Exact Chirped Soliton Solutions for the One-Dimensional Gross–Pitaevskii Equation with Time-Dependent ParametersZhenyun Qina and Gui Mub

a School of Mathematics and LMNS, Fudan University, Shanghai 200433, PR Chinab College of Mathematics and Information Science, Qujing Normal University, Qujing 655011,

PR China

Reprint requests to Z. Q.; E-mail: zyqin@fudan.edu.cn

Z. Naturforsch. 67a, 141 – 146 (2012) / DOI: 10.5560/ZNA.2011-0070Received June 21, 2011 / revised October 17, 2011

The Gross–Pitaevskii equation (GPE) describing the dynamics of a Bose–Einstein condensate atabsolute zero temperature, is a generalized form of the nonlinear Schrodinger equation. In this work,the exact bright one-soliton solution of the one-dimensional GPE with time-dependent parametersis directly obtained by using the well-known Hirota method under the same conditions as in S. Ra-jendran et al., Physica D 239, 366 (2010). In addition, the two-soliton solution is also constructedeffectively.

Key words: Hirota Method; Gross–Pitaevskii Equation; Chirped Soliton Solution.PACS numbers: 03.75.Lm; 05.30.Jp; 67.40.Fd

1. Introduction

The nonlinear Schrodinger equation (NLSE) hasbeen triggered immense interest in the modelling ofmany physical phenomena [1] such as propagationof laser beams in nonlinear media [2, 3], plasmadynamics [4], mean field dynamics of Bose–Einsteincondensates [5 – 16], condensed matter [17], etc.The Gross–Pitaevskii equation (GPE), describing thedynamics of a Bose–Einstein condensate (BEC) atabsolute zero temperature [18], is a generalized formof the NLSE [19 – 21]. On the other hand, the NLSEis a completely integrable soliton system [19] whereasthe GPE is not integrable in general, but can admitexact solutions only in very special cases [22, 23].TheGPE can be reduced to the effective one-dimensionl(1D) GPE by assuming the kinetic energy of thelongitudinal excitations and the two-body interactionenergy of the atoms.

The 1D GPE is given as [24]

iΦt = − 12

Φxx− R(t)|Φ |2Φ +Ω 2(t)

2x2

Φ

+ iγ(t)

2Φ ,

(1)

which arises as a model for the dynamics of Bose–Einstein condensates in the mean field approximation,

where the nonlinear coefficient R(t) is physicallycontrolled by acting on the so-called Feschbachresonances, Ω(t) stands for the strength of thequadratic potential as a function of time, and γ(t)is the gain/loss term which is phenomenologicallyincorporated to account for the interaction of theatomic or the thermal cloud. Equation (1) can betransformed into the standard NLSE by means ofthe similar transformation [24]. Under the conditionsΩ(t) = γ(t) = β = const, the N-solitary solution of (1)has been obtained by Darboux transformation [25].Modulation instability and solitons on a continuous-wave background in inhomogeneous optical fibermedia have been discussed in [15]. Additionally, whenΩ 2(t) = 2sech2(t)− 1, R(t) = sech(t), γ(t) = tanh(t)or Ω 2(t) = 2sech2(t + t0), R(t) = sech(t),γ(t) = tanh(t), the N-solitary solution of (1) hasbeen also obtained by Darboux transformation [26].

It is well known that the Hirota bilinear method isa established method for obtaining multi-soliton ex-pressions in nonlinear evolution equations [27, 28].But the bilinear form guarantees only the existence oftwo-soliton solutions. The purpose of this work is todevelop the usage of this method to the GPE with theaim of producing one- and new two-soliton solutions,see (22) and (36) below. The new two-soliton solutionsare obtained by a novel factorization procedure, which

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142 Z. Qin and G. Mu · Chirped Soliton Solutions for 1D Gross–Pitaevskii Equation

will reduce the algebraic manipulations involved in theintermediate calculations considerably.

The structure of the paper can be explained as fol-lows. In terms of the reasonable assumption, the GPEcan be decoupled into two equations, and then theone-soliton solution of (1) is presented in Section 2.A comparison of our results with those from [24] isalso made. The new two-soliton solution is analyti-cally constructed in Section 3. Conclusions are drawnin Section 4.

2. Chirped One-Soliton Solution

First, the main idea of the Hirota bilinear method isintroduced briefly. Applying the transformation

Φ(x, t) = exp

[∫γ(t)

2dt

]φ(x, t) , (2)

(1) can be rewritten in terms of the new variable φ(x, t)as

iφt +12

φxx−12

x2Ω

2(t)φ +1R(t)|φ |2φ = 0, (3)

where R(t) is a time-dependent parameter and can beexpressed as

R(t) = exp

[∫γ(t)dt

]R(t). (4)

Now, we consider the following transformation:

φ(x, t) =G(x, t)F(x, t)

, (5)

where G(x, t) is a complex function, and F(x, t) isa real function. With this transformation, (3) is decou-pled as bilinear form(

iDt +12

D2x−

x2

2Ω

2)

G ·F = 0, (6)

12

D2xF ·F = RGG, (7)

where G is the conjugate function of G, and the D-operator is defined by

Dmx Dn

t F(x, t) ·G(x, t) =(

∂

∂x− ∂

∂x′

)m(∂

∂ t− ∂

∂ t ′

)n

·[F(x, t)G

(x′, t ′

)]∣∣x′=x,t ′=t

in which the spatial and time-dependent terms x2Ω 2(t)and R(t) will give a difficulty for getting solutions asshown below.

In order to obtain the bright chirped one-soliton so-lution of (1), we proceed in the standard assumption

F(x, t) = 1+ ε2F1(x, t), G(x, t) = εG1(x, t), (8)

where ε is an arbitrary parameter which will be ab-sorbed in expressing the soliton solution in the follow-ing sections. Substituting (8) into (6) – (7), then col-lecting the coefficients with same power in ε yields

ε : iG1t +12

G1xx−12

x2Ω

2(t)G1 = 0, (9)

ε2 : F1xx−RG1G1 = 0, (10)

ε3 :

(iDt +

12

D2x−

12

x2Ω

2(t))

G1 ·F1 = 0, (11)

ε4 : D2

xF1 ·F1 = 0. (12)

Firstly, we can assume G1(x, t) has the form

G1 = eη , η = ip(t)x2 +q(t)x+ω(t), (13)

where p(t),q(t),ω(t) are the time-dependent functionsto be determined, and p(t) is a chirp function. Substi-tuting (13) into (9), a system of algebraic or first-orderordinary differential equations for the parameter func-tions is achieved:

2pt +4p2 +Ω2 = 0, qt +2pq = 0,

ωt =i2

q2− p.(14)

From (10), we get F1(x, t) in the following form:

F1(x, t) = c(t)eη+η , (15)

where c(t) is a time-dependent function. Inserting thisform into (10) – (13), the relations between these pa-rameters are given by

R = c(q+q)2, ct −2pc = 0. (16)

Solving above equations shows that

q =q0RR0

, c =c0R0

R, (17)

ω = ω0 +∫ t

0

(i2

q2− p

)dz, (18)

R0 = c0(q0 +q0)2, (19)

p(t) =− Rt

2R. (20)

Z. Qin and G. Mu · Chirped Soliton Solutions for 1D Gross–Pitaevskii Equation 143

Fig. 1 (colour online). Temporal evolution of one-soliton solution (22) for variable coefficient 1D GPE, (a) with the parametersγ(t) = 0, R0 = 1, q0 = 0.5 + i, ω0 = 0, c0 = 1, R(t) = sech(0.1t). (b) under conditions identical to those of (a) except thatγ(t) = 2cos(2t).

Here the subscript 0 denotes the value of the givenfunction at z = 0. Equation (20) is thought to be a con-dition constraint of the initial value for the parameterfunctions. p(t) is a normalized chirp function; it is re-lated to the nonlinear coefficient R(t) and the gain/losscoefficient γ(t). It is noted that R(t) and Ω(t) satisfythe following Riccati equation:

ddt

(Rt

R

)−(

Rt

R

)2

−Ω2(t) = 0. (21)

It is not possible to find explicit solutions of this Ric-cati equation, however, for a given form of R(t), thecorresponding form of the trap frequency Ω(t) can

be calculated as Ω(t) =√

ddt

(RtR

)−(Rt

R

)2. From (17)

to (20), the other functions can be figured out when thefunction R(t) is fixed. Mathematically, the choices ofR(t) must allow for the term

∫ t0 q2 dt in (18). After tak-

ing ε = 1, the exact chirped one-soliton solution of (1)can be derived as

Φ(x, t) =e∫ γ(t)

2 dt eη

1+ c(t)eη+η, (22)

where η = ip(t)x2 +q(t)x+ω(t), the parameters p(t),q(t), ω(t), and c(t) are determined by (17) to (21).In (22), after taking

q0 =√

2a+ i(c+2br0), R0 =1r0

,

w0 =−12

ln

(R0

8a2

),

(23)

the one soliton solution is reduced to (23) [24]. Herethe soliton evolutions under the condition (23) areomitted deliberately. So here we only want to show thedifferent phenomenon when R(t) = sech(0.1t), γ(t) =0 and γ(t) = 2cos(2t). Figure 1a and 1b show that thesoliton amplitude and width are affected by the ex-ternal potential and the nonlinear term. Figure 1a isdepicted with vanishing gain/loss term. Figure 1b il-lustrates that the soliton will breathe more and morelightly with the propagation distance. From the physi-cal point of view, the collapse and revival of the atomiccondensate through periodic exchange of atoms withthe background is due to the gain/loss term γ(t) =2cos(2t). It is noted here that Strecker et al. have ex-perimentally observed the formation of the bright soli-ton of a lithium atom in a quasi-1D optical trap by mag-netically turning the interactions in a stable BEC fromrepulsive to attractive [29].

3. New Chirped Two-Soliton Solution

In this section, we show how to obtain analyticallya chirped two-soliton solution of the GPE with time-dependent parameters. Similar to Section 2, to obtaintwo-soliton, we now assume that

F(x, t) = 1+ ε2F1(x, t)+ ε

4F2(x, t),

G(x, t) = ε G1(x, t)+ ε3G2(x, t).

(24)

Inserting (24) into (6) and (7), requiring the coeffi-cients of power term of ε be equal to zero, a series of

144 Z. Qin and G. Mu · Chirped Soliton Solutions for 1D Gross–Pitaevskii Equation

equations are obtained

ε : iG1t +12

G1xx−12

x2Ω

2(t)G1 = 0, (25)

ε2 : F1xx−RG1G1 = 0, (26)

ε3 : iG2t +

12

G2xx−12

x2Ω

2(t)G2

+(iDt +12

D2x−

12

x2Ω

2(t))G1 ·F1 = 0,(27)

ε4 :

12

D2xF1 ·F1 +F2xx−RG2G1−RG1G2 = 0, (28)

ε5 : (iDt +

12

D2x−

12

x2Ω

2(t))

· (G1 ·F2 +G2 ·F1) = 0,(29)

ε6 : D2

xF1 ·F2−RG2G2 = 0, (30)

ε7 : (iDt +

12

D2x−

12

x2Ω

2(t))G2 ·F2 = 0, (31)

ε8 : D2

xF2 ·F2 = 0. (32)

We set

G1(x, t) = eη1 + eη2

together with

η1 = ip(t)x2 +q1(t)x+ω1(t),

η2 = ip(t)x2 +q2(t)x+ω2(t).

By employing the same procedure as in Section 2,we can choose F1, G2, and F2 according to (25) – (28)as

F1(x, t) = c1(t)eη1+η1 + c2(t)eη2+η1

+ c3(t)eη1+η2 + c4(t)eη2+η2 ,(33)

G2(x, t) = b1(t)eη1+η1+η2 +b2(t)eη2+η2+η1 , (34)

F2(x, t) = c5(t)eη1+η1+η2+η2 . (35)

In this way, a system of algebraic or fist-order ordinarydifferential equations for the parameter functions is ob-tained. Solving the obtained system, after a long andtedious calculation and taking ε = 1, the new chirpedtwo-soliton solution of (1) with time-dependent param-eters reads as

Φ(x, t) =e∫ γ(t)

2 dt(

eη1 + eη2 +b1 eη1+η1+η2 +b2 eη2+η2+η1)

1+ c1 eη1+η1 + c2 eη2+η1 + c3 eη1+η2 + c4 eη2+η2 + c5 eη1+η1+η2+η2, (36)

where

η1 = ipx2 +q1x+ω1 , η2 = ipx2 +q2x+ω2 ,

c1 =c10R0

R, c2 =

c20R0

R, c3 =

c30R0

R,

c4 =c40R0

R,

q1 =q10RR0

, q2 =q20RR0

, p =− Rt

2R,

ω1 = ω10 +∫ t

0

(ı2

q21− p

)dz ,

ω2 = ω20 +∫ t

0

(ı2

q22− p

)dz ,

ddt

(Rt

R

)−(

Rt

R

)2

−Ω2(t) = 0 ,

b1 =c20(q10−q20)2R0

(q10 +q10)2R,

b2 =(q10−q20)2R0c30c40

c10(q10 +q10)2R,

c5 =c30c40R2

0(q10−q20)2(q10−q20)2

(q10 +q10)2(q20 +q10)2R2 .

Here the subscript 0 denotes the value of the givenfunction at z = 0. The choices for the parameters c10,c20, c30, c40, q10, and q20 should satisfy

c10(q10 +q10)2 = c20(q20 +q10)2 = c30(q10 +q20)2

= c40(q20 +q20)2 = R0, c20 = c30 . (37)

An interaction between two solitons is interesting forthe 1D GPE with time-dependent parameters. Fig-ure 2a and 2b show that the interaction between the twosolitons is affected by the periodic nonlinear gain orloss. Similarly to Figure 1, Figure 2a is depicted withvanishing gain/loss term. Figure 2b illustrates that thenew chirped two-soliton will breathe more and morelightly with the propagation distance. From the physi-cal point of view, the collapse and revival of the atomiccondensate through periodic exchange of atoms withthe background is due to the gain/loss term γ(t) =2cos(2t). The two-soliton does not collide or attract

Z. Qin and G. Mu · Chirped Soliton Solutions for 1D Gross–Pitaevskii Equation 145

Fig. 2 (colour online). Temporal evolution of two-soliton solution (36) for 1D GPE, (a) under conditions γ = 0, R0 = 1, q10 =−0.5+ i, q20 = 1+ i, ω10 = 0, ω20 = 0, R(t) = sech(0.1t), (b) with the same parameters as in (a) except that γ(t) = 2cos(2t).

each other and propagate parallel as the time evolves.Furthermore, from the above phenomenon, it is worth-while to note that the gain/loss term does not affect thewidth and motion of the soliton but changes its peakheight.

4. Conclusions

In this work, the soliton solutions with chirp of theGPE with time-dependent parameters are investigated.A developed Hirota method is applied carefully to theGPE. In terms of this technique, we decoupled the GPEinto two equations. Additionally, with a reasonable as-sumption, the exact chirped one- and new two-solitonsolutions are constructed effectively. The finding ofa new mathematical algorithm to discover soliton so-

lutions in nonlinear dispersive systems with parametervariations is helpful on future research. On the otherhand, the results are useful not only in Bose–Einsteincondensates, but also in nonlinear optical systems.

Acknowledgements

The work was supported by the National Nat-ural Science Foundation of China (No. 10801037,No. 11061028), the Science foundation for theYouth Scholars of the Youth Scholars of the Doc-toral Fund of the Ministry of Education of China(No. 200802461007), the Young Teachers Foundation(No. 1411018) of Fudan University and the YoungTeacher Fundation (No. 2010QN018) of QujingNormal University.

[1] L. Vazquez, L. Streit, and V. M. Perez-Garcıa (Eds.),Nonlinear Klein–Gordon and Schrodinger Systems:Theory and Applications, World Scientific, Singapore1997.

[2] Y. S. Kivshar and G. P. Agrawal, Optical Solitons:From Fibers to Photonic Crystals, Academic Press,California 2003.

[3] A. Hasegawa, Optical Solitons in Fibers, Springer-Verlag, Berlin 1989.

[4] R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Mor-ris, Solitons and Nonlinear Wave Equations, AcademicPress, New York 1982.

[5] M. H. Anderson, J. R. Ensher, M. R. Matthews,C. E. Wieman, and E. A. Cornell, Science 269, 198(1995).

[6] C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G.Hulet, Phys. Rev. Lett. 75, 1687 (1995).

[7] K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. vanDruten, D. S. Durfee, D. M. Kurn, and W. Ketterle,Phys. Rev. Lett. 75, 3969 (1995).

[8] A. L. Fetter and J. D. Walecka, Quantum Theory ofMany-Particle Systems, McGraw-Hill, New York 1971.

[9] Z. D. Li, Q. Y. Li, X. H. Hu, Z. X. Zheng, and Y. B. Sun,Ann. Phys. (NY) 322, 2545 (2007).

146 Z. Qin and G. Mu · Chirped Soliton Solutions for 1D Gross–Pitaevskii Equation

[10] Z. D. Li and Q. Y. Li, Ann. Phys. (NY) 322, 1961(2007).

[11] X. Lu, B. Tian, T. Xu, K. J. Cai, and W. J. Liu, Ann.Phys. (NY) 323, 2554 (2008).

[12] H. W. Xiong, S. J. Liu, W. P. Zhang, and M. S. Zhan,Phys. Rev. Lett. 95, 120401 (2005).

[13] L. P. Pitaevskii and S. Stringari, Bose–Einstein Con-densation, Clarendon Press, Oxford 2003.

[14] M. I. Rodas-Verde, H. Michinel, and V. M. Perez-Garcıa, Phys. Rev. Lett. 95, 153903 (2005).

[15] M. Salerno, V. V. Konotop, and Y. V. Bludov, Phys.Rev. Lett. 101, 030405 (2008).

[16] C. Q. Dai, D. S. Wang, L. L. Wang, J. F. Zhang, andW. M. Liu, Ann. Phys. (NY) 326, 2356 (2011).

[17] A. Scott, Nonlinear Science: Emergence and Dynam-ics of Coherent Structures, in: Oxford Appl. and Eng.Mathematics, Vol. 1, Oxford University Press, Oxford1999.

[18] C. J. Pethick and H. Smith, Bose–Einstein Conden-sation in Dilute Gases, Cambridge University Press,Cambridge, England 2002.

[19] J. Ablowitz and P. A. Clarkson, Nonlinear EvolutionEquations and Inverse Scattering, Cambridge Univer-sity Press, Cambridge 1991.

[20] G. P. Agrawal, Nonlinear Fiber Optics, AcademicPress, New York 1995.

[21] M. Lakshmanan and S. Rajasekar, Nonlinear Dynam-ics: Integrability, Chaos and Patterns, Springer-Verlag,Berlin 2003.

[22] L. Salasnich, A. Parola, and L. Reatto, Phys. Rev. A 65043614 (2002).

[23] V. A. Brazhnyi and V. V. Konotop, Phys. Rev. A 68043613 (2003).

[24] S. Rajendran, P. Muruganandam, and M. Lakshmanan,Physica D 239, 366 (2010).

[25] L. Li, Z. H. Li, S. Q. Li, and G. S. Zhou, Opt. Commun.234, 169 (2004).

[26] R. Y. Hao and G. S. Zhou, Opt. Commun. 281, 4474(2008).

[27] S. Homma and S. Takeno, J. Phys. Soc. Jpn. 56, 3480(1987).

[28] W. J. Liu, B. Tian, T. Xu, K. Sun, and Y. Jiang, Ann.Phys. (NY) 325, 1633 (2010).

[29] K. E. Strecker, G. B. Partridge, A. G. Truscott, andR. G. Hulet, Nature 417, 150 (2002).