Solutions of multi-component NLS models and spinor Bose-Einstein condensates

Solutio n s of m ulti-comp onen t N LS mo de ls and s pinor Bos e -Einstei n condens ates V. S. Gerdjiko v, N. A. Kostov, T. I. V alchev Institute for Nu cle ar R ese ar ch and Nucle ar Ener gy, Bulgarian A c ademy of Scienc es, 72 Tsarigr adsko chausse e 1784 Sofia, Bulgar ia Abstract A thr ee- an d fiv e-comp onen t nonlinear S c h r o dinger-t yp e mod els, wh ic h describ e spinor Bose-Einstein conden s ates (BEC’s) with h yp erfine s tructures F = 1 and F = 2 resp ectiv ely , are studied. These mod els for particular v alues of the cou- pling constan ts are int egrable by the in v erse scattering metho d. Th ey are related to symmetric spaces of BD . I -type ≃ SO(2r + 1) / SO(2) × SO(2r − 1) for r = 2 and r = 3. Using conv enien tly mo difi ed Zakharov-Shabat dressing p ro cedure we obtain differen t typ es of soliton solutions. Key wor ds: Bose-Einstein condensates, in tegrable systems, soliton mo dels 1 In t ro duction The dynamics of spinor BECs is described by a three-comp o nent Gross-Pitaevskii (GP) system of equations. In the o ne-dimensional a ppro ximation the GP sys- tem go es in to t he follow ing m ulticomp onen t nonlinear Sc hr¨ odinger (MNLS) equation in 1 D x -space [1]: i∂ t Φ 1 + ∂ 2 x Φ 1 + 2( | Φ 1 | 2 + 2 | Φ 0 | 2 )Φ 1 + 2Φ ∗ − 1 Φ 2 0 = 0 , i∂ t Φ 0 + ∂ 2 x Φ 0 + 2( | Φ − 1 | 2 + | Φ 0 | 2 + | Φ 1 | 2 )Φ 0 + 2Φ ∗ 0 Φ 1 Φ − 1 = 0 , (1) i∂ t Φ − 1 + ∂ 2 x Φ − 1 + 2( | Φ − 1 | 2 + 2 | Φ 0 | 2 )Φ − 1 + 2Φ ∗ 1 Φ 2 0 = 0 . W e consider BECs of alk ali atoms in the F = 1 hyperfine state, elongated in x direction and confined in the transv erse directions y , z b y purely optical means. Th us the assem bly of atoms in t he F = 1 h yp erfine state can b e described b y a normalized spinor wa v e v ector Φ ( x, t ) = (Φ 1 ( x, t ) , Φ 0 ( x, t ) , Φ − 1 ( x, t )) T whose comp onents are lab eled b y the v alues o f m F = 1 , 0 , − 1. The ab ov e mo del is in tegrable b y means of the in v erse scattering tra nsform method [1]. Preprint s ubmitted to Elsevier Prepr in t 22 Octob er 2018 It also allows an exact description of the dynamics and in teraction of brigh t solitons with spin degrees of freedom. Matter-w a v e solitons are exp ected to b e useful in atom laser, atom interferometry a nd coheren t atom transp ort. It could contribute to the realization of quantum informatio n pro cessing or computation, as a part of new fie ld of ato m optics. Lax pairs and g eometrical in terpretation of the MNLS mo dels related to sym- metric spaces (including the mo del ( 1 )) are giv en in [2]. Darb oux t ransforma- tion for this sp ecial in tegrable model is dev elop ed in [3 ]. In [4] the authors study soliton s olutions for the multicomponent Gross-Pitaevs kii equation for F = 2 spinor condensate b y t wo differen t metho ds assuming single-mo de amplitudes and by generalizing Hirota’s direct metho d for m ulticomp onent systems . They p oin t out the imp orta nce of in tegrable cases, whic h tak e place for particular c hoices of the coupling constan t s. The aim of presen t pap er is to show that b oth systems men tioned ab ov e are in- tegrable by the inv erse scattering metho d and are related to symmetric spaces [5] BD . I -type: ≃ SO(2r + 1 ) / SO(2) × SO(2r − 1) with r = 2 and r = 3 re- sp ectiv ely . In Section 2 we formulate the Lax represen tations for the mo dels. Section 3 is dev oted to the F = 2 BEC model. In Section 4 w e construct the fundamen tal ana lytic solutions of the corres p onding Lax op erator L and reduce the inv erse scattering problem (ISP) f o r L to a Riemann-Hilb ert prob- lem (RHP). Using the special prop erties o f the BD . I symmetric space s w e also o btain the minimal sets of scattering data T i eac h of whic h allo w one to reconstruct b oth the sc attering m atrix T ( λ ) and the correspo nding p oten- tial Q ( x, t ). This allo ws us to deriv e in Section 5 their soliton solutions using suitable mo dification o f the Z akharo v-Shabat dressing metho d, prop osed in [13,6]. 2 Multicomp onen t nonlinear Sc hr¨ odinger equations for BD.I. se- ries of symmetric spaces MNLS equations for the BD.I. series of symmetric spaces (algebras of the t yp e so (2 r + 1) and J dual to e 1 ) ha v e the Lax represen tation [ L, M ] = 0 as follo ws Lψ ( x, t, λ ) ≡ i∂ x ψ + ( Q ( x, t ) − λJ ) ψ ( x, t, λ ) = 0 . (2) M ψ ( x, t, λ ) ≡ i∂ t ψ + ( V 0 ( x, t ) + λV 1 ( x, t ) − λ 2 J ) ψ ( x, t, λ ) = 0 , (3) V 1 ( x, t ) = Q ( x, t ) , V 0 ( x, t ) = i ad − 1 J dQ dx + 1 2 h ad − 1 J Q, Q ( x, t ) i . (4) 2 where Q =        0 ~ q T 0 ~ p 0 s 0 ~ q 0 ~ p T s 0 0        , J = dia g (1 , 0 , . . . 0 , − 1) . (5) The 2 r − 1-vec tors ~ q and ~ p ha v e the form ~ q = ( q 2 , . . . , q r , q r +1 , q r +2 , . . . , q 2 r ) T , ~ p = ( p 2 , . . . , p r , p r +1 , p r +2 , . . . , p 2 r ) T , while the matrix s 0 represen ts t he metric in volv ed in the definition of so ( 2 r − 1), therefore it is related to the metric S 0 asso ciated with so (2 r + 1) in the following manner S 0 = 2 r +1 X k =1 ( − 1) k +1 E k , 2 r +2 − k =        0 0 1 0 − s 0 0 1 0 0        , ( E k n ) ij = δ ik δ nj (6) Next w e will use ~ E ± 1 = ( E ± ( e 1 − e 2 ) , . . . , E ± ( e 1 − e r ) , E ± e 1 , E ± ( e 1 + e r ) , . . . , E ± ( e 1 + e 2 ) ) , (7) W e w ill use also the ”sc alar product” ( ~ q · ~ E + 1 ) = r X k =2 ( q k ( x, t ) E e 1 − e k + q 2 r − k +2 ( x, t ) E e 1 + e k ) + q r +1 ( x, t ) E e 1 . Then the g eneric form of the p otentials Q ( x, t ) related to the se t yp e of sym- metric spaces is Q ( x, t ) = ( ~ q ( x, t ) · ~ E + 1 ) + ( ~ p ( x, t ) · ~ E − 1 ) , (8) where E α are the W eyl generators of the corresp onding Lie algebra ( see [5 ] for details) and ∆ + 1 is the set of all positive ro ots of so (2 r + 1) suc h that ( α, e 1 ) = 1. In fact ∆ + 1 = { e 1 , e 1 ± e k , k = 2 , . . . , r } . In terms of these notations the g eneric MNLS type equations connected to BD . I . acquire the fo r m i~ q t + ~ q xx + 2( ~ q , ~ p ) ~ q − ( ~ q , s 0 ~ q ) s 0 ~ p = 0 , i~ p t − ~ p xx − 2( ~ q , ~ p ) ~ p − ( ~ p, s 0 ~ p ) s 0 ~ q = 0 , (9) In the case of r = 2 if we imp ose the reduction p k = q ∗ k and in tro duce the new v a riables Φ 1 = q 2 , Φ 0 = q 3 / √ 2, Φ − 1 = q 4 then w e repro duce the equations (1). 3 3 F=2 spinor Bose-Einstein c ondensate, in tegrable case Let us introduce Hamiltonian for MNLS e quations (9) with ~ p = ǫ~ q ∗ , ǫ = ± 1 H MNLS = Z ∞ −∞ dx  ( ∂ x ~ q , ∂ x ~ q ∗ ) − ǫ ( ~ q , ~ q ∗ ) 2 + ǫ ( ~ q , s 0 ~ q ) ( ~ q ∗ , s 0 ~ q ∗ )  , (10) Define the num b er dens it y and the s inglet-pair amplitude b y [7,8,4] n = ( ~ Φ , ~ Φ ∗ ) = X α = − 2 ... 2 Φ α Φ ∗ α , Θ = ( ~ Φ , s 0 ~ Φ ) . (11) where Φ 2 = q 2 , Φ 1 = q 3 , Φ 0 = q 4 , Φ − 1 = q 5 ,Φ − 2 = q 6 . Then the singlet-pair amplitude ta ke t he form [7,8 ,4] Θ = 2Φ 2 Φ − 2 − 2Φ 1 Φ − 1 + Φ 2 0 . (12) The phys ical meaning of Θ is a measure of formation of spin-singlet ” pa irs” of b osons. The a ssem bly o f at oms in the F = 2 hy p erfine state can b e described b y a normalized spinor w a v e v ector Φ ( x, t ) = (Φ 2 ( x, t ) , Φ 1 ( x, t ) , Φ 0 ( x, t ) , Φ − 1 ( x, t ) , Φ − 2 ( x, t )) T , (13) whose comp onen t s ar e lab eled b y the v a lues of m F = 2 , 1 , 0 , − 1 , − 2. Here the energy functional within mean-field theory [9,10,7,8,4] is defined b y E GP [ Φ ] = Z ∞ −∞ dx ~ 2 2 m | ∂ x Φ | 2 + c 0 2 n 2 + c 2 2 f 2 + c 4 2 | Θ | 2 ! . (14) The coupling constan ts c i are real and can b e expressed in terms of a transv erse confinemen t radius and a linear com binat io n o f the s -wa v e scattering lengths of ato ms [1,11,12] and f describ e spin densities [4]. Cho osing c 2 = 0, c 4 = 1 and c 0 = − 2 w e obtain inte grable b y the in v erse scattering method mo del with the Hamiltonian. W e set for simplicit y ~ = 1 , 2 m = 1 without an y loss of generalit y . The evolution equation is described b y t he multi-compo nent Gross-Pitaevskii equation in one dimension [4 ] i ∂ Φ ∂ t = δ E GP [ Φ ] δ Φ ∗ . (15) Then w e hav e 4 i ~ Φ t + ~ Φ xx = − 2 ǫ ( ~ Φ , ~ Φ ∗ ) ~ Φ + ǫ ( ~ Φ , s 0 ~ Φ ) s 0 ~ Φ ∗ , (16) or in explicit form b y componen ts w e hav e i∂ t Φ ± 2 + ∂ xx Φ ± 2 = − 2 ǫ ( ~ Φ , ~ Φ ∗ )Φ ± 2 + ǫ (2Φ 2 Φ − 2 − 2 Φ 1 Φ − 1 + Φ 2 0 )Φ ∗ ∓ 2 , i∂ t Φ ± 1 + ∂ xx Φ ± 1 = − 2 ǫ ( ~ Φ , ~ Φ ∗ )Φ ± 1 − ǫ (2Φ 2 Φ − 2 − 2Φ 1 Φ − 1 + Φ 2 0 )Φ ∗ ∓ 1 , i∂ t Φ 0 + ∂ xx Φ 0 = − 2 ǫ ( ~ Φ , ~ Φ ∗ )Φ ± 0 + ǫ (2Φ 2 Φ − 2 − 2 Φ 1 Φ − 1 + Φ 2 0 )Φ ∗ 0 . 4 In verse scattering metho d and rec onst r uction of p oten tial from minimal scatter ing data Herein w e remind some basic features of the inv erse scattering theory appro- priate for t he special case of F = 2 spinor BEC equations. Solving the direct and the inv erse scattering problem (ISP) for L uses the Jost solutions whic h are defined b y , see [1 6] and the refe rences the rein lim x →−∞ φ ( x, t, λ ) e iλJ x = 1 1 , lim x →∞ ψ ( x, t, λ ) e iλJ x = 1 1 (17) and the scattering matrix T ( λ, t ) ≡ ψ − 1 φ ( x, t, λ ). Due to the sp ecial c hoice of J and to the fact that the Jo st solutions and the scattering matrix take v alues in the gr oup S O (2 r + 1) we can use the following blo ck -matrix structure of T ( λ, t ) T ( λ, t ) =        m + 1 − ~ b − T c − 1 ~ b + T 22 − s 0 ~ b − c + 1 ~ b + T s 0 m − 1        , (18) where ~ b ± ( λ, t ) are 2 r − 1-comp onen t v ectors, T 22 ( λ ) is a 2 r − 1 × 2 r − 1 blo c k and m ± 1 ( λ ), c ± 1 ( λ ) are scalar functions satisfying c ± 1 = 1 / 2( ~ b ± · s 0 ~ b ± ) /m ± 1 . Imp ortan t to o ls for reducing the ISP to a Riemann-Hilb ert pro blem (RHP) are the fundamen tal analytic solutio n ( F AS) χ ± ( x, t, λ ). Their construction is based on the gene ralized Gauss decomp osition of T ( λ, t ) χ ± ( x, t, λ ) = φ ( x, t, λ ) S ± J ( t, λ ) = ψ ( x, t, λ ) T ∓ J ( t, λ ) D ± J ( λ ) . (19) Here S ± J , T ± J upp er- and lo w er- blo c k-triangular matrices, while D ± J ( λ ) are blo c k-diagonal matrices with the same blo c k structure as T ( λ, t ) ab ov e. Skip- ping the details w e giv e the explicit expressions of the G auss factors in terms of the matrix ele men ts of T ( λ, t ) 5 S ± J ( t, λ ) = exp  ± ( ~ τ ± ( λ, t ) · ~ E ± 1 )  , T ± J ( t, λ ) = exp  ∓ ( ~ ρ ± ( λ, t ) · ~ E ± 1 )  , D + J =        m + 1 0 0 0 m + 2 0 0 0 1 /m + 1        , D − J =        1 /m − 1 0 0 0 m − 2 0 0 0 m − 1        , (20) where ~ τ ± ( λ, t ) = ~ b ∓ /m ± 1 , ~ ρ ± ( λ, t ) = ~ b ± /m ± 1 and m + 2 = T 22 + ~ b + ~ b − T m + 1 , m − 2 = T 22 + s 0 ~ b − ~ b + T s 0 m − 1 . If Q ( x, t ) evolv es according to (1) t hen the scattering matrix and its elemen ts satisfy the follo wing linear evolution equations i d ~ b ± dt ± λ 2 ~ b ± ( t, λ ) = 0 , i dm ± 1 dt = 0 , i d m ± 2 dt = 0 , (21) so the blo c k-diagonal matrices D ± ( λ ) can b e considered as generating func- tionals of the integrals of motion. The fact that a ll (2 r − 1) 2 matrix elemen ts of m ± 2 ( λ ) for λ ∈ C ± generate in tegrals of motio n reflect the sup erin tegrabilit y of the mo del and are due to the degeneracy of the disp ersion law of (1). W e remind that D ± J ( λ ) allow analytic e xtension for λ ∈ C ± and that their zeroes and p oles de termine the disc rete e igen v a lues of L . The F AS for real λ are linearly related χ + ( x, t, λ ) = χ − ( x, t, λ ) G J ( λ, t ) , G 0 ,J ( λ, t ) = S − J ( λ, t ) S + J ( λ, t ) . (22) One can rewrite eq. (22) in an equiv alen t form for the F AS ξ ± ( x, t, λ ) = χ ± ( x, t, λ ) e iλJ x whic h satisfy a lso the relatio n lim λ →∞ ξ ± ( x, t, λ ) = 1 1 . (23) Then these F AS satisfy ξ + ( x, t, λ ) = ξ − ( x, t, λ ) G J ( x, λ , t ) , G J ( x, λ , t ) = e − iλJ x G − 0 ,J ( λ, t ) e iλJ x . (24) Ob viously the sewing function G j ( x, λ , t ) is uniquely determined b y the Gauss factors S ± J ( λ, t ). In view of eq. (20) w e ar riv e to the follo wing Lemma 1. L et the p otential Q ( x, t ) is such that the L ax op er ator L has no discr ete eigenvalues. Th en as minimal set of sc attering data w hich determines uniquely the sc attering matrix T ( λ, t ) and the c orr esp o nding p otential Q ( x, t ) one c an c onsider either o ne of the sets T i , i = 1 , 2 T 1 ≡ { ~ ρ + ( λ, t ) , ~ ρ − ( λ, t ) , λ ∈ R } , T 2 ≡ { ~ τ + ( λ, t ) , ~ τ − ( λ, t ) , λ ∈ R } . (25) 6 Pr o of. i) F ro m the fact that T ( λ, t ) ∈ S O (2 r + 1) one can deriv e that 1 m + 1 m − 1 = 1 + ( ~ ρ + , ~ ρ − ) + 1 4 ( ~ ρ + , s 0 ~ ρ + )( ~ ρ − , s 0 ~ ρ − ) (26) for λ ∈ R . Using the analyticit y prop erties of m ± 1 w e can reco ver them from eq. (2 6) using Cauc h y-Plemelji formulae. Giv en T i and m ± 1 one easily recov ers ~ b ± ( λ ) and c ± 1 ( λ ). In order to reco ver m ± 2 one ag ain uses their analyticity prop erties, only now the problem reduces to a RHP f or functions on S O (2 r +1). The details will be pr esen ted elsew here. ii) Ob viously , given T i one unique ly reco v ers the sewing function G J ( x, t, λ ). In order to reco v er the corresp onding p o ten tial Q ( x, t ) one can use the fact that the RHP (24) with canonical normalization has unique solution. Giv en that solution χ ± ( x, t, λ ) one recov ers Q ( x , t ) via the form ula Q ( x, t ) = lim λ →∞ λ  J − χ ± J b χ ± ( x, t, λ )  . (27) whic h is w ell kno wn. W e imp ose also the standard r eduction, namely assume tha t Q ( x, t ) = Q † ( x, t ), or in comp onents p k = q ∗ k . As a consequence w e ha v e ~ ρ − ( λ, t ) = ~ ρ + , ∗ ( λ, t ) a nd ~ τ − ( λ, t ) = ~ τ + , ∗ ( λ, t ). 5 Dressing metho d and soliton solutions The ma in goal of the dressin g metho d [17,18,19,20,21] is, starting from a kno wn solutions χ ± 0 ( x, t, λ ) of L 0 ( λ ) with p oten tia l Q (0) ( x, t ) to construct new singular solutions χ ± 1 ( x, t, λ ) of L with a p otential Q (1) ( x, t ) with tw o a dditina l singularities lo cated a t pre scrib ed p ositions λ ± 1 ; the reduction ~ p = ~ q ∗ ensures that λ − 1 = ( λ + 1 ) ∗ . It is r elated to the regular one b y a dressing factor u ( x, t, λ ) χ ± 1 ( x, t, λ ) = u ( x, λ ) χ ± 0 ( x, t, λ ) u − 1 − ( λ ) . u − ( λ ) = lim x →−∞ u ( x, λ ) (28) Note that u − ( λ ) is a blo c k-diagonal matrix. The dres sing f actor u ( x, λ ) m ust satisfy the e quation i∂ x u + Q (1) ( x ) u − uQ (0) ( x ) − λ [ J, u ( x, λ )] = 0 , (29) and the normalizatio n conditio n lim λ →∞ u ( x, λ ) = 1 1 . Besides χ ± i ( x, λ ), i = 0 , 1 and u ( x, λ ) m ust b elong to the corresp onding Lie group S O (2 r + 1 , C ); in addition u ( x, λ ) by construction has p oles and zero es at λ ± 1 . 7 The construction of u ( x, λ ) is based on an appropriate anzats sp ecifying ex- plicitly the form of its λ -dependence [6,20] and the r eferences the rein. u ( x, λ ) = 1 1 + ( c ( λ ) − 1) P ( x, t ) + 1 c ( λ ) − 1 ! P ( x, t ) , P = S − 1 0 P T S 0 , (30) where P ( x, t ) a nd P ( x, t ) are pro jectors whose ra nk s can not exceed r a nd whic h satisfy P P ( x, t ) = 0. Giv en a set of s linearly indep enden t polar ization v ectors | n k i spanning the cor r esp onding eigensubspase of L one can define P ( x, t ) = s X a,b =1 | n a ( x, t ) i M − 1 ab h n † b ( x, t ) | , M ab ( x, t ) = h n † b ( x, t ) | n a ( x, t ) i , | n a ( x, t ) i = χ + 0 ( x, t, λ + ) | n 0 ,a i , c ( λ ) = λ − λ + λ − λ − , h n 0 ,a | S 0 | n 0 ,b i = 0 . (31) T aking the limit λ → ∞ in eq. (29) w e get that Q (1) ( x, t ) − Q (0) ( x, t ) = ( λ − 1 − λ + 1 )[ J, P ( x, t ) − P ( x, t )] . Belo w w e list the explicit expres sions only f or the o ne- solito n solutions. T o this end w e assum e Q (0) = 0 and put λ ± 1 = µ ± iν . As a result we g et q (1s) k ( x, t ) = − 2 iν  P 1 k ( x, t ) + ( − 1) k P ¯ k , 2 r +1 ( x, t )  , (32) where ¯ k = 2 r + 2 − k . Rep eating the abov e pro cedu re N times w e can obtain N soliton solutions. 5.1 The c ase of r ank one solitons In this case s = 1 so that the gene ric (arbitrary r ) o ne-soliton solution reads q k = − iν e − iµ ( x − vt − δ 0 ) cosh 2 z + ∆ 2 0  α k e z − iφ k + ( − 1) k α ¯ k e − z + iφ ¯ k  , v = ν 2 − µ 2 µ , u = − 2 µ , z ( x, t ) = ν ( x − ut − ξ 0 ) , (33) ξ 0 = 1 2 ν ln | n 0 , 2 r + 1 | | n 0 , 1 | , α k = | n 0 ,k | q | n 0 , 1 || n 0 , 2 r + 1 | , ∆ 2 0 = P 2 r k =2 | n 0 ,k | 2 2 | n 0 , 1 n 0 , 2 r + 1 | , and δ 0 = a r g n 0 , 1 /µ = − arg n 0 , 2 r + 1 /µ , φ k = arg n 0 ,k . The polarizatio n v ectors satisfy the follo wing relation r X k =1 2( − 1) k +1 n 0 ,k n 0 , ¯ k + ( − 1) r n 2 0 ,r +1 = 0 . (34) 8 Th us f o r r = 2 w e iden tify Φ 1 = q 2 , Φ 0 = q 3 / √ 2 and Φ 3 = q 4 and w e obtain the following solutio ns for the equation (1) Φ ± 1 = − 2 iν √ α 2 α 4 e − iµ ( x − vt − δ ± 1 ) cosh 2 z + ∆ 2 0 (cos φ ± 1 cosh z ± 1 − i sin φ ± 1 sinh z ± 1 ) , (35) δ ± 1 = δ 0 ∓ φ 2 − φ 4 2 µ , φ ± 1 = φ 2 + φ 4 2 z ± 1 = z ∓ 1 2 ln α 4 α 2 , Φ 0 = − √ 2 iν α 3 e − iµ ( x − vt − δ 0 ) cosh 2 z + ∆ 2 0 (cos φ 3 sinh z − i sin φ 3 cosh z ) . (36) F or r = 3 w e iden tify Φ 2 = q 2 , Φ 1 = q 3 , Φ 0 = q 4 , Φ − 1 = q 5 and Φ − 2 = q 6 , so that the one-soliton solution for equation (16) reads Φ ± 2 = − 2 iν √ α 2 α 6 e − iµ ( x − vt − δ ± 2 ) cosh 2 z + ∆ 2 0 (cos φ ± 2 cosh z ± 2 − i sin φ ± 2 sinh z ± 2 ) , (37) Φ ± 1 = − 2 iν √ α 3 α 5 e − iµ ( x − vt − δ ± 1 ) cosh 2 z + ∆ 2 0 (cos φ ± 1 sinh z ± 1 − i sin φ ± 1 cosh z ± 1 ) , (38) δ ± 2 = δ 0 ∓ φ 2 − φ 6 2 µ , φ ± 2 = φ 2 + φ 6 2 z ± 2 = z ∓ 1 2 ln α 6 α 2 , δ ± 1 = δ 0 ∓ φ 3 − φ 5 2 µ , φ ± 1 = φ 3 + φ 5 2 , z ± 1 = z ∓ 1 2 ln α 5 α 3 , Φ 0 = − 2 iν α 4 e − iµ ( x − vt − δ 0 ) cosh 2 z + ∆ 2 0 (cos φ 4 cosh z − i sin φ 4 sinh z ) . (39) Cho osing appropriately the polarizat io n vec tors | n i w e are able to repro duce the solito n solutions obtained by W adati et al. b oth for F = 1 and F = 2 BEC. 5.2 The c ase of r ank two solitons Here s = 2 and w e ha v e tw o linearly indep enden t p ola rization v ectors | n a i , a = 1 , 2 . F r o m eq. (31) w e get P ( x, t ) = 1 det M  | n 1 ( x, t ) i M 22 h n † 1 ( x, t ) | − | n 2 ( x, t ) i M 12 h n † 1 ( x, t ) | − | n 1 ( x, t ) i M 21 h n † 2 ( x, t ) | + | n 2 ( x, t ) i M 11 h n † 2 ( x, t ) |  , det M ( x, t ) = M 11 M 22 − M 12 M 21 , M ab ( x, t ) = h n † a ( x, t ) | n b ( x, t ) i , (40) The corr esp onding expressions for the rank 2 soliton solution are obtained b y inserting eq. (40) in to ( 3 2) and are rather inv olv ed. W e r emark here that the 9 reduction Q † = Q may not b e sufficien t t o ensure tha t det M is p ositiv e for all x and t , so for certain c hoices of | n a i w e ma y hav e singular solitons. These and other prop erties of the rank 2 soliton solutions will b e analyzed elsewhere. 6 Conclusions and discussion The main result of the presen t pap er is that a sp ecial v ersion o f the mo del describing F = 2 spinor Bose-Einstein condensate is in tegrable by the ISM. The corresp onding Lax represen tation is naturally related to the symmetric space BD . I . ≃ SO(7) / SO(2) × SO(5), see [5]. F or a generic h yp erfine spin F , the dynamics within the mean field theory is describ ed by the 2 F + 1 com- p onen t Gross-Pitaevs kii equation in one dime nsion. If all the spin dep enden t in teractio ns v anish and only intensit y in teraction exists, the multi-component Gross-Pitaevskii equation in one dimension is equiv alent to the ve ctor nonlin- ear Sc hr¨ odinger equation with 2 F + 1 comp onen ts [22]. Then equations (9) with the reduction p = ǫq ∗ , ǫ = ± 1 are natural general- ization of the v ector no nlinear Sc hr¨ odinger equation, whic h adequately mo del the spinor Bo se-Einstein condensates for v alues of F = r equal to 1 and 2. W e exp ect that for generic F these equations ma y b e useful in describing BECs with higher hyperfine structure. Here we deriv ed only generic one-soliton solutions. 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