3
2α0 (β0 − α0)
]|φ|4 , (4e)
where α0 6= 0 and β0 two arbitrary real parameters. The choice of Eq. (4a) is made to preserve the scaling. The realfunction θ that satisfies the conditions (4d) and (4e) is called the phase-imprint on the old order parameter ψ(x, t)[31]. Ansatz (3) under the conditions (4a)–(4e) converts the GP Eq. (2) to the following HO-NLS equation withself-steepening and self-frequency shift
i∂φ
∂T+
1
2
∂2φ
∂X2+ g0 |φ|2 φ+ iα0
∂
∂X
(|φ|2 φ
)+ i (β0 − 2α0)φ
∂ |φ|2∂X
+1
2(β0 − α0) (β0 − 2α0) |φ|4 φ = 0. (5)
In the context of nonlinear optics, the HO-NLS Eq. (5) models the propagation of ultrashort (femtosecond) pulses ina single-mode optical fiber [32, 33]. In this context, φ(X,T ) is the normalized complex envelope of an optical pulse, Tand X are the distance and retarded time, respectively. The coefficient 1
2 of the second derivative with respect withX points to the GVD effect, parameter g0 of the cubic nonlinearity denotes the Kerr nonlinearity, α0 accounts for thepulse self-steepening effect, while β0 is the self-frequency shift coefficient and relates to the nonlinearity dispersion,and the combination 1
2 (β0 − α0) (β0 − 2α0) is the quintic nonlinearity. In the context of fiber optics, the cubic term
|φ|2 φ is often referred to as self-phase modulation, and then its coefficient g0 can be scaled out to the GVD term,which will be termed anomalous dispersion for positive g0 and normal dispersion if g0 < 0 [32, 34].
The HO-NLS equation (5) is rather general, as it can reduce to a series of well-established integrable equations ofSchrodinger type such as Chen–Lee–Liu type NLS (CLL–NLS) equation if β0 − α0 = 0 [35], the Kaup–Newell typeNLS (KN–NLS) equation if β0 − 2α0 = 0 [36], and the Gerdjikov–Ivanov (GI) equation if β0 = 0 [37]. It is importantto point out that the GP Eq. (2) is reduced into the HO-NLS (5) when the nonlinearity parameter g(t), the harmonictrapping potential parameter α(t), and the gain/loss parameter γ(t) satisfy Eqs. (4b) and (4c). Throughout thiswork, Eqs. (4b) and (4c) will the be referred to as the integrable conditions of the GP Eq. (2). Thus, the virtueof the lens-type transformation (3) is that, without much complicated calculation, we not only find the integrableconditions (4b) and (4c) for the GP Eq. (2), but we also retrieve a higher-order nonlinear Schrodinger equationwith self-steepening and self-frequency shift terms whose solitons solutions and double-kink solitons solutions withnonlinear chirp can be obtained under certain parametric conditions.
Integrating Eq. (4b) yields g(t) = λ0 exp[−4∫ t
0γ(τ)dτ
]. This means that the absolute value of the nonlinearity
parameter g(t) will be an increasing function of time t if and only, if γ(τ) is negative. Therefore, BECs with lossof atoms is associated with increasing (in absolute value) nonlinearity parameter g(t), while BECs with feeding ofatoms correspond to decreasing (in absolute value) nonlinearity parameter g(t). As well as we know, the density
|ψ(x, t)|2 of a BEC with the increasing of the absolute value of the s-wave scattering length (with loss of atoms) has
an increase (decrease) in the peak value; also, the density |ψ(x, t)|2 of a BEC with the decreasing of the absolutevalue of the s-wave scattering length (with feeding of atoms) has a decrease (increase) in the peak value [13, 15]. Wethus conclude that Eq. (4b) can lead to a compensation process so that for BECs that satisfy condition (4b), the
4
density |ψ(x, t)|2 will have a constant in the peak value for all time t. The compensation process consists of a balanceof loss or gain effects with the effects of the s-wave scattering length on the condensate. Such a compensation processensures the stability of the condensates over a longer interval of time. Equation (4b) can thus be referred to as the“compensation condition”.It is important to notice that parameter χ0 will not have any effect of the density |ψ(x, t)|2. As we can see from
Eq. (4e), our above ansatz is made so that the factor χ0 of the three-body interatomic interaction will affect the wavephase through the phase-imprint θ.
B. Chirped femtosecond and double-kink solitonlike solutions of the HO-NLS Eq. (5)
Because Eq. (5) in the context of nonlinear optics models the propagation of ultrashort (femtosecond) pulses, itssolitonlike solutions will be referred to as the femtosecond or double-kink solitonlike solutions. In this subsection, wefocus ourselves to the traveling wave solutions of the HO-NLS Eq. (5), that is, the solutions of the form
φ(X,T ) = ρ(ξ) exp [i (χ(ξ)− ΩT )] , (6)
where ρ and χ are reals functions of the traveling coordinate ξ = X−υT ; here, υ is a real parameter given in terms of
the group velocity of the wave packet. The corresponding frequency chirp is given by δω(X,T ) = − ∂∂X
(χ(ξ)− ΩT ) =
− ddξχ(ξ) = −χ′(ξ). Inserting Eq. (6) into ansatz (3) yields the following solution of the GP Eq. (2)
ψ(x, t) = ρ(ξ) exp[i(γx2 + θ(X,T ) + χ(ξ)− ΩT
)]. (7)
with the corresponding chirp δω(x, t) = − ∂∂x
[γx2 + θ(X,T ) + χ(ξ)− ΩT
]. Using Eq. (4d), the frequency chirp
associated to solution (7) is found to be
δω(x, t) = −[2γx+
√g
g0
(α0ρ
2(ξ) + χ′(ξ))]
ξ=√
g
g0x− υ
g0
∫
t
0g(τ)dτ
. (8)
Substituting Eq. (6) into Eq. (5) and separating the real and imaginary parts yield
Ωρ+ υdχ
dξρ− 1
2
(dχ
dξ
)2
ρ+1
2
d2ρ
dξ2− α0
dχ
dξρ3 + g0ρ
3 +1
2(β0 − α0) (β0 − 2α0) ρ
5 = 0, (9a)
−υdρdξ
+1
2
d2χ
dξ2ρ+
dχ
dξ
dρ
dξ+ (2β0 − α0) ρ
2 dρ
dξ= 0. (9b)
One easily verifies that
dχ
dξ=α0 − 2β0
2ρ2 + υ. (10)
satisfies Eq. (9b). Inserting Eq. (10) into Eqs. (8) and (9a) yields
δω(x, t) = −[2γx+
√g
g0
(υ +
3α0 − 2β02
ρ2(ξ)
)]
ξ=√
g
g0x− υ
g0
∫
t
0g(τ)dτ
, (11)
and
d2ρ
dξ2+ b5ρ
5 + b3ρ3 + b1ρ = 0, (12a)
b5 =3
2α0 (α0 − β0) , b3 = 2 (g0 − α0υ) , b1 = 2Ω + υ2, (12b)
respectively. It is seen from Eq. (11) that the frequency chirp δω(x, t) depends on different coefficients of the HO-NLSEq. (5) such as Kerr nonlinearity, self-steepening, and self-frequency shift, as well as on the nonlinearity parameterg(t) and the gain/loss parameter γ(t). This means that for given either the nonlinearity parameter g(t) or the gain/loss
5
parameter γ(t), the amplitude of chirping can be controlled by varying the three free parameters g0, α0, and β0. Itis also seen from Eq. (11) that the frequency chirp evolves in space and time.
Rewriting Eq. (12a) in the form
(dr
dξ
)2
= α1r4 + 4β1r
3 + 6γ1r2 + 4δ1r, r = ρ2, (13a)
α1 = −8
3b5, β1 = −b3
2, γ1 = −2
3b1, (13b)
all its traveling-wave solutions can be expressed in a generic form by means of the Weierstrass function ℘ [38–40] (here,δ1 is a constant of integration). In this paper, we limit ourselves to localized solutions of Eq. (12a). In the special caseof the integrable CLL–NLS (β0−α0 = 0), Eq. (12a) reduces to a cubic nonlinear equation that admits bright and darksoliton solutions. In the special case when υ = g0/α0, Eq. (12a) can be solved for localized solutions by using a Eq.(13a). For 2Ω+υ2 = 0, Eq. (12a) will admit a Lorentzian-type solution. When α0 (α0 − β0) (g0 − α0υ)
(2Ω + υ2
)6= 0,
Eq. (13a) is useful to find double-kink-type and bright- and dark-soliton solutions [40]. To apply Eq. (13a) for findinglocalized solutions of Eq. (12a), we first introduce the invariants g2 and g3 of Weierstrass’ elliptic function ℘ whichare related to the coefficients of the polynomial R(r) = α1r
4 + 4β1r3 + 6γ1r
2 + 4δ1r according to
g2 = −4β1δ1 + 3γ21 , g3 = 2β1γ1δ1 − α1δ21 − γ31 .
If the discriminant ∆ = g32 − 27g23 of ℘ and R is zero, g2 ≥ 0 and g3 ≤ 0, then r(ξ) is solitary wavelike and given[39, 40]
r(ξ) = r0 +R′(r0)
4
sinh2[√
3e1ξ]
3e1 +(e1 − R′′(r0)
24
)sinh2
[√3e1ξ
] , (14)
where e1 = 3√−g3 and r0 is any simple zero of polynomial R(r). In what follows, we discuss, dependent on the behavior
of the parameters b5, b3, and b1, the solitonlike solutions of Eq. (12a). We will mainly build such solutions with the
help of solitonlike solutions of Eq. (13a). From the relationship r(ξ) = ρ2(ξ) yields. ρ(ξ) = ±√r(ξ). Since for the
GP Eq. (2) we are interesting in the density |ψ(x, t)|2 = ρ2(√
gg0x− υ
g0
∫ t
0g(τ)dτ
), we will consider only sign “+”
in the expression for ρ(ξ). In each of the cases b5b3b1 = 0 and b5b3b1 6= 0, we will limit ourselves to some interestingsolutions of Eq. (13a) that lead to localized chirped solitonlike solutions of the GP Eq. (2).
1. Case b5 = 0 and b3b1 6= 0
In the situation when b5 = 0 and b3b1 6= 0, we consider the two interesting situations δ1 = 0 and δ1 =9γ2
1
16β1
leading
to ∆ = 0. When b5 = 0, R(r) admits one simple zero r0 = − 3γ1
2β1
= 2Ω+υ2
α0υ−g0, if δ1 = 0. In this special case, ∆ = 0.
Inserting r0 into Eq. (14) yields r(ξ) = − 32γ1
β1
[1− tanh2
[√32γ1ξ
]]. Therefore,
ρ(ξ) =
√
− 2Ω + υ2
g0 − α0υ
1
cosh[√
− (2Ω + υ2)ξ] , 2Ω + υ2 < 0, g0 − α0υ > 0.
Inserting ρ(ξ) into Eqs. (7) and (11) yields, under the conditions that 2Ω + υ2 < 0 and g0 − α0υ > 0, the followingsolution of the GP Eq. (2) with the corresponding frequency chirp
ψ(x, t) =
√
− 2Ω + υ2
g0 − α0υ
1
cosh
[√− (2Ω + υ2)
(√g(t)g0x− υ
g0
∫ t
0g(τ)dτ
)] exp[i(γx2 + θ(X,T ) + χ(ξ)− ΩT
)],(15a)
δω(x, t) = −
2γ(t)x+
√g(t)
g0
υ − (3α0 − 2β0)
(2Ω + υ2
)
2 (g0 − α0υ) cosh2
[√− (2Ω + υ2)
(√g(t)g0x− υ
g0
∫ t
0g(τ)dτ
)]
, (15b)
6
respectively.
If δ1 =9γ2
1
16β1
, then ∆ = 0 and polynomial R(r) admits one double root r0 = − 3γ1
4β1
and one simple zero r0 = 0.
Seeking a localized solution of Eq. (13a) associated with the double zero r0 = − 3γ1
4β1
in the form r = A tanh2 [Bξ]
yields
ρ(ξ) =
√2Ω + υ2
2 (α0υ − g0)tanh
[√2Ω + υ2
2ξ
], 2Ω + υ2 > 0, g0 − α0υ < 0. (16a)
For the solution of Eq. (13a) corresponding to the simple zero r0 = 0, we use Eq. (14) and obtain the following kinksolitonlike solution of Eq. (12a)
ρ(ξ) =1
2
√3 (2Ω + υ2)
α0υ − g0
sinh[√
2Ω + υ2ξ]
√3 + 2 sinh2 [x]
, 2Ω + υ2 > 0, g0 − α0υ < 0. (16b)
Using now Eqs. (7) and (11), we obtain from Eqs. (16a) and (16b) the following exact solutions of the GP Eq. (2)with the corresponding chirp
ψ(x, t) =
√2Ω + υ2
2 (α0υ − g0)tanh
[√2Ω + υ2
2
(√g(t)
g0x− υ
g0
∫ t
0
g(τ)dτ
)]exp
[i(γx2 + θ + χ(ξ)− ΩT
)], (17a)
δω(x, t) = −[2γ(t)x+
√g(t)
g0
(υ +
(3α0 − 2β0)(2Ω + υ2
)
4 (α0υ − g0)
)
× tanh2
[√2Ω + υ2
2
(√g
g0x− υ
g0
∫ t
0
g(τ)dτ
)]], (17b)
and
ψ(x, t) =1
2
√3 (2Ω + υ2)
α0υ − g0
sinh
[√2Ω + υ2
(√g(t)g0x− υ
g0
∫ t
0g(τ)dτ
)]
√3 + 2 sinh2
[√2Ω + υ2
(√g(t)g0x− υ
g0
∫ t
0g(τ)dτ
)]
× exp[i(γx2 + θ(X,T ) + χ(ξ)− ΩT
)], (18a)
δω(x, t) = −[2γ(t)x+
√g(t)
g0
(υ +
3 (3α0 − 2β0)(2Ω + υ2
)
4 (α0υ − g0)
×sinh2
[√2Ω + υ2
(√g(t)g0x− υ
g0
∫ t
0g(τ)dτ
)]
3 + 2 sinh2[√
2Ω + υ2(√
g(t)g0x− υ
g0
∫ t
0g(τ)dτ
)]
, (18b)
respectively, where α0 6= 0, g0 6= 0, υ, and Ω are four arbitrary real parameters satisfying the conditions 2Ω + υ2 > 0and g0 − α0υ < 0.
2. Case b3 = 0, b5b1 6= 0
When b3 = 0 and b5b1 6= 0, we have ∆ = −27α1
(α1δ
21 + 2γ31
)δ21 = 0 if and only, if either δ1 = 0
or δ1 = ±γ1√
−2 γ1
α1
= ± 23
(2Ω + υ2
)√− 2Ω+υ2
3α0(α0−β0). Here, we focus ourselves to the special case when δ1 =
7
± 23
(2Ω + υ2
)√− 2Ω+υ2
3α0(α0−β0), under the condition that α0 (α0 − β0)
(2Ω + υ2
)< 0. In this case, r0 = 0 is a sim-
ple zero of R(r). Inserting r0 = 0 into Eq. (14) and using the relationship ρ2 = r(ξ) leads to the kink solution
ρ2(ξ) =2
3
√− 2Ω + υ2
3α0 (α0 − β0)
sinh2[√
2 (2Ω + υ2)ξ]
2 + sinh2[√
2 (2Ω + υ2)ξ] ,
(2Ω + υ2
)> 0 and α0 (α0 − β0) < 0.
Inserting ρ(ξ) into Eqs. (7) and (11) leads to the following exact solitonlike solution of the GP Eq. (2) with thecorresponding frequency chirp
ψ(x, t) =
√√√√√√√2
3
√− 2Ω + υ2
3α0 (α0 − β0)
sinh2[√
2 (2Ω + υ2)
(√g(t)g0x− υ
g0
∫ t
0g(τ)dτ
)]
2 + sinh2[√
2 (2Ω + υ2)
(√g(t)g0x− υ
g0
∫ t
0g(τ)dτ
)]
× exp[i(γx2 + θ(X,T ) + χ(ξ)− ΩT
)], (19a)
δω(x, t) = −[2γ(t)x+
√g(t)
g0
(υ +
3α0 − 2β03
√− 2Ω + υ2
3α0 (α0 − β0)
×sinh2
[√2 (2Ω + υ2)
(√g(t)g0x− υ
g0
∫ t
0g(τ)dτ
)]
2 + sinh2[√
2 (2Ω + υ2)
(√g(t)g0x− υ
g0
∫ t
0g(τ)dτ
)]
, (19b)
where α0 6= 0, β0, Ω, and υ are four arbitrary real parameters satisfying the conditions 2Ω+υ2 > 0 and α0 (α0 − β0) <0.
3. Case b1 = 0 and b5b3 6= 0
We now focus on the situation when b1 = 0 and b5b3 6= 0. In this special case, we have ∆ = −(27α2
1δ1 + 64β31
)δ31 = 0
if and only, if either δ1 = 0 or δ1 = − 6427
β3
1
α2
1
, the interesting possibility been δ1 = − 6427
β3
1
α2
1
(since δ1 = 0 leads to the
constant solution r = g0−α0υα0(β0−α0)
, if α0 (β0 − α0) (g0 − α0υ) > 0). In this special case, r0 = 0 is a simple zero of
polynomial R(r). Inserting r0 = 0 into Eq. (14) yields
r(ξ) =α0υ − g0
3α0 (α0 − β0)
sinh2[2 (g0 − α0υ)
√− 1
3α0(α0−β0)ξ]
3 + sinh2[2 (g0 − α0υ)
√− 1
3α0(α0−β0)ξ]
, α0 (α0 − β0) < 0, g0 − α0υ > 0.
To obtain the corresponding exact solution of the GP Eq. (2) with the corresponding chirp, we recall that r(ξ) = ρ2(ξ)and use Eqs. (7) and (11). We then obtain
ψ(x, t) =
√√√√√√√α0υ − g0
3α0 (α0 − β0)
sinh2[2 (g0 − α0υ)
√− 1
3α0(α0−β0)
(√g(t)g0x− υ
g0
∫ t
0g(τ)dτ
)]
3 + sinh2[2 (g0 − α0υ)
√− 1
3α0(α0−β0)
(√g(t)g0x− υ
g0
∫ t
0g(τ)dτ
)]
× exp[i(γx2 + θ(X,T ) + χ(ξ)− ΩT
)], (20a)
8
FIG. 1: (Color online) Amplitude profile of the soliton solution in Eq. (21) for different values of the solution parameters. (a):
α1 = −1, β1 = 1, γ1 = 23
β2
1
α1, and δ10 = 8
27
β3
1
α2
1
; (b): γ1 = δ10 = 1, α1 = 0.248872, and β1 = 0.56137.
δω(x, t) = −[2γ(t)x+
√g(t)
g0
(υ +
(3α0 − 2β0) (α0υ − g0)
6α0 (α0 − β0)
×sinh2
[2 (g0 − α0υ)
√− 1
3α0(α0−β0)
(√g(t)g0x− υ
g0
∫ t
0g(τ)dτ
)]
3 + sinh2[2 (g0 − α0υ)
√− 1
3α0(α0−β0)
(√g(t)g0x− υ
g0
∫ t
0g(τ)dτ
)]
, (20b)
where α0 6= 0, β0, g0 6= 0, and υ are four real parameters satisfying the conditions α0 (α0 − β0) < 0 and g0 −α0υ > 0.
4. Case b5b3b1 6= 0
Now, we consider the general case of Eq. (12a) when b5b3b1 6= 0. Comparing Eqs. (12b) and (13b), conditionb5b3b1 6= 0 leads to α1β1γ1 6= 0. In this general case,
∆ = −[27α2
1δ21 + 4β1
(16β2
1 − 27α1γ1)δ1 + 18γ21
(3α1γ1 − 2β2
1
)]δ21 .
To the solution δ1 6= 0 of equation ∆ = 0, we associate the simple zero r0 = 0 of polynomial R(r). Let δ10 be any
positive root of equation ∆ = 0 that satisfies the conditions γ31 −2β1γ1δ10+α1δ210 > 0 and 3
√γ31 − 2β1γ1δ10 + α1δ210−
12γ1 > 0. Inserting r0 = 0 into Eq. (14) yields
r(ξ) =δ1
e1 − 12γ1
sinh2[√
3e1ξ]
3e1e1−
1
2γ1
+ sinh2[√
3e1ξ] , e1 = 3
√γ31 − 2β1γ1δ10 + α1δ210.
Using the relationship r(ξ) = ρ2(ξ) leads to the following double-kink-type soliton solution of Eq. (12a)
ρ(ξ) =
√δ10
e1 − 12γ1
sinh[√
3e1ξ]
√3e1
e1−1
2γ1
+ sinh2[√
3e1ξ] , e1 = 3
√γ31 − 2β1γ1δ10 + α1δ210. (21)
It is important to point out that the interesting double-kink feature of the solution given by Eq. (21) exists only forsufficiently large values of 3e1
e1−1
2γ1
. The amplitude profile of the soliton solution (21) for different α1, β1, γ1, and δ10
is shown in Fig. 1. For Fig. 1(a), 3e1e1−
1
2γ1
= 65 , while for Fig. 1(b), 3e1
e1−1
2γ1
= 1003. The data used in Fig. 1(b) means
that δ1 = 1, υ = g0+0.56137α0
, Ω = − 3+2υ2
4 , and α0 (α0 − β0) + 0.62 218 = 0.
9
Going back to Eqs. (7) and (11) and using Eq. (21), we obtain the following double-kink-type soliton solution ofthe GP Eq. (2) with the corresponding chirp
ψ(x, t) =
√3δ10
3e1 + 2Ω + υ2
sinh
[√3e1
(√g(t)g0x− υ
g0
∫ t
0g(τ)dτ
)]
√9e1
3e1+2Ω+υ2 + sinh2[√
3e1
(√g(t)g0x− υ
g0
∫ t
0g(τ)dτ
)]
× exp[i(γx2 + θ(X,T ) + χ(ξ)− ΩT
)], (22a)
δω(x, t) = −[2γ(t)x+
√g(t)
g0
(υ +
3δ10 (3α0 − 2β0)
2 (3e1 + 2Ω + υ2)
×sinh2
[√3e1
(√g(t)g0x− υ
g0
∫ t
0g(τ)dτ
)]
9e13e1+2Ω+υ2 + sinh2
[√3e1
(√g(t)g0x− υ
g0
∫ t
0g(τ)dτ
)]
; (22b)
here, e1 = − 23
3
√(2Ω + υ2)
3+ 9
2 (g0 − α0υ) (2Ω + υ2) δ10 +272 α0 (α0 − β0) δ210, δ10 is any root of equation 27α2
1δ21 +
4β1(16β2
1 − 27α1γ1)δ1 + 18γ21
(3α1γ1 − 2β2
1
)= 0, and α0 6= 0, g0 6= 0, β0, υ, and Ω are five real parameters to be
taken from conditions γ31 − 2β1γ1δ10 + α1δ210 > 0 and 3
√γ31 − 2β1γ1δ10 + α1δ210 − 1
2γ1 > 0, α1, β1, and γ1 being givenby Eq. (13b)When δ1 = 0, polynomial R then takes the form R(r) = α1r
4 + 4β1r3 + 6γ1r
2 and admits two double zeros, r0 = 0
and, under the condition γ1 =2β2
1
3α1
, r0 = − 2β1
α1
; for γ1 6= 2β2
1
3α1
, R(r) admits two simple zeros, r0 =−2β1±
√
2(2β2
1−3α1γ1)
α1
.
Interesting solitonlike solutions of Eq. (13a) with δ1 = 0 can be sought using the direct method. Here we focusourselves on the kink and bright solitonlike solutions of Eq. (13a). As an example of kink solitonlike solution of Eq.
(13a), we obtain, under the conditions α0 (β0 − α0) > 0, g0 − α0υ > 0, (g0 − α0υ)2 − 4α0
(2Ω + υ2
)(α0 − β0) = 0,
r(ξ) =g0 − α0υ
4α0 (β0 − α0)
(1± tanh
[g0 − α0υ
2√α0 (β0 − α0)
ξ
]).
Going back in Eqs. (7) and (11) and knowing that r(ξ) = ρ2(ξ) yield the following kink solitonlike solution of the GPEq. (2) and the corresponding frequency chirp
ψ(x, t) =1
2
√√√√ g0 − α0υ
α0 (β0 − α0)
(1± tanh
[g0 − α0υ
2√α0 (β0 − α0)
(√g(t)
g0x− υ
g0
∫ t
0
g(τ)dτ
)])
× exp[i(γx2 + θ(X,T ) + χ(ξ)− ΩT
)], (23a)
δω(x, t) = −[2γx+
√g
g0
(υ +
3α0 − 2β02
g0 − α0υ
4α0 (β0 − α0)
×(1± tanh
[g0 − α0υ
2√α0 (β0 − α0)
(√g(t)
g0x− υ
g0
∫ t
0
g(τ)dτ
)]))]; (23b)
here, α0 6= 0, g0 6= 0, β0, υ, and Ω are five real parameters satisfying the conditions α0 (β0 − α0) > 0, g0 − α0υ > 0,
4α0 (α0 − β0)(2Ω + υ2
)− (g0 − α0υ)
2= 0.
If γ1 > 0, β1 < 0, 2β21 − 3γ1α1 > 0, then Eq. (13a) admits a bright solitonlike solution
r(ξ) = −3γ1β1
1
1 +√
6β2
1−9γ1α1
6β2
1
cosh[√
6γ1ξ]
10
FIG. 2: (Color online) The dynamics of a double-kink soliton in a time-independent harmonic trapping potential given byequation (22a) and the corresponding frequency chirp given by Eq. (22b). (a) and (c): Spatiotemporal evolution of a double-kink soliton and the corresponding chirp for g(t) = λ0 exp [4λt]. (b) and (d): Evolution of a double-kink soliton and the
corresponding chirp for g(t) = λ0 exp [−4λt]. Different plots are generated with υ = g0+0.56137α0
, Ω = − 3+2υ2
4, δ1 = 1,
β0 =α2
0+0.62 218
α0, g0 = 1.8, α0 = 0.5, λ0 = 0.1, and λ = 0.05.
that leads to the below bright solitonlike solution of the GP Eq. (2) with the associated frequency chirp
ψ(x, t) = 2
√√√√√2Ω + υ2
2
(α0υ − g0 − λ0 cosh
[2√− (2Ω + υ2)
(√g(t)g0x− υ
g0
∫ t
0g(τ)dτ
)])
× exp[i(γx2 + θ(X,T ) + χ(ξ)− ΩT
)], (24a)
δω(x, t) = −[2γ(t)x+
√g(t)
g0(υ + (2β0 − 3α0)
×(2Ω + υ2
)
g0 − α0υ + λ0 cosh
[2√− (2Ω + υ2)
(√g(t)g0x− υ
g0
∫ t
0g(τ)dτ
)]
(24b)
where λ0 =
√(g0 − α0υ)
2 − 4α0 (α0 − β0) (2Ω + υ2), and α0 6= 0, g0 6= 0, β0, υ, and Ω are five real parameters
satisfying the conditions 2Ω + υ2 < 0, g0 − α0υ > 0, (g0 − α0υ)2 − 4α0 (α0 − β0)
(2Ω + υ2
)> 0.
11
FIG. 3: (Color online) Spatial evolution of the (a, c) the double-kink soliton given by Eq. (22a) and (b, d) the correspondingfrequency chirp defined by Eq. (22b) for the same parameters as in Fig. 2. Plots of the top panels are generated withg(t) = λ0 exp [4λt], while those of the bottom panels are obtained with g(t) = λ0 exp [−4λt] . In different plots, the solid line,the dotted line, and the dashed line show the wave evolution at time t = 0, t = 2, and t = 5, respectively.
III. DYNAMICS OF CHIRPED FEMTOSECOND SOLITONS AND DOUBLE-KINK SOLITONS INBECS WITH TIME-DEPENDENT ATOMIC SCATTERING LENGTH IN A COMPLEX POTENTIAL
With the help of the above exact solitonlike solutions of Eq. (2), we now turn to the analytical investigation of thedynamics of chirped femtosecond solitons and double-kink solitons in BECs described by the GP Eq. (2). It followsfrom different expressions for the wave function ψ(x, t) and the corresponding frequency chirp δω(x, t) found in the
previous section that the centre of different solitons corresponding to ψ(x, t) and δω(x, t) is ζ = υg0
√g0g(t)
∫ t
0g(τ)dτ,
which satisfies the equation d2ζdt2
+2αζ = 0. This means that the centre of mass of the macroscopic wave packet behaveslike a classical particle, and allows us to manipulate the motion of chirped femtosecond solitons and double-kink solitonsin BEC systems by controlling the external harmonic trapping potential. It follows from the above equation of thecentre of matter waves and the integrability conditions that (i) the found chirped femtosecond solitons and double-
kink solitons move with the speed dζdt
= υg0
(2γ∫ t
0g(τ)dτ + g
)√g0g(t) which depends on both the s-scattering length
and the feeding/loss parameter, while the width is proportional to√
g0g(t) . Therefore, the soliton width decreases
(increases) during the wave propagation when g(t) increases (decreases) as time t increases; also, the soliton widthincreases when the nonlinearity parameter g0 increases. In what follows, we take some examples to demonstrate thedynamics of femtosecond solitons and double-kink solitons in one-dimensional BEC systems with different kinds ofscattering length, harmonic trapping potential, and feeding or loss parameter. For the demonstration of the dynamicsof femtosecond solitons and double-kink solitons in the BEC systems, we will limit ourselves to the use of the double-kink soliton solution (22a) and the bright femtosecond soliton solution (24a) and their corresponding chirps (22b)and (24b), respectively. For the double-kink soliton solution (22a) with the associating chirp (22b), we will use the
parameters δ1 = 1, υ = g0+0.56137α0
, Ω = − 3+2υ2
4 , and β0 =α2
0+0.62 218
α0
, while the dynamics of bright femtosecond
solitons and the corresponding chirp associated to solution (24a) with chirp (24b) will be investigated with the use of
either the parameters α20 = 1, β0 = 1
6α0
, υ = 1α0
, g0 > 1, (g0 − 1)2+5 > 0, and Ω = − 3+2υ2
4 , or the parameters α20 = 1,
12
FIG. 4: (Color online) Effects of the self-steepening coefficient α0 on the dynamics of a bright femtosecond soliton in a time-independent harmonic trapping potential given by equation Eq. ( 24a) and the corresponding chirp defined by Eq. (24b) forg(t) = λ0 exp [4λt] and g0 = 1.8 with two values of α0, α0 = 1 for plots of the top panels and α0 = −1 for plots (c) and (d).Other values of different parameters are given in the text.
g0 > −1, β0 = 1α0
, υ = − 1α0
, and Ω = − 3α2
0+2
4α2
0
. As we will see in the below examples, frequency chirp corresponding
to femtosecond solitons and double-kink solitons will be localized; moreover, chirp associated to double-kink solitonwill have a double-kink feature dark or bright dependent on the sign of the self-steepening coefficient α0, while thatcorresponding to bright femtosecond soliton will have, dependent on the sign of the self-steepening coefficient α0,either a bright or dark soliton feature.
A. BECs with time-independent harmonic potential
As the first example, we consider the time-independent harmonic potential which was used in the creation ofbright BEC solitons [41]. For that experience, the strength of the harmonic potential was α = −2λ2 with λ ≈ 0.05.From the integrability conditions (4b) and (4c), we obtain γ(t) = ±λ and g(t) = λ0 exp [∓4λt] , where λ0 6= 0 isan arbitrary real parameter having the same sign as g0, that is, λ0g0 > 0. The dynamics of a double-kink solitonin the harmonic trapping potential and the corresponding chirp are shown in Fig. 2. Figures 2(a) and 2(c) areobtained with g(t) = λ0 exp [4λt] and correspond to BECs with the loss of atoms associating to the loss parameterγ(t) = −λ, while Figs. 2(b) and 2(d) correspond to BECs with the gain of atoms for the feeding parameter γ(t) = λcorresponding to a time decreasing s-wave scattering length with g(t) = λ0 exp [−4λt]. Figures 2(c) and (d) revealthat the frequency chirp associated with the double-kink soliton has double-kink feature. We can see from plots ofFig. 2 that with the increasing (decreasing) of the absolute value of the s-wave scattering length, the double-kinksoliton keeps the same absolute depth and has a compression (broadening) in its width, while the correspondingfrequency chirp has an increase (decrease) in the absolute depth value and a compression (broadening) in its width.
For a better understanding, we show in Fig. 3(a) and 3(b) the time profile of respectively the wave density |ψ(x, t)|2and the corresponding chirp δω(x, t) for different values of time t.
For the set of parameters α20 = 1, β0 = 1
6α0
, υ = 1α0
, g0 > 1, (g0 − 1)2+ 5 > 0, Ω = − 3+2υ2
4 with g0 = 1.8 and
13
FIG. 5: (Color online) Plot of the trajectory ζ = υg0
√
g0g(t)
∫ t
0g(τ)dτ of the centre of (a) the double-kink soliton and (b) the
bright femtosecond soliton with m = 0.4 and = 2 for different sign of the self-steepening coefficient α0. (a): Trajectory ofthe double-kink soliton for λ0 = −0.1, g0 = −1.8 with two values of α0, α0 = 0.5 (solid line) and negative α0 = −0.5 (dottedline); (b): Trajectory of the bright femtosecond soliton for λ0 = 2.1, g0 = 1.8, with two different values of α0, α0 = 1 (solidline) and negative α0 = −1 (dotted line). Other parameters used to generate different plots are given in the text.
FIG. 6: (Color online) Dynamics of the double-kink soliton with the corresponding chirp associated to solution (22a) with chirp(22b) of the GP Eq. (2) for the self-steepening coefficient having different sign. Plots (a) and (b) are generated with α0 = 0.5,while (c) and (d) are obtained with α0 = −0.5. Other parameters used in different plots are the same as in Fig. 5(a).
α0 = ±1, we show in Fig. 4 the effects of the self-steepening coefficient α0 on the dynamics of the femtosecond solitonin the harmonic trapping potential defined by Eq. (24a) and the corresponding chirp given by Eq. (24b). As we cansee from Figs. 4(a) and 4(b) showing the evolution of respectively a bright femtosecond soliton and the correspondingfrequency chirp, reveal that for the chirp has bright soliton feature for a positive self-steepening coefficient (α0 > 0).Fig. 4(c) showing the dynamics of the frequency chirp associated to the bright femtosecond soliton showed in Fig.4(c) with a negative self-steepening coefficient (α0 < 0) has dark soliton feature. Independently of the sign of the self-
14
FIG. 7: (Color online) Spatiotemporal evolution of the bright femtosecond soliton with the corresponding chirp associated tosolution (24a) with chirp (24b) of the GP Eq. (2) for the self-steepening coefficient having different sign. Plots (a) and (b) areobtained with α0 = 1, while (c) and (d) are generated with α0 = 1. Other parameters used in different plots are the same asin Fig. 5(b) except for = 1.
steepening coefficient α0, the bright femtosecond soliton has an constant in the peak value although the absolute valueof the s-wave scattering length increases as a function of time (Figs. (a) and (d)). As a consequence of the increasingof the absolute value of the s-wave scattering length, the bright bright femtosecond soliton and the correspondingchirp during their propagation have a compression in their width. It is seen from Fig. 4(b) [4(d)] that the frequencychirp associated to a positive [negative] self-steepening coefficient α has an increase in the peak value [absolute depth].It is also seen from plots of Fig. 4 that for positive (negative) self-steepening coefficient α0, the wave propagates inthe −x−direction (+x−direction). Therefore, the self-steepening coefficient α0 can be used to manipulate the motionof the matter wave solitons in the BEC systems.
B. BECs with a temporal periodic modulation of the s-wave scattering length
As the second example, we consider the temporal periodic modulation of the s-wave scattering length [42] and thenonlinearity parameter takes the form g(t) = λ0 (1 +m sin [t]) with 0 < m < 1, λ0 6= 0 being any real constant.
Using the integrability conditions (4b) and (4c) yields γ(t) = −m4
cos[t]1+m sin[t] and α(t) = −m2
8
m(2+cos2[t])+2 sin[t]
(1+m sin[t])2.
For this second example, we first depict the wave trajectory in Fig. 5 for different sign of the self-steepening coefficientα0. Due to the temporal periodic modulation of the s-wave scattering length and trapping potential, these trajectoriesoscillate. Plots of Fig. 5 reveal that the direction of the wave propagation depends on the sign of the self-steepeningcoefficient α0.Figures 6 and 7 show the dynamics of respectively a double-kink soliton and a bright femtosecond soliton with the
15
FIG. 8: (Color online) Spatiotemporal evolution of (left panels) the double-kink soliton given by Eq. (22a) and (right panels)the corresponding chirp defined by Eq. (22b) for BEC systems with a time-varying hyperbolic s-wave scattering length withg(t) = λ0 (1 + tanh [µt]). Plots (a) and (b) are generated with a positive self-steepening coefficient α0 (α0 = 0.5), while (c) and(d) are obtained with a negative self-steepening coefficient α0 (α0 = −0.5). Other parameter used in plots (a)–(d) are λ0 = 0.3,g0 = 1.8, and µ = 2, and m = 0.99. Other parameters are given in the text.
corresponding frequency chirp for BEC system with a temporal periodic modulation of the s-wave scattering length inthe temporal periodic modulation of the trapping potential a temporal periodic loss and gain of atoms. Plots of thetop panels correspond to positive self-steepening coefficient α0, while those of the bottom panels are obtained withnegative self-steepening coefficient α0. It is seen from the Chirping dynamics (plots of the bottom panels) that dueto the temporal periodic modulation of the s-wave scattering length and trapping potential, the trajectories of thefrequency chirp oscillate and the direction of the wave motion depends on the sign of the self-steepening coefficientα0. For positive self-steepening coefficient α0 (plots of the top panels), the frequency chirp corresponding to thedouble-kink soliton has bright double-kink feature, while for the negative self-steepening coefficient α0, the chirp hasdouble-kink feature, as we can see from Fig. 6(d). It is seen from Fig. 7(b) that the frequency chirp associated withthe bright femtosecond soliton has dark (bright) soliton feature for positive (negative) self-steepening coefficient α0.
C. BECs with a time-varying hyperbolic s-wave scattering length
Following Xue [43], we consider for the last example a BEC in the time-dependent trapping potential with thetime-varying hyperbolic s-wave scattering length that corresponds to g(t) = λ0 (1 +m tanh [µt]) for any real constantsλ0 6= 0 and µ 6= 0, and 0 < m < 1. Using the integrability conditions (4b) and (4c) yields γ(t) = −µ
41
(1+tanh[µt]) cosh2[µt]
and α(t) = −µ2
83+4 cosh[µt] sinh[µt](1+tanh[µt])
cosh4[µt](1+tanh[µt])2. Figures 8 and 9 report the spatiotemporal evolution of respectively the
16
FIG. 9: (Color online) The dynamics of (left panels)a bright femtosecond soliton defined by Eq. (24a) and (right panel) thecorresponding frequency chirp given by Eq. (24b) for BEC systems with a time-varying hyperbolic s-wave scattering lengthwith g(t) = λ0 (1 + tanh [µt]). Plots (a) and (b) of the top panels are obtained with a positive self-steepening coefficient α0
(α0 = 1), while (c) and (d) are generated with a negative self-steepening coefficient α0 (α0 = −1). Other parameter used inplots (a)–(d) are λ0 = −1.1, g0 = −0.95, and µ = 2, and m = 0.99. Other parameters are given in the text.
double-kink soliton and the femtosecond soliton in BEC with a time-varying hyperbolic s-wave scattering length forrespectively a positive and negative nonlinearity parameter g0. For the double-kink soliton and the femtosecondsoliton, we have used respectively the above set of data and the set of data α2
0 = 1, g0 > −1, β0 = 1α0
, υ = − 1α0
,
and Ω = − 3α2
0+2
4α2
0
. As in the previous two examples, the direction of the wave motion and the feature of the frequency
chirp depend on the sign of the self-steepening coefficient α0. The chirp has a dark feature for positive self-steepeningcoefficient α0 (plots of the top panels) and a bright feature for negative self-steepening coefficient α0, as we can seefrom plots of the bottom panels. Figures 8 and 9 show that the width of the waves decreases at the beginning ofthe wave propagation and then reaches its minimum value, which no longer changes. It is sure that for negative µ,the width of the waves will increase at the beginning of the wave propagation and then reaches its maximum value,which no longer changes as the wave propagates. For a better understanding, we show in Fig. 10 the evolution plotof both double-kink soliton (top panels) and femtosecond soliton (bottom panels) and their corresponding chirp for anegative µ.
IV. CONCLUSION
In this work, we have considered the cubic-quintic GP equation in a complex potential consisting of parabolic andcomplex terms which describes the dynamics of the BEC matter waves with the time-dependent s-wave scatteringlength, with loss/gain of atoms, and with both the two- and three-body interatomic interactions in a time-dependentharmonic trapping potential. Using the modified phase-imprint transformation, we have obtained both the integrabil-ity conditions and the compensation condition, and converted the GP equation under consideration to a NLS equationwith self-steepening and self-frequency shift. We have demonstrated that the competing cubic-quintic nonlinearity in-duces propagating solitonlike dark (bright) solitons and double-kink solitons in the NLS equation with self-steepening
17
FIG. 10: (Color online) a) and (b): Evolution plots of respectively the double-kink soliton given by Eq. (22a) and the
corresponding chirp defined by Eq. (22b) for δ1 = 1, υ = g0+0.56137α0
, Ω = − 3+2υ2
4, and β0 =
α2
0+0.62 218
α0with α0 = 0.5,
λ0 = −0.3, g0 = −1.8, m = 0.8, and µ = −0.1. (c) and (d): The evolution plots of a bright femtosecond soliton defined byEq. (24a) and the corresponding frequency chirp given by Eq. (24b), respectively for α0 = 1, λ0 = 1.1, g0 = 0.95, m = 0.8,
µ = −0.1, β0 = 1α0
, υ = − 1α0
, and Ω = −3α2
0+2
4α2
0
. Other parameters are showed in the text.
and self-frequency shift. Parameter domains are delineated in which these BEC solitons exist. It is showed that thenonlinear chirp associated with each of these BEC solitons is formed of three terms, one term proportional to theloss/gain parameter γ(t) with a spatial coefficient, one term proportional to the absolute value of the s-wave scatteringlength, and one term which is directly proportional to the intensity of the wave with a time-dependent parameter.We found that the chirp associated to each BEC soliton is localized and its feature (dark or bright) depends onthe sign of the self-steepening coefficient α0. We showed that the motion of both the waves and the correspondingchirp depend on the self-steepening coefficient α0. More interesting, our results reveal that the solitons amplitudeis constant during the wave propagation (this result comes from the compensation process), while the amplitude ofthe chirp strongly depends on the s-wave scattering length. It i also for that for BECs having an increasing in timeabsolute s-wave scattering length, our exact solutions can be used to describe the compression of solitons in the BECsunder consideration. The methodology presented in this work is powerful for systematically finding an infinite numberof BEC bright and dark solitonlike solutions by exactly matching the s-wave scattering length, the strength of theloss/feeding parameter and external harmonic trapping potential.
Declaration of Competing Interest: The authors declare that they have no known competing financial interestsor personal relationships that could have appeared to influence the work reported in this paper.
18
CRediT authorship contribution statement: Emmanuel Kengne: Conceptualization, Project administration,Methodology, Software, Writing - original draft, Investigation, Visualization, and Writing - review & editing. Ahmed
Lakhssassi: Writing - review & editing, Investigation, Validation.Data Availability Statement: Data sharing is not applicable to this article as no new data were created or
analyzed in this study.
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