Accepted Manuscript

Prolongation structures and matter-wave solitons in F = 1 spinor Bose-Einsteincondensate with time-dependent atomic scattering lengths in an expulsive har-monic potential

Deng-Shan Wang, Yu-Quan Ma, Xiang-Gui Li

PII: S1007-5704(14)00087-2DOI: http://dx.doi.org/10.1016/j.cnsns.2014.02.019Reference: CNSNS 3107

To appear in: Communications in Nonlinear Science and Numer-ical Simulation

Please cite this article as: Wang, D-S., Ma, Y-Q., Li, X-G., Prolongation structures and matter-wave solitons inF = 1 spinor Bose-Einstein condensate with time-dependent atomic scattering lengths in an expulsive harmonicpotential, Communications in Nonlinear Science and Numerical Simulation (2014), doi: http://dx.doi.org/10.1016/j.cnsns.2014.02.019

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customerswe are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, andreview of the resulting proof before it is published in its final form. Please note that during the production processerrors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Prolongation structures and matter-wave solitons in

F = 1 spinor Bose-Einstein condensate with

time-dependent atomic scattering lengths in an

expulsive harmonic potential

Deng-Shan Wang∗, Yu-Quan Ma and Xiang-Gui LiSchool of Applied Science, Beijing Information Science and Technology University, Beijing 100192, China

Abstract In this paper, the integrability and matter-wave solitons in a system of three compo-nent Gross-Pitaevskii equation arising from the context of F = 1 spinor Bose-Einstein condensatewith time-dependent atomic scattering lengths in an expulsive harmonic potential are investigatedby similarity transformation, prolongation technique and Riemann-Hilbert formulation. As a re-sult, some new exact nonautonomous matter-wave soliton solutions with varying amplitudes andspeeds are obtained. It is shown that there exist two integrable systems and exact N -matter-wavesolitons in spin-1 Bose-Einstein condensates with time-dependent s-wave scattering lengths. Thecollision dynamics of the two matter-wave solitons are analyzed and the shape changing interactionphenomena are found.

Keywords: Similarity transformation, Prolongation structure, Linear spectral problem, Gross-Pitaevskii equation, Matter-wave soliton.

1 Introduction

Over the last decade, Bose-Einstein condensation of ultra-cold atomic gases has attracted a renewedtheoretical and experimental interest in quantum many-body systems at low temperatures [1]-[3].One of significant development of atomic Bose-Einstein condensates (BEC) is the intense study ofmacroscopic nonlinear excitations, such as solitons [4]-[8] and vortices [8, 9]. In fact, the emergenceof such macroscopic coherent structures in the many body physics establishes a close connectionbetween BEC and other branches of physics, such as nonlinear optics and fluid mechanics. Recently,there has been an increasing focus on the study of spinor BEC [9]-[12], which have been realizedwith the help of far-off resonant optical techniques [13] for trapping ultra-cold atomic gases.

At sufficiently low temperatures and in the framework of the mean-field approximation, thespinor BEC with F = 1 is described by the vector order parameter [Φ1(r),Φ0(r), Φ−1(r)] with thefollowing energy functional

EGP [Φ] =∫

d3r

Φ†m(− ~2m

∇2 + Vext)Φm +d0

2Φ†mΦ†jΦjΦm +

d2

2Φ†mΦ†jFmkFjlΦlΦk

, (1)

∗E-mail: dswang@bistu.edu.cn

1

where the trapping potential Vext is a harmonic potential and is spin independent, i.e. Vext =Mω2

xx2/2 + Mω2⊥(y2 + z2)/2, terms F = (F x, F y, F z) with F x,y,z are 3 × 3 spin-1 matrices, the

parameters d0 and d2 which describe binary elastic collisions of spin-1 atoms in the combinedsymmetric channel of total spin 0 and 2, are expressed as d0 = 4π~2(a0 + 2a2)/3M and d2 =4π~2(a2 − a0)/3M with M the atomic mass and a0, a2 the s-wave scattering lengths. Note thatthe F = 1 spinor condensates may be either ferromagnetic for d2 < 0 such as the 87Rb atoms [14],or polar d2 > 0 such as the 23Na atoms [15].

Assuming that the condensate is kept in a highly anisotropic trap, with the longitudinal andtransverse trapping frequencies chosen so that ω⊥ À ωx, we may assume approximately separablewave functions Φm(t, r) = ψm(t, x)ψm⊥(y, z) with the transverse components ψm⊥(y, z) determinedby the ground state of the tight harmonic trap. Then averaging of the underlying system of thecoupled three-dimensional Gross-Pitaevskii (GP) equation in the transverse plane (y, z) leads to aquasi-one-dimensional three component GP equation for ψm(t, x) as

i∂ψ0

∂t= Lψ0 + b0|ψ0|2ψ0 + (b0 + b2)(|ψ+1|2 + |ψ−1|2)ψ0 + 2b2ψ

∗0ψ+1ψ−1, (2a)

i∂ψ±1

∂t= Lψ±1 + (b0 + b2)(|ψ±1|2 + |ψ0|2)ψ±1 + (b0 − b2)|ψ∓1|2ψ±1 + b2ψ

∗∓1ψ

20, (2b)

where L = −12∂2/∂x2 + V (x) with V (x) = −1

2ω2x2, an expulsive harmonic potential. The lengthand time are measured in units of a⊥ =

√~/mω⊥ and 1/ω⊥, respectively. The Feshbach-managed

nonlinear coefficients b0 = 2(a0 + 2a2)/(3a⊥) and b2 = 2(a2 − a0)/(3a⊥). Recently, there has beenan increased interest in studying the properties of BEC with time varying parameters includingtemporal variation of atomic scattering lengths [16, 17], time modulation of trap frequencies [16]-[18] and time-dependent gain or loss terms [19]. So in this paper, we assume the s-wave scatteringlengths a0 and a2 depend on time, which can be adjusted experimentally by the optical Feshbachresonance [20]. Therefore, the nonlinear coefficients b0 and b2 can also be time-dependent.

The investigation of matter-wave soliton solutions of the three component GP equation (2)may reveal many novel phenomena in spinor BEC. The theoretical researchers have been primarilyconcerned with finding linear spectral problems associated with the nonlinear equations like Eq.(2). So far, there have been no logical procedures for writing down the linear spectral problems ofnonlinear wave equations, except the Wahlquist-Estabrook (WE) prolongation structure technique[21, 22] using exterior differential form [23], Cartan-Ehresmann connections [24] and Lie algebra[25]. The essence of the WE prolongation procedure for a given nonlinear system is to introducenew dependent variables, also defined by partial differential equations, in such a way that theintegrability condition for the existence of the new variables is precisely the equation originallygiven. The equations for the new variables are linear equations in which the original variable playsa role like that of a potential. Prolongation resembles the classical Backlund transformations [26],except that the number of new variables is not specified in advance and in particular may be morethan one. Moreover, WE prolongation technique is very effective to find linear spectral problemsto the (1+1)-dimensional nonlinear systems including nonlinear Schrodinger (NLS) type equations[27, 28]. Once we find the linear spectral problems associated with the nonlinear systems, theCauchy problem for the nonlinear system is then solved via the so-called inverse scattering transform(IST) method [29]. The original IST method was based on the Gel’fand-Levitan-Marchenko integralequations. Later on, a Riemann-Hilbert formulation [30, 31] was developed which streamlines and

2

simplifies the IST method considerably.In the present paper, we study the integrability and matter-wave solitons of the three component

GP equation with time-varying atomic scattering lengths arising from the context of spinor BECin an expulsive harmonic potential. The paper is organized as follows. In Section 2, by using thesimilarity transformation we convert the three component GP equation (2) into the three coupledNLS equation without external potential, subject to the constraints on the trap frequency andnonlinear coefficients. We investigate the prolongation structures of the three coupled NLS equationand find two completely integrable three coupled NLS equations with linear spectral problems inSection 3. Next, in Section 4 we derive the generalized N -bright soliton solutions of the integrablethree coupled NLS equation by Riemann-Hilbert formulation. The dynamics of the matter-wavesolitons in the F = 1 spinor BEC are studied by analyzing their interaction behaviors. We presenta summary of the conclusions in Section 5. Finally, in the Appendix, we show the commutationrelations of the incomplete Lie algebra L in Section 3.

2 Similarity transformation

In this section, we adopt the similarity transformation to map the three component GP equation(2) onto the three coupled NLS equation without external potential. To do so, the complex wavefunctions ψ±1 and ψ0 can be written as

ψm = βmϕm (T, X) ei(α1x2+α2x+α3), m = ±1, 0, (3)

where X = η1x + η2, and T, η1, η2, β±1, β0, α1, α2 and α3 are functions of time t, and the functionsϕ±1(T, X) and ϕ0(T, X) satisfy [4]

i∂ϕ0

∂T= −∂2ϕ0

∂X2+ (c0 + c2)(|ϕ+1|2 + |ϕ−1|2)ϕ0 + 2c0|ϕ0|2ϕ0 + 2c2ϕ

∗0ϕ+1ϕ−1, (4a)

i∂ϕ±1

∂T= −∂2ϕ±1

∂X2+ (c0 + c2)(|ϕ±1|2 + 2|ϕ0|2)ϕ±1 + (c0 − c2)|ϕ∓1|2ϕ±1 + 2c2ϕ

∗∓1ϕ

20, (4b)

where c0 and c2 are constants.Substituting Eq. (3) into the three component GP equation (2) and letting ϕ±1 and ϕ0 sat-

isfy Eq. (4), the terms T, η1, η2, β±1, β0, α1, α2 and α3 should obey the following set of ordinarydifferential equations

α1t − 12ω2 + 2 α1

2 = 0, (5a)

β+12 (b0 − b2) = β−1

2 (b0 − b2) = Tt (c0 − c2) , (5b)

β+12 (b0 + b2) =

12

β02 (b0 + b2) = β−1

2 (b0 + b2) = Tt (c0 + c2) , (5c)

η12 = 2 Tt, β+1α1 + β+1t = 0, β−1α1 + β−1t = 0, (5d)

α2t + 2 α2α1 = 0, 2 c0Tt − b0β02 = 0, α2

2 + 2 α3t = 0, (5e)η1t + 2 η1α1 = 0, β0t + β0α1 = 0, η2t + η1α2 = 0, (5f)

b2β−1β02 = 2 c2β+1Tt, β−1β+1b2 = c2Tt, b2β+1β0

2 − 2 c2β−1Tt = 0, (5g)

3

where the subscript of t denotes the derivative with respective to time t.With the aid of symbolic computation, solving Eqs. (5b)-(5g) we have

η2 = −α0

ωtanh (ω t) , α1 = ω tanh (ω t) /2, α2 = α0sech (ω t) , α3 = −α0

2

2ωtanh (ω t) ,

T =tanh (ω t)

2ω, b0 = c0sech (ω t) , b2 = c2sech (ω t) , β+1 = β−1 =

√2

2β0, η1 = sech (ω t) , (6)

where β0 =√

sech (ω t) and α0, c0, c2 are constants.

3 Prolongation structures of the three coupled NLS equation

In nonlinear mathematical physics, there are various distinct definitions of integrability such as Li-ouville integrability, Lax integrability, inverse scattering integrability, Painleve integrability, sym-metry integrability, geometric integrability, etc. Among them, the standard notion of completeintegrability is the so called Liouville integrability. However, as far as finding exact solutions isconcerned, we prefer to investigate the Lax integrability of nonlinear wave equations. Thus in thispaper, we claim that a nonlinear wave equation is integrable if it owns a Lax pair.

In general, the three coupled NLS equation (4) is not integrable for arbitrary parameters c0 andc2. However, it is integrable in some sense for special c0 and c2. Wadati et al. [32] have studied thePainleve integrability of the three coupled NLS equation (4) and found that it passes the Painlevetest only for two specific parametric choices, namely, c2 = 0 and c2 = c0. Moreover, Wadati et al.[4] have proved that the equation (4) with c2 = c0 < 0 is completely integrable and has multiplebright soliton solutions.

In this section, we investigate the prolongation structures of the three coupled NLS equation(4) by means of prolongation technique [21, 22, 28]. In order to do this, the complex conjugates ofthe dependent variables are denoted as (here we write ϕ+1 as ϕ1 for simplify)

ϕ∗1 = φ1, ϕ∗0 = φ0, ϕ∗−1 = φ−1. (7)

Then the equations in (4) become six nonlinear equations

iϕ1T + ϕ1XX − (c0 + c2)(ϕ1φ1 + 2ϕ0φ0)ϕ1 − (c0 − c2)ϕ−1φ−1ϕ1 − 2c2φ−1ϕ20 = 0, (8a)

−iφ1T + φ1XX − (c0 + c2)(ϕ1φ1 + 2ϕ0φ0)φ1 − (c0 − c2)ϕ−1φ−1φ1 − 2c2ϕ−1φ20 = 0, (8b)

iϕ0T + ϕ0XX − (c0 + c2)(ϕ1φ1 + ϕ−1φ−1)ϕ0 − 2c0ϕ0φ0ϕ0 − 2c2φ0ϕ1ϕ−1 = 0, (8c)−iφ0T + φ0XX − (c0 + c2)(ϕ1φ1 + ϕ−1φ−1)φ0 − 2c0ϕ0φ0φ0 − 2c2ϕ0φ1φ−1 = 0, (8d)

iϕ−1T + ϕ−1XX − (c0 + c2)(ϕ−1φ−1 + 2ϕ0φ0)ϕ−1 − (c0 − c2)ϕ1φ1ϕ−1 − 2c2φ1ϕ20 = 0, (8e)

−iφ−1T + φ−1XX − (c0 + c2)(ϕ−1φ−1 + 2ϕ0φ0)φ−1 − (c0 − c2)φ1ϕ1φ−1 − 2c2ϕ1φ20 = 0. (8f)

Next we introduce six new variables

ϕ1X = u, φ1X = v, ϕ0X = m, φ0X = n, ϕ−1X = p, φ−1X = q,

4

and define a set of differential 2-form I = θ1, θ2, θ3, θ4, θ5, θ6, θ7, θ8, θ9, θ10, θ11, θ12 on solutionmanifold M = T, X, u, v, m, n, p, q, ϕ1, φ1, ϕ0, φ0, ϕ−1, φ−1 , where

θ1 = dϕ1 ∧ dT + udT ∧ dX, θ2 = dφ1 ∧ dT + vdT ∧ dX,

θ3 = dϕ0 ∧ dT + mdT ∧ dX, θ4 = dφ0 ∧ dT + ndT ∧ dX,

θ5 = dϕ−1 ∧ dT + pdT ∧ dX, θ6 = dφ−1 ∧ dT + qdT ∧ dX,

θ7 = idϕ1 ∧ dX − du ∧ dT − ρ1dT ∧ dX, θ8 = −idφ1 ∧ dX − dv ∧ dT − ρ2dT ∧ dX,

θ9 = idϕ0 ∧ dX − dm ∧ dT − ρ3dT ∧ dX, θ10 = −idφ0 ∧ dX − dn ∧ dT − ρ4dT ∧ dX,

θ11 = idϕ−1 ∧ dX − dp ∧ dT − ρ5dT ∧ dX, θ12 = −idφ−1 ∧ dX − dq ∧ dT − ρ6dT ∧ dX,

withρ1 = (c0 + c2)(ϕ1φ1 + 2ϕ0φ0)ϕ1 + (c0 − c2)ϕ−1φ−1ϕ1 + 2c2φ−1ϕ

20,

ρ2 = (c0 + c2)(ϕ1φ1 + 2ϕ0φ0)φ1 + (c0 − c2)ϕ−1φ−1φ1 + 2c2ϕ−1φ20,

ρ3 = (c0 + c2)(ϕ1φ1 + ϕ−1φ−1)ϕ0 + 2c0ϕ0φ0ϕ0 + 2c2φ0ϕ1ϕ−1,

ρ4 = (c0 + c2)(ϕ1φ1 + ϕ−1φ−1)φ0 + 2c0ϕ0φ0φ0 + 2c2ϕ0φ1φ−1,

ρ5 = (c0 + c2)(ϕ−1φ−1 + 2ϕ0φ0)ϕ−1 + (c0 − c2)ϕ1φ1ϕ−1 + 2c2φ1ϕ20,

ρ6 = (c0 + c2)(ϕ−1φ−1 + 2ϕ0φ0)φ−1 + (c0 − c2)φ1ϕ1φ−1 + 2c2ϕ1φ20.

When these differential 2-forms are restricted on the solution manifold M , they become zero, sowe recover the original (1+1)-dimensional six coupled NLS equations (8). Moreover, it is easy toverify that I is a differential closed idea, i.e. dI ⊂ I.

We further introduce l differential 1-forms

Ωi = dζi − F idX − GidT, (9)

where i = 1, 2, · · · , l, F i and Gi are functions of u, v, m, n, p, q, ϕ1, φ1, ϕ0, φ0, ϕ−1, φ−1 and ζi. Weassume that both F i and Gi are linearly dependent on ζi, namely F i =

∑lj=1 F i

j ζj , Gi =

∑lj=1 Gi

jζj

where F ij and Gi

j are functions of u, v, m, n, p, q, ϕ1, φ1, ϕ0, φ0, ϕ−1, φ−1. For the sake of simplifica-tion, we write F and G for the matrices (F i

j )l×l and (Gij)l×l, respectively, and ζ = (ζ1, ζ2, · · · , ζ l)

′

where ′ denotes transpose. When restricting on solution manifold, the differential 1-forms Ωi arenull, i.e. Ωi = 0 which is just the linear spectral problem ζX = Fζ and ζT = Gζ.

Following the well-known prolongation technique [21, 22, 27, 28], the extended set of differentialform I = I

⋃Ωi

must be a closed ideal under exterior differentiation, i.e. dI ⊂ I . Because

dI ⊂ I ⊂ I , we only need dΩi

⊂ I , which denotes that

dΩi =12∑

j=1

f ijθ

j + ηi ∧ Ωi, i = 1, 2, · · · , l, (10)

where ηi = gi(T,X)dX + hi(T, X)dT are differential 1-forms.

5

When Eq. (10) is written out in detail, we have the following partial differential equationsabout F and G as

Fu = Fv = Fm = Fn = Fp = Fq = 0, iGu + Fϕ1 = 0, iGv − Fφ1 = 0,

iGm + Fϕ0 = 0, iGn − Fφ0 = 0, iGp + Fϕ−1 = 0, iGq − Fφ−1 = 0, (11)

Gϕ1u+Gφ1v+Gϕ0m+Gφ0n+Gϕ−1p+Gφ−1q+ρ1Gu+ρ2Gv+ρ3Gm+ρ4Gn+ρ5Gp+ρ6Gq−[F, G] = 0,

with [F, G] = FG−GF.Solving Eq. (11), the functions F and G may be expressed as

F = x0 + x1ϕ1 + x2ϕ0 + x3ϕ−1 + x4φ1 + x5φ0 + x6φ−1, (12)

G = i(x1u+x2m+x3p−x4v−x5n−x6q−|ϕ0|2x18−ϕ1φ−1x16−φ0ϕ−1x21+ϕ1x8−ϕ0φ−1x19−ϕ0φ1x17

+ϕ−1x10 − φ0x12 − φ1x11 − |ϕ−1|2x22 − φ0ϕ1x15 + ϕ0x9 − x13φ−1 − |ϕ1|2x14 − φ1ϕ−1x20) + x7,

where L = x0, x1, x2, · · · , x22 is an incomplete Lie algebra which is called prolongation algebraand this algebra satisfies the commutation relations in the Appendix. The key point is to find thematrix (or operator) representations of this prolongation algebra. Before doing this, we first givethe definition of nontrivial prolongation structure.

Definition. There exists nontrivial prolongation structure in a nonlinear wave equation if theprolongation algebra of the prolongation structure has nontrivial matrix (or operator) representa-tions, i.e. the representations of the prolongation algebra can formulate a Lax pair of the nonlinearwave equation by combining with the prolonged differential 1-forms.

In the case of prolongation algebra L = x0, x1, x2, · · · , x22 above, if [xi, xj ] = 0 for alli, j = 0, 1, 2, · · · , 22 we say it has trivial matrix (or operator) representations, but if there existnonzero representations of elements x0, x1, x2, · · · , x22 such that the prolonged differential 1-forms(9) bring up a Lax pair of the three coupled NLS equation (4) we say the prolongation algebra L hasnontrivial matrix (or operator) representations. Thus if a nonlinear wave equation has nontrivialprolongation structure it is Lax integrability.

Theorem. Only for two specific parametric choices, namely, c2 = c0 or c2 = 0 the three coupledNLS equation (4) is Lax integrability.

Proof. From the Appendix, it is seen that the following special commutation relations hold

[x1, x14] = −(c0 + c2)x1, [x4, x14] = (c0 + c2)x4, [x1, x4] = x14,

[x2, x18] = −2x2c0, [x5, x18] = 2c0 x5, [x2, x5] = x18, (13)

[x3, x22] = −(c0 + c2)x3, [x6, x22] = (c0 + c2)x6, [x3, x6] = x22.

We note that the spans of x1, x4, x14, x2, x5, x18 and x3, x6, x22 are three subalgebra ofthe Lie algebra sl(n,C) for n > 2, which are isomorphic to sl(2, C). Thus it is seen thatx1, x2, x3, x4, x5, x6 are nilpotent elements and x14, x18, x22 are neutral elements. In order to find

6

the matrix representations of the prolongation algebra L, we try to embed L into the Lie alge-bra sl(l, C). Starting from the case of l = 2, it is obvious that sl(2, C) is not the whole alge-bra. After some computation, we find that sl(3, C) is also not be the whole algebra. Subse-quently, embedding L into the Lie algebra sl(4, C) and using the commutation relations (13) and[x1, x2] = 0, [x1, x3] = 0, [x2, x3] = 0, [x4, x6] = 0, [x5, x6] = 0, [x4, x5] = 0 in the Appendix, weobtain only two cases of 4× 4 matrix representations of the elements x1, x2, x3, x4, x5, x6 .

Case 1. The 4× 4 matrix representations of x1, x2, x3, x4, x5, x6 are

x1 =

0 0 1 0

0 0 0 0

0 0 0 0

0 0 0 0

, x2 =

0 0 0 1

0 0 1 0

0 0 0 0

0 0 0 0

, x3 =

0 0 0 0

0 0 0 1

0 0 0 0

0 0 0 0

,

x4 =

0 0 0 0

0 0 0 0c0+c2

2 0 0 0

0 0 0 0

, x5 =

0 0 0 0

0 0 0 0

0 c0 0 0

c0 0 0 0

, x6 =

0 0 0 0

0 0 0 0

0 0 0 0

0 c0+c22 0 0

. (14)

Case 2. The 4× 4 matrix representations of x1, x2, x3, x4, x5, x6 are also

x1 =

0 1 0 0

0 0 0 0

0 0 0 0

0 0 0 0

, x2 =

0 0 1 0

0 0 0 0

0 0 0 0

0 0 0 0

, x3 =

0 0 0 1

0 0 0 0

0 0 0 0

0 0 0 0

,

x4 =

0 0 0 0c0+c2

2 0 0 0

0 0 0 0

0 0 0 0

, x5 =

0 0 0 0

0 0 0 0

c0 0 0 0

0 0 0 0

, x6 =

0 0 0 0

0 0 0 0

0 0 0 0c0+c2

2 0 0 0

. (15)

Substituting the matrix representations (14) of the elements x1, x2, x3, x4, x5, x6 into thecommutation relations in the Appendix, it is found that only when parameter c2 = c0 thesecommutation relations satisfy, which denotes that when c2 = c0 Eq. (4) has nontrivial prolongationstructure. In this case, the nontrivial 4 × 4 matrix representations of the other elements in theprolongation algebra L can also be obtained immediately. Then substituting all these relations intoEqs. (12) and (9), the linear spectral problem of Eq. (4) with c2 = c0 is found as

ζX = Fζ, ζT = Gζ, ζ = (ζ1, ζ2, ζ3, ζ4)′, (16)

7

where ′ denotes transpose and F, G satisfy

F =

−iλ 0 ϕ1 ϕ0

0 −iλ ϕ0 ϕ−1

c0ϕ∗1 c0ϕ

∗0 iλ 0

c0ϕ∗0 c0ϕ

∗−1 0 iλ

=

( −iλI2 Q

c0Q† iλI2

),

G = 2λ

( −iλI2 Q

c0Q† iλI2

)+ i

( −c0QQ† QX

−c0Q†X c0Q

†Q

),

where I2 is a 2× 2 unit matrix and Q is a 2× 2 matrix of form

Q =

(ϕ1 ϕ0

ϕ0 ϕ−1

). (17)

In addition, it is noted that when parameter c2 = c0, Eq. (4) is transformed into the followingthree coupled NLS equation

i∂ϕ±1

∂T= −∂2ϕ±1

∂X2+ 2c0(|ϕ±1|2 + 2|ϕ0|2)ϕ±1 + 2c0ϕ

∗∓1ϕ

20, (18a)

i∂ϕ0

∂T= −∂2ϕ0

∂X2+ 2c0(|ϕ+1|2 + |ϕ0|2 + |ϕ−1|2)ϕ0 + 2c0ϕ

∗0ϕ+1ϕ−1, (18b)

which is equivalent to the following matrix NLS equation [4]

iQT + QXX − 2c0QQ†Q = 0. (19)

Next substituting the matrix representations (15) of the elements x1, x2, x3, x4, x5, x6 intothe commutation relations in the Appendix, we find that only when parameter c2 = 0 thesecommutation relations satisfy, which denotes that when c2 = 0 Eq. (4) has nontrivial prolongationstructure. In this case, Eq. (4) is transformed into the well-known three component Manakovequation [33]

i∂ϕ±1

∂T= −∂2ϕ±1

∂X2+ c0(|ϕ±1|2 + 2|ϕ0|2 + |ϕ∓1|2)ϕ±1, (20a)

i∂ϕ0

∂T= −∂2ϕ0

∂X2+ c0(|ϕ+1|2 + 2|ϕ0|2 + |ϕ−1|2)ϕ0. (20b)

Here the nontrivial 4× 4 matrix representations of the other elements in the prolongation algebraL can also be obtained immediately. For simplify, we don’t list them here one by one. So the linear

8

spectral problem for Eq. (20) is

ζX =

−iλ ϕ1 ϕ0 ϕ−1

12c0ϕ

∗1 iλ 0 0

c0ϕ∗0 0 iλ 0

12c0ϕ

∗−1 0 0 iλ

ζ, (21a)

ζT =

A1 2λϕ1 + iϕ1X 2λϕ0 + iϕ0X 2λϕ−1 + iϕ−1X

λ c0 ϕ∗1 − 12 ic0ϕ

∗1X

i2c0|ϕ1|2 + 2 iλ2 i

2c0 ϕ0ϕ∗1

i2c0ϕ

∗1ϕ−1

2λ c0 ϕ∗0 − ic0ϕ∗0X ic0ϕ

∗0ϕ1 ic0|ϕ0|2 + 2 iλ2 ic0ϕ

∗0ϕ−1

λ c0 ϕ∗−1 − i2c0 ϕ∗−1X

i2c0 ϕ1ϕ

∗−1

i2c0 ϕ0ϕ

∗−1 A2

ζ, (21b)

with A1 = − i2c0(|ϕ1|2 + 2|ϕ0|2 + |ϕ−1|2)− 2 iλ2 and A2 = i

2c0 |ϕ−1|2 + 2 iλ2.We have tried to embed the prolongation algebra L into the Lie algebra sl(l, C) for l > 4 by

using the nilpotent elements x1, x2, x3, x4, x5, x6 and neutral elements x14, x18, x22, but it is stillfound that when parameter c2 6= c0 or c2 6= 0, the prolongation algebra L has trivial matrixrepresentations, which corresponds to trivial prolongation structures of the three coupled NLSequation (4). Therefore, in this case the three coupled NLS equation (4) doesn’t have linearspectral problem and it is not completely integrable.

Thus we conclude that only when c2 = c0 or c2 = 0 the three coupled NLS equation (4) is Laxintegrability. This finishes the proof of the theorem.

4 Matter-wave solitons in F = 1 spinor Bose-Einstein condensate

In this section, we propose the explicit N -bright soliton solutions to the three coupled NLS equation(4) for the case of c2 = c0 and c2 = 0, respectively. From these solutions, the exact matter-wavesoliton solutions of the F = 1 spinor BEC in an expulsive harmonic potential are constructed. Inaddition, the dynamics of the matter-wave solitons in the F = 1 spinor BEC are examined. Itis shown that the spinor BEC with time-dependent atomic scattering lengths in contrast to BECwith homogeneous atomic scattering lengths exhibits certain shape changing soliton collisions.

4.1 Exact N-bright soliton solutions to the integrable case b2 = b0

We first study the exact N -bright soliton solutions of the three coupled NLS equation (18) which isjust the matrix NLS equation (19). Since the matrix NLS equation is integrable, the linear spectralproblem of this equation can be solved exactly by the IST method. Recently, the initial valueproblem of the matrix NLS equation (19) was solved and the general expression for the N -brightsoliton solution was obtained under the constraint c0 = −1 and the vanishing boundary conditionϕm(T, X) → 0,m = 0,±1 as X → ±∞, which is [4]

(ϕ1 ϕ0

ϕ0 ϕ−1

)= (I2 · · · I2)S−1

Π1eχ1

· · ·ΠNeχN

, (22)

9

where S is a 2N × 2N matrix with Sij satisfying

Sij = δijI2 +N∑

l=1

Πi ·Π†l(ki + k∗l )(kj + k∗l )

eχi+χ∗l , 1 ≤ i, j ≤ N, (23)

and Πj , χj are given by

Πj =

(θj σj

σj γj

), 2|σj |2 + |θj |2 + |γj |2 = 1, χj = kjX + ik2

j T − νj . (24)

Here Πj is called polarization matrix, the complex constant kj is a discrete eigenvalue of the j-thsoliton, which determines a bound state by the potential Q in the context of ISM, and νj is a realconstant which can be used to modulate the initial displacement of a soliton.

Therefore, the exact nonautonomous N -bright soliton solution to the quasi-one-dimensionalthree component GP equation (2) with b0 = b2 = −sech(ωt) is

ψm = βmϕm(T, X)ei(α1x2+α2x+α3), m = 0,±1, (25)

where X = η1x + η2 and T, η1, η2, βm, α1, α2 and α3 are given in Eq. (6) with c2 = c0 = −1,and ϕ±1(T, X) and ϕ0(T, X) satisfy the three coupled NLS equation (4) with c2 = c0, which areexpressed by Eq. (22) with Eqs. (23) and (24).

When N = 2, exact solution (25) is a two-bright soliton solution which can describe the collisionsbetween two matter-wave bright solitons. In Fig. 1, we demonstrate the collision dynamics betweentwo matter-wave bright solitons in the components ψ1(t, x), ψ0(t, x) and ψ−1(t, x) of the quasi-one-dimensional three component GP equation (2) with b0 = b2 = −sech(ωt), where the parametersare k1 = −0.5 + 0.45i, k2 = 0.5− 0.45i, σ1 = σ2 = 4/17, γ1 = θ2 = 1/17, θ1 = γ2 = 16/17, ν1 = ν2 =0, α0 = 1, ω = 0.01.

Fig. 1. Collision dynamics of the two matter-wave solitons in the F = 1 spinor BEC with nonlinearitycoefficients b0 = b2 = −sech(ωt) trapped in an expulsive harmonic potential V (x) = − 1

2ω2x2.

10

4.2 Exact N-bright soliton solutions to the integrable case b2 = 0

As pointed above, the scattering lengths a0 and a2 for 87Rb atoms [14] in the F = 1 hyperfine stateare nearly equal so that the dimensionless nonlinearity coefficients b2 = −6.94× 10−5 which is verysmall and can be ignored. So the three component GP equation (2) with b2 = 0 can approximatelydescribe the dynamics of 87Rb condensate. From Eq. (6) we have b2 = c2sech(ωt), thus in this casec2 can be approximately zero and it is important to solve the three coupled Manakov equation (20)to obtain the exact matter-wave soliton solutions of the the quasi-one-dimensional three componentGP equation (2) with b2 = 0.

In what follows, we fist examine the N -bright soliton solutions of the coupled NLS equation(20) using the Riemann-Hilbert formulation [30, 31] based on the linear spectral problem (21).To do so, we consider the localized solutions of Eq. (20) with zero boundary conditions, i.e. thepotentials ϕ1, ϕ0 and ϕ−1 in Eq. (21) decay to zero sufficiently fast as X → ±∞. From Eq. (21),we see that when X → ±∞, ζ ∝ e−iλΛX−2iλ2ΛT with Λ = diag(1,−1,−1,−1). Introduce variabletransformation

ζ = Je−iλΛX−2iλ2ΛT , (26)

so that J is (X,T )-independent at infinity, then the linear spectral problem (21a) and (21b) become

JX = −iλ[Λ, J ] + WJ, (27a)

JT = −2iλ2[Λ, J ] + V J, (27b)

with

W =

0 ϕ1 ϕ0 ϕ−1

12c0ϕ

∗1 0 0 0

c0ϕ∗0 0 0 0

12c0ϕ

∗−1 0 0 0

,

V = 2λW +

− i2c0(|ϕ1|2 + 2|ϕ0|2 + |ϕ−1|2) iϕ1X iϕ0X iϕ−1X

−12 ic0ϕ

∗1X

i2c0|ϕ1|2 i

2c0 ϕ0ϕ∗1

i2c0ϕ

∗1ϕ−1

−ic0ϕ∗0X ic0ϕ

∗0ϕ1 ic0|ϕ0|2 ic0ϕ

∗0ϕ−1

− i2c0 ϕ∗−1X

i2c0 ϕ1ϕ

∗−1

i2c0 ϕ0ϕ

∗−1

i2c0 |ϕ−1|2

.

Here [Λ, J ] = ΛJ − JΛ is the commutator, tr(W ) = tr(V ) = 0 and

W † = −BWB−1, (28)

where † represents the Hermitian of a matrix, and B = diag(−c0, 2, 1, 2).In the scattering problem, we first introduce matrix Jost solutions J±(x, λ) of (27a) with the

asymptotic conditionJ± → I, when X → ±∞, (29)

11

where I is a 4 × 4 unit matrix. Here the subscripts in J± refer to which end of the X-axis theboundary conditions are set. Then due to tr(Q) = 0 and Abel’s formula we have det(J±) = 1 forall X. Next we denote E = e−iλΛX . Since Ω ≡ J+E and Θ ≡ J−E are both solutions of (27a),they must be linearly related, Θ = ΩS(λ), i.e.

J−E = J+ES(λ), λ ∈ R (30)

where

S(λ) =

s11 s12 s13 s14

s21 s22 s23 s24

s31 s32 s33 s34

s41 s42 s43 s44

, λ ∈ R

is the scattering matrix, and R is the set of real numbers. Notice that det(S(λ)) = 1 sincedet(J±) = 1. In addition, (Θ, Ω) satisfy the spectral equation (21a), i.e.

ζX + iλΛζ = Wζ. (31)

If we treat the Wζ term in the above equation as an inhomogeneous term and notice that thesolution to the homogeneous equation on its left hand side is E, then using the method of variationof parameters as well as the boundary conditions (29), we can turn (31) into Volterra integralequations for (Θ, Ω). These equations can be cast in terms of J± as

J−(λ,X) =I +∫ X

−∞eiλΛ(Y−X)W (Y )J−(λ, Y )eiλΛ(X−Y )dY, (32a)

J+(λ,X) =I −∫ ∞

XeiλΛ(Y−X)W (Y )J+(λ, Y )eiλΛ(X−Y )dY. (32b)

Thus J± allow analytical continuations off the real axis λ ∈ R as long as the integrals on theirright hand sides converge. We can easily see that the integral equation for the first column ofJ− contains only the exponential factor eiλ(X−Y ) which decays when λ is in the upper half planeC+, and the integral equation for the last three columns of J+ contain only the exponential factoreiλ(Y−X) which also decays when λ is in the upper half plane C+. Thus, these three columns canbe analytically continued to the upper half plane λ ∈ C+. Similarly, we find that the last threecolumns of J− and the first column of J+ can be analytically continued to the lower half planeλ ∈ C−. If we express (Θ, Ω) as a collection of columns

Θ = [Θ1,Θ2,Θ3, Θ4], Ω = [Ω1, Ω2, Ω3, Ω4],

then the Jost solutions

P+ = [Θ1, Ω2, Ω3, Ω4]eiλΛX = J+H2 + J−H1, (33)

are analytic in λ ∈ C+, and the Jost solutions [Ω1, Θ2, Θ3, Θ4]eiλΛX are analytic in λ ∈ C−. HereH1 = diag(1, 0, 0, 0),H2 = diag(0, 1, 1, 1). In addition, from the Volterra integral equations (32), wefind that

P+(X, λ) → I, as λ ∈ C+ →∞,

12

[Ω1, Θ2, Θ3, Θ4]eiλΛX → I, as λ ∈ C− →∞.

Next we construct the analytic counterpart of P+ in the C−. Noting that the adjoint equationof (27a) reads

KX = −iλ[Λ,K]−KW. (34)

The inverse matrices J−1± solve this adjoint equation. If we express Θ−1 and Ω−1 as a collection of

rows as

Θ−1 =

Θ1

Θ2

Θ3

Θ4

, Ω−1 =

Ω1

Ω2

Ω3

Ω4

,

then by similar techniques as used above, we can show that adjoint Jost solutions

P− = e−iλΛX

Θ1

Ω2

Ω3

Ω4

= H2J

−1+ + H1J

−1− (35)

are analytic for λ ∈ C−. In the same way, we see that

P−(X, λ) → I, as λ ∈ C− →∞.

Now we have constructed two matrix functions P+ and P− which are analytic in C+ and C−respectively. On the real line, they are related by

P−(X, λ)P+(X,λ) = G(X, λ), λ ∈ R, (36)

where

G(X,λ) =E(H1 + H2S)(H1 + S−1H2)E−1

=E

1 s12 s13 s14

s21 1 0 0

s31 0 1 0

s41 0 0 1

E−1. (37)

Equation (36) with (37) is just a matrix Riemann-Hilbert problem. The asymptotics

P±(X, λ) → I, as λ →∞ (38)

provides the canonical normalization condition for this Riemann-Hilbert problem.The solution to the Riemann-Hilbert problem (36) with (37) will not be unique unless the zeros

of det P+ and detP− in the upper and lower half of the λ plane are also specified, and the kernelstructures of P± at these zeros are provided. From the definitions of P± as well as the scatteringrelations between J+ and J−, we see that

detP+(X,λ) = s11(λ), detP−(X, λ) = s11(λ), (39)

13

where s11 =(S−1

)11

= (s33s44 − s34s43) s22 + (s43s24 − s23s44) s32 + (s23s34 − s33s24) s42. Supposes11 has zeros λk ∈ C+, 1 ≤ k ≤ N, and s11 has zeros λk ∈ C−, 1 ≤ k ≤ N. For simplicity,we assume that all zeros (λk, λk), k = 1, . . . , N are simple zeros of (s11, s11), which is the genericcase. In this case, each of kerP+(λk) and kerP−(λk) contains only a single column vector vk androw vector vk, respectively

P+(λk)vk = 0, vkP−(λk) = 0, 1 ≤ k ≤ N. (40)

If the Riemann-Hilbert problem (36) with the normalization condition (38) and zero structures(40) can be solved, then one can readily reconstruct the potential W as follows. Notice that P+ isthe solution of the spectral problem (27a). Thus if we expand P+ at large λ as

P+(x, λ) = I +1λ

P+1 (x) + O(λ−2), λ →∞, (41)

and inserting this expansion into (27a), then comparing O(1) terms, we find that

W = i[Λ, P+1 ] =

0 2iP12 2iP13 2iP14

−2iP21 0 0 0−2iP31 0 0 0−2iP41 0 0 0

. (42)

Thus, the potentials ϕ1, ϕ0 and ϕ−1 are reconstructed as

ϕ1 = 2iP12, ϕ0 = 2iP13, ϕ−1 = 2iP14, (43)

where P+1 = (Pij).

The symmetry property (28) of the potential W gives rise to symmetry properties in the scat-tering matrix as well as in the Jost functions. Taking the Hermitian of the spectral equation (27a),we have

(J†±)X = −iλ[Λ, J†±] + J†±W †. (44)

Right-multiplying this equation by B and using (28) yields

(J†±B)X = −iλ[Λ, J†±B]− J†±BW. (45)

This equation shows that J†±B is also a fundamental matrix of the adjoint equation (34). Recallingthat J−1

± satisfies (34) as well, we see that J†±B must be linearly related to J−1± , i.e. J†±B = DJ−1

± ,where D is x-independent. Using the large-x boundary conditions of J±, we find that D = B. Sowe find that J± satisfies the involution property

J†± = BJ−1± B−1.

From this property as well as the definitions (33) and (35) for P±, we can see that the analyticsolutions P± satisfy the involution property

(P+)†(λ∗) = BP−(λ)B−1. (46)

14

Then in view of the relation J−E = J+ES, we see that S satisfies the involution property

S†(λ∗) = BS−1(λ)B−1. (47)

Due to this involution property, we have the symmetry relation

λk = λ∗k (48)

for the zeros of s11(λ) and s11(λ). To obtain the symmetry properties for the eigenvectors vk andvk, we take the Hermitian of the first equation in (40). Upon the use of the involution properties(46) and (48), we get

v†kBP−(λk) = 0. (49)

Then comparing it with the second equation in (40), we see that eigenvectors (vk, vk) satisfy theinvolution property

vk = v†kB. (50)

To find the spatial evolutions for vectors vk(x, t), we take the x-derivative to equation P+vk = 0.By using (27a), one gets

P+(λk, X)(dvk

dX+ iλkΛvk) = 0, (51)

thusdvk

dX= −iλkΛvk. (52)

In what follows, we determine the time dependence of the scattering data vk and vk in a similarway to that for the x-dependence of vk and vk. Taking the t-derivative to equation P+vk = 0 andusing (27b), one gets

P+(λk, X, T )(dvk

dT+ 2iλ2

kΛvk) = 0, (53)

thusdvk

dT= −2iλ2

kΛvk. (54)

Combining these results, we get

vk(X,T ) =e−iλkΛX−2iλ2kΛT vk0, (55a)

vk(X,T ) =vk0eiλ∗kΛX+2iλ∗2k ΛT B, (55b)

where (vk0, vk0) are now constants.The N -soliton solutions for Eq. (20) can be obtained by solving the Riemann-Hilbert problem

(36) with G = I and the discrete scattering data λk, λk, vk0, vk0. The solutions to this specialRiemann-Hilbert problem (36) have been derived in [30], and the result is

P+1 (λ,X, T ) =

N∑

j,k=1

vj

(M−1

)jk

vk, (56)

15

where vectors vj and vj are given by (55a) and (55b), and the matrix M is given by

Mjk =vjvk

λ∗j − λk, 1 ≤ j, k ≤ N.

The N -soliton solutions for Eq. (20) are then given from (40) as

ϕ1 = 2iP12 = 2i

N∑

j,k=1

vj(M−1)jkvk

12

, (57)

ϕ0 = 2iP13 = 2i

N∑

j,k=1

vj(M−1)jkvk

13

, (58)

ϕ−1 = 2iP14 = 2i

N∑

j,k=1

vj(M−1)jkvk

14

. (59)

Thus the general N -soliton solution for Eq. (20) can be written out in vector form as

ϕ1

ϕ0

ϕ−1

= 2i

N∑

j,k=1

2δjτ∗k

δjε∗k

2δj

eθj−θ∗k(M−1)jk, (60)

withMjk =

1λ∗j − λk

[(2τ∗j τk + ε∗jεk + 2

)e−(θk+θ∗j ) − c0δ

∗j δke

θk+θ∗j], (61)

where θk = −iλkX − 2iλ2kT , and we have chosen vk0 = [δk, τk, εk, 1]T .

If setting N = 1 in Eq. (60) with (61), we have

ϕ1 =4i(λ∗1 − λ1)δ1τ

∗1 eθ1−θ∗1

(2|τ1|2 + |ε1|2 + 2)e−(θ1+θ∗1) − c0|δ1|2eθ1+θ∗1, (62a)

ϕ0 =2i(λ∗1 − λ1)δ1ε

∗1e

θ1−θ∗1

(2|τ1|2 + |ε1|2 + 2)e−(θ1+θ∗1) − c0|δ1|2eθ1+θ∗1, (62b)

ϕ−1 =4i(λ∗1 − λ1)δ1e

θ1−θ∗1

(2|τ1|2 + |ε1|2 + 2)e−(θ1+θ∗1) − c0|δ1|2eθ1+θ∗1, (62c)

where θ1 = −iλ1X − 2iλ21T. It is found that the quantity c0 must always negative to avoid the

singularity in Eq. (62). Thus letting λ1 = λ11 + λ12i and −c0|δ1|2/(2|τ1|2 + |ε1|2 + 2) = e2ξ1 , thebright single soliton solution (62) can be written as

ϕ1

ϕ0

ϕ−1

=

λ12

2|τ1|2 + |ε1|2 + 2r1e−ξ1eθ1−θ∗1 sech(θ1 + θ∗1 + ξ1), (63)

16

with r1 = [4δ1τ∗1 , 2δ1ε

∗1, 4δ1]T . From the notation above, we have

θ∗1 + θ1 = 2 λ12 (X + 4 λ11 T ), θ1 − θ∗1 = −2 i[λ11X − 2(λ212 − λ2

11) T ].

Therefore, the bright single soliton solution (63) can be rewritten as

ϕ1

ϕ0

ϕ−1

=

λ12

2|τ1|2 + |ε1|2 + 2r1e−ξ1sech[2λ12 (X + 4 λ11 T ) + ξ1]e−2 i[λ11X−2(λ2

12−λ211) T ], (64)

which indicates that the bright single solitons in the integrable three coupled NLS equation (20)have the shape of hyperbolic secant functions with peak amplitudes λ12r1e−ξ1/(2|τ1|2 + |ε1|2 + 2),and they have the same velocity 4λ11 depending only on the real part of the eigenvalue λ1.

Fig. 2. Collisions of the two matter-wave solitons in the F = 1 spinor BEC with nonlinearity coefficientsb0 = c0sech(ωt) and b2 = 0 trapped in an expulsive harmonic potential V (x) = − 1

2ω2x2 with parameters in(67).

When N = 2, the soliton solutions (60) with (61) is a two-bright soliton solution to the coupledNLS equation (20)

ϕ1

ϕ0

ϕ−1

= 2i

2∑

j,k=1

2δjτ∗k

δjε∗k

2δj

eθj−θ∗k(M−1)jk, (65)

withMjk =

1λ∗j − λk

[(2τ∗j τk + ε∗jεk + 2

)e−(θk+θ∗j ) − c0δ

∗j δke

θk+θ∗j], 1 ≤ j, k ≤ 2, (66)

where θk = −iλkX − 2iλ2kT with λk = λk1 + iλk2, k = 1, 2.

17

So the exact nonautonomous soliton solution to the quasi-one-dimensional three component GPequation (2) with b0 = c0sech(ωt) and b2 = 0 is expressed by Eq. (25) where X = η1x + η2 andT, η1, η2, βm, α2 and α3 satisfy Eq. (6) with c2 = 0, and ϕ±1(T, X) and ϕ0(T, X) satisfy the threecoupled NLS equation (20). In this case, the single matter-wave soliton solution is expressed byEq. (25) with Eqs. (6) and (64), the two matter-wave soliton solution is expressed by Eq. (25)with Eqs. (6) and (65) with (66), and the N -matter-wave soliton solution is expressed by Eq. (25)with Eqs. (6) and (60) with (61).

In Figs. 2 and 3, we display the collision dynamics and bound states of the matter-wave solitonsin the F = 1 spinor BEC with nonlinearity coefficients b0 = c0sech(ωt) and b2 = 0 trapped in anexpulsive harmonic potential V (x) = −1

2ω2x2. The parameters in Fig. 2 are

δ1 = 1 + 1.5i, δ2 = 1.5− 1.2i, τ1 = 0.2− 0.2i, τ2 = 0.5 + 0.6i,

ε1 = 0.5− i, ε2 = 1 + 0.5i, λ1 = 0.2 + 0.5i, λ2 = −0.3 + 0.4i, (67)

and the parameters in Fig. 3 are

δ1 = δ2 = 3, τ1 = τ2 = 2, ε1 = ε2 = 1, λ1 = 0.6i, λ2 = 0.4i. (68)

In both figures, the other parameters are c0 = −1, ω0 = 0.01, α0 = 1 and N = 2. It is observedfrom Fig. 2 that shape changing interaction occurs in this two-bright matter-wave soliton, wherematter waves redistribute energy in each component. Fig. 3 demonstrates the two-soliton boundstates with vanishing background. The two constituent solitons stay together and form a boundstate which moves at equal velocities. From their figures we find that this bound state is spatiallylocalized and its width changes periodically with time, thus we also call it a “breather” followingthe literatures. Thus bound states here are different with the usual breathes with plane wavebackground. Moreover, our numerical simulation shows that the bound states here are stable andthey don’t split into fundamental solitons.

18

Fig. 3. Bound states of the two matter-wave solitons in the F = 1 spinor BEC with nonlinearity coefficientsb0 = c0sech(ωt) and b2 = 0 trapped in an expulsive harmonic potential V (x) = − 1

2ω2x2 with parameters in(68).

5 Conclusions

In conclusion, we have studied the integrability and exact matter-wave soliton solutions of thethree component GP equation from spinor BEC with time-dependent atomic scattering lengths inan expulsive harmonic potential. We first transform the three component GP equation into a threecoupled NLS equation, then examine the Lax integrability of the three coupled NLS equation byprolongation technique and find two integrable systems, finally we construct the exact N-brightsoliton solutions of the two integrable systems to derive some exact nonautonomous matter-wavesoliton solutions of the original three component GP equation. Our results show that when thes-wave scattering lengths are time-modulated, there also exist various matter-wave solitons in theF=1 spinor BEC trapped in an expulsive harmonic potential, and the time evolutions of thematter-wave solitons are sharply changed by the time-dependent nonlinear coefficients. This re-search provides an understanding of the possible mechanism for the fraction of atoms transformbetween the three components in the spinor BEC. The shape changing interactions and two-solitonbound states found in this paper will encourage the investigation of novel matter-wave solitons bysuitable control of time-dependent atomic scattering lengths in real spinor BEC experiments.

Acknowledgments

This work is supported by NSFC under Grant Nos. 11001263, 11171032, 11271362 and 11375030,Beijng Natural Science Foundation under Grant No. 1132016 and Beijing Nova program No.Z131109000413029.

Appendix.

The commutation relations of the prolongation algebra L = x0, x1, x2, · · · , x22 are shown asfollows

[x1, x2] = 0, [x1, x3] = 0, [x2, x3] = 0, [x4, x6] = 0, [x5, x6] = 0, [x4, x5] = 0, [x0, x1] = x8, (A.1)

[x1, x16] = 0, [x0, x2] = x9, [x0, x4] = x11, [x0, x3] = x10, [x0, x5] = x12, [x0, x6] = x13, (A.2)

[x1, x4] = x14, [x1, x5] = x15, [x1, x6] = x16, [x2, x4] = x17, [x2, x5] = x18, [x2, x6] = x19, (A.3)

[x3, x4] = x20, [x3, x5] = x21, [x3, x6] = x22, 2 x4c2 = [x5, x21], [x5, x15] = 2x6c2, (A.4)

[x2, x17] + 2x3c2 = 0,−x1(c0 − c2) = [x3, x16] + [x1, x22], [x0, x16] + [x1, x13] = [x6, x8], (A.5)

[x3, x12] + [x0, x21] = [x5, x10], x5(c0 + c2) = [x6, x21] + [x5, x22], [x1, x8] = 0, [x4, x17] = 0, (A.6)

[x4, x20] = 0, [x1, x11] + [x0, x14] = [x4, x8], x5(c0 + c2] = [x5, x14] + [x4, x15], [x1, x15] = 0, (A.7)

19

[x4, x9] = [x2, x11] + [x0, x17], [x4, x16]− x6(c0 − c2) + [x6, x14] = 0, [x4, x12] + [x5, x11] = 0, (A.8)

x4(c0 − c2) = [x6, x20] + [x4, x22], [x6, x7] = i[x0, x13], [x5, x7] = i[x0, x12], [x6, x19] = 0, (A.9)

[x4, x21] + [x5, x20] = 0, [x3, x11] + [x0, x20] = [x4, x10], [x0, x22] + [x3, x13] = [x6, x10], (A.10)

[x5, x19] + [x6, x18]− 2(c0 + c2) x6 = 0, (c0 + c2)x6 = [x6, x22], [x0, x10] = i[x3, x7], (A.11)

[x2, x22] + [x3, x19] + (c0 + c2)x2 = 0, [x3, x17] + [x2, x20] = 0, [x0, x11] + i[x4, x7] = 0, (A.12)

2x5c2 = [x6, x17]+[x4, x19], [x2, x21]+[x3, x18]+2x3(c0+c2) = 0, [x6, x16] = 0, [x0, x7] = 0, (A.13)

2x4(c0+c2) = [x5, x17]+[x4, x18], [x6, x15]+[x5, x16] = 0, 2x1c2+[x2, x19] = 0, [x5, x12] = 0, (A.14)

[x4, x13] = −[x6, x11], [x1, x20]+ [x3, x14] = x3(c2− c0), (c0 + c2)x2 +[x2, x14]+ [x1, x17] = 0, (A.15)

[x6, x12] + [x5, x13] = 0, [x5, x9] = [x2, x12] + [x0, x18], [x1, x21] + 2x2c2 + [x3, x15] = 0, (A.16)

[x2, x13] + [x0, x19] = [x6, x9], [x0, x15] + [x1, x12] = [x5, x8], x4(c0 + c2) = [x4, x14], (A.17)

[x2, x8] = −[x1, x9], [x3, x22] + (c0 + c2)x3 = 0, [x3, x10] = [x6, x13] = [x4, x11] = 0, (A.18)

[x2, x18] + 2x2c0 = 0, [x1, x7] = −i[x0, x8], [x3, x20] = 0, [x3, x8] = −[x1, x10], (A.19)

(c0 + c2)x1 + [x1, x14] = 0, [x2, x7] = −i[x0, x9], 2x1(c0 + c2) + [x2, x15] + [x1, x18] = 0, (A.20)

2x5c0 = [x5, x18], [x1, x19] + [x2, x16] = 0, [x2, x9] = 0, [x2, x10] + [x3, x9] = 0, [x3, x21] = 0. (A.21)

References

[1] Pethick CJ, Smith H. Bose-Einstein Condensation in Dilute Gases. Cambridge UniversityPress, Cambridge; 2002.

[2] Pitaevskii LP, Stringari S. Bose-Einstein Condensation. Oxford University Press, Oxford; 2003.

[3] Stenger J, Inouye S, Stamper-Kurn DM, Miesner HJ, Chikkatur AP, Ketterle W. Nature.1998;396: 345.

[4] Ieda J, Miyakawa T, Wadati M. Phys Rev Lett 2004;93: 194102.

[5] Li L, Li Z, Malomed BA, Mihalache D, Liu WM. Phys Rev A 2005;72: 033611.

[6] Nistazakis HE, Frantzeskakis DJ, Kevrekidis PG, Malomed BA, Carretero-Gonzalez R, PhysRev A 2008;77: 033612.

[7] Dabrowska-Wuster BJ, Ostrovskaya EA, Alexander TJ, Kivshar YS. Phys Rev A 2007;75:023617.

[8] Kevrekidis PG, Frantzeskakis DJ, Carretero-Gonzalez R. Emergent Nonlinear Phenomena inBose-Einstein Condensates. Theory and Experiment. Springer-Verlag, Berlin; 2008.

[9] Leanhardt AE, Shin Y, Kielpinski D, Pritchard DE, Ketterle W. Phys Rev Lett 2003;90:140403.

20

[10] Szankowski P, Trippenbach M, Infeld E, Rowlands G. Phys Rev Lett 2010;105: 125302.

[11] Ho TL. Phys Rev Lett 1998;81: 742.

[12] Ohmi T, Machida K. J Phys Soc Jpn 1998;67: 1822.

[13] Stamper-Kurn DM, Ketterle W. Coherent Atomic Matter Waves, edited by R. Kaiser, C. West-brook, and F. David, Vol. 72 of Les Houches Summer School, p. 137, 2000, e-print arXiv:cond-mat/0005001.

[14] van Kempen EG, Kokkelmans SJ, Heinzen DJ, Verhaar BJ. Phys Rev Lett 2002; 88: 093201.

[15] Klausen NN, Bohn JL, Greene CH. Phys Rev A 2001; 64: 053602.

[16] Rajendrana S, Muruganandamb P, Lakshmanana M. Physica D 2010; 239: 366-386.

[17] Zhang XF, Hu XH, Liu XX, Liu WM. Phys Rev A 2009;79: 033630.

[18] Janis J, Banks M, Bigelow NP. Phys Rev A 2005;71: 013422.

[19] Atre R, Panigrahi PK, Agarwal GS. Phys Rev E 2006;73: 056611.

[20] Yamazaki R, Taie S, Sugawa S, Takahashi Y. Phys Rev Lett 2010;105: 050405.

[21] Wahlquist HD, Estabrook FB. J Math Phys 1975;16: 1.

[22] Dodd R, Fordy AP. Phys Lett A 1982;89: 168.

[23] Chern SS, Griffiths PA, Bryant RL. Exterior differential systems. Beijing: China ScientificPress; 1982.

[24] Hermann R. Gauge Fields and Cartan-Ehresmann Connections. Math. Sci. Press; 1975.

[25] Humphreys JE. Introduction to Lie algebra and representation theory, GTM 9, Springer; 1972.

[26] Rogers C, Shadwick WF. Backlund Transformations and Their Applications. Academic Press;1982.

[27] Estabrook FB, Wahlquist HD. J Math Phys 1976;17: 1293.

[28] Wang DS. Optical Solitons in a Nonlinear Fiber Medium with Higher-Order Effects. in thebook “Recent Progress in Optical Fiber Research” edited by Moh. Yasin, Sulaiman W. Harunand Hamzah Arof, InTech, pp.49-64; 2012.

[29] Ablowitz MJ, Segur H. Solitons and the Inverse Scattering Transform. SIAM; 1981.

[30] Shchesnovich VS, Yang J. J Math Phys 2003;44: 4604.

[31] Wang DS, Zhang D, Yang J. J Math Phys 2010;51: 023510.

[32] Kanna T, Sakkaravarthi K, Kumar CS, Lakshmanan M, Wadati M. J Math Phys 2009;50:113520.

[33] Vijayajayanthi M, Kanna T, Lakshmanan M. Phys Rev A 2008;77: 013820.

21

Research Highlights

March 25, 2014

Integrability classification of the three component Gross-Pitaevskii equation from spinor-

1 Bose-Einstein condensate is proposed. Some new exact nonautonomous matter-wave soli-

ton solutions with varying amplitudes and speeds are obtained. The collision dynamics of

the two matter-wave solitons are analyzed and the shape changing interaction phenomena

are found.

1