Bose-Einstein condensates with F=1 and F=2. Reductions and soliton interactions of multi-component NLS models

Bose-Einstein condensates with F = 1 and F = 2 . Reductions and soliton in teractions of m ulti-comp onen t NLS mo dels. V. S. Gerdjik o v , N. A. K osto v and T. I. V alc hev Institute for Nuclear Researc h and Nuclear Energy , Bulgarian academy of science s 72 Tsarigradsk o c haussee , 1 784 Sofia, Bulgaria ABSTRA CT W e analyz e a class o f m ulticomp onen t nonlinear Schr¨ odinger equatio ns (MNLS) r elated to the symmetr ic BD.I - t yp e symmetric spa ces and their reductio ns . W e briefly outline the dire ct a nd the inverse scattering method for the relev an t Lax oper ators and the soliton solutions. W e use the Zakharov-Shabat dressing metho d to obtain the t wo-soliton solution and analy z e the soliton interactions of the MNLS equa tions and some of their reductions. Keyw ords: Bose-Eins tein co ndens ates, Multicomp onen t nonlinear Schr¨ odinger equations , Soliton s olutions, Soliton interactions 1. INTRODUCTION Bose-Eins tein condensate (BE C) of alk ali ato ms in the F = 1 hyper fine sta te, e longated in x direction and confined in the tra nsv erse directions y , z by pur ely optical means are des cribed by a 3 - component normalized spinor wa ve vector Φ ( x, t ) = (Φ 1 , Φ 0 , Φ − 1 ) T ( x, t ) sa tisfying the nonlinear Schr¨ odinger (MNLS) equation 1 see also: 2–6 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 . spinor BEC with F = 2 for r ather sp ecific choices of the s c a ttering lengths in dimensio nless co ordinates takes the for m: 7 i∂ t Φ ± 2 + ∂ xx Φ ± 2 + 2( ~ Φ , ~ Φ ∗ )Φ ± 2 − (2Φ 2 Φ − 2 − 2Φ 1 Φ − 1 + Φ 2 0 )Φ ∗ ∓ 2 , i∂ t Φ ± 1 + ∂ xx Φ ± 1 + 2( ~ Φ , ~ Φ ∗ )Φ ± 1 − (2Φ 2 Φ − 2 − 2Φ 1 Φ − 1 + Φ 2 0 )Φ ∗ ∓ 1 , (2) i∂ t Φ 0 + ∂ xx Φ 0 + 2( ~ Φ , ~ Φ ∗ )Φ 0 − (2Φ 2 Φ − 2 − 2Φ 1 Φ − 1 + Φ 2 0 )Φ ∗ 0 . Both models have natur a l Lie algebraic int erpre ta tion and are related to the sy mm etric spaces BD . I ≃ SO(n + 2) / SO(n) × SO(2) with n = 3 a nd n = 5 resp ectiv ely . They are integrable b y means of in verse scattering transform metho d. 8, 9, 12 Using a mo dification o f the Za k haro v-Shabat ‘dres sing metho d’ we desc r ibe the solito n solutions 1, 10 and the effects of the reductions on them. Sections 2 and 3 contain the basic details on the dir ect and inv erse scattering problems fo r the Lax op erator. Section 4 o ut lines the effects of the algebra ic r eductions of the MNLS. In Sec tion 5 us ing the Za kharov-Shabat dressing metho d we derive the one- and tw o-so liton so lutions of the MNLS and discus s their prop erties. Sec tio n 6 is de dica ted to the analysis of the soliton interactions of the MNLS. T o this end we ev aluate the limits of the generic t wo-soliton solution for t → ±∞ . As a result we establish that the effect o f the interactions on the soliton parameters is analo gous to the one for the scalar NLS equa tion and consis ts in shifts of the ‘center o f mass’ and shift in the phase. F urther author information: (Send correspond ence to V. S. Gerdjiko v) V. S. Gerdjiko v: E-mail: g erjiko v@inrne.bas.bg, T elephone: +3592 979 5638 2. THE MET HOD FOR SOL VING MNLS WI TH F = 1 AND F = 2 MNLS equatio ns for the BD.I. series of sy mmet ric s paces (algebras of the type so ( n + 2) and J dual to e 1 ) have the La x r epresen tation [ L, M ] = 0 as follows 8, 11 , 12 Lψ ( x, t, λ ) ≡ i∂ x ψ + U ( x, t, λ ) ψ ( x , t, λ ) = 0 , M ψ ( x , t, λ ) ≡ i∂ t ψ + V ( x, t, λ ) ψ ( x , t, λ ) = 0 , U ( x, t, λ ) = Q ( x, t ) − λJ, V ( x, t, λ ) = V 0 ( x, t ) + λV 1 ( x, t ) − λ 2 J, V 1 ( x, t ) = Q ( x, t ) , V 0 ( x, t ) = i ad − 1 J dQ dx + 1 2  ad − 1 J Q, Q ( x, t )  . (3) where a d J X = [ J, X ] and a d − 1 J is w ell defined on the imag e o f ad J in g ; Q =   0 ~ q T 0 ~ p ∗ 0 s 0 ~ q 0 ~ p † s 0 0   , J = diag (1 , 0 , . . . 0 , − 1) . (4) The vector ~ q for F = 1 (r esp. F = 2 ) is 3- (r e sp. 5 - ) co mponent and has the fo rm ~ q = (Φ 1 , Φ 0 , Φ − 1 ) T , ~ q = (Φ 2 , Φ 1 , Φ 0 , Φ − 1 , Φ − 2 ) T , (5) and the corr e sponding matrices s 0 ent er in the definition of so (2 r + 1) with r = 2 and r = 3: X ∈ so (2 r + 1) , X + S 0 X T S 0 = 0 , S 0 = 2 r + 1 X s =1 ( − 1) s +1 E s,n +1 − s , S 0 =   0 0 1 0 − s 0 0 1 0 0   . (6) By E sp ab o ve we mean 2 r + 1 × 2 r + 1 matrix with matrix elemen ts ( E sp ) ij = δ si δ pj . With the definition of orthogo nalit y used in (6) the C a rtan g enerators H k = E k,k − E n +3 − k,n +3 − k are repr esen ted by diago nal matric e s. If we make use of the t ypical reduction Q = Q † (or ~ p ∗ = ~ q ) the generic MNLS type equatio ns related to BD . I . ac quire the for m: i~ q t + ~ q xx + 2( ~ q † , ~ q ) ~ q − ( ~ q , s 0 ~ q ) s 0 ~ q ∗ = 0 . (7) The Ha milto nians for the MNLS eq uations (7) are given by H MNLS = Z ∞ −∞ dx  ( ∂ x ~ q † , ∂ x ~ q ) − ( ~ q † , ~ q ) 2 + 1 2 | ( ~ q T , s 0 ~ q ) | 2  . (8) 3. THE DIREC T AND THE I NVERSE SC A TTERING PR OBLEM 3.1 T he fundamental analytic solution W e remind some basic features of the inv erse scattering theor y for the Lax op erators L , see. 11, 12 There we hav e made use of the gener a l theory developed in 13–16, 18 and the reference s therein. The Jo st solutions o f L are defined by: lim x →−∞ φ ( x, t, λ ) e iλJ x = 1 1 , lim x →∞ ψ ( x, t, λ ) e iλJ x = 1 1 (9) and the scattering matrix T ( λ, t ) ≡ ψ − 1 φ ( x, t, λ ). The sp ecial choice of J a nd the fact that the Jos t s o lutions and the scatter ing matrix take v alues in the gr oup S O (2 r + 1) we can use the following blo c k-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    , ˆ T ( λ, t ) =    m − 1 ~ B − T c − 1 − ~ B + ˆ T 22 s 0 ~ b − c + 1 − ~ b + T s 0 m + 1    , (10) where ~ b ± ( λ, t ) and ~ B ± ( λ, t ) are 2 r − 1-comp onen t vectors, T 22 ( λ ) is 2 r − 1 × 2 r − 1 blo c k matr ix, and m ± 1 ( λ ), and c ± 1 ( λ ) are scalar functions. Suc h parametriza tio n is compatible with the g e neralized Gauss deco mpositions of T ( λ ) which read as follows: T ( λ, t ) = T − J D + J ˆ S + J , T ( λ, t ) = T + J D − J ˆ S − J , T ∓ J = e ± ( ~ ρ ± , ~ E ∓ 1 ) , S ± J = e ± ( ~ τ ± , ~ E ± 1 ) , D ± J = diag  ( m ± 1 ) ± 1 , m ± 2 , ( m ± 1 ) ∓ 1  , (11) where  ~ ρ + , ~ E + 1  = r − 1 X k =1 ( ρ + k E e 1 − e k +1 + ρ + ¯ k E e 1 + e k +1 ) + ρ + r E e 1 ,  ~ ρ − , ~ E − 1  = r − 1 X k =1 ( ρ − k E − e 1 + e k +1 + ρ − ¯ k E − e 1 − e k +1 ) + ρ − r E − e 1 , (12) and s imila r ex pressions for  ~ τ ± , ~ E ± 1  . The functions m ± 1 and n × n matrix -v alued functions m ± 2 are a na lytic for λ ∈ C ± . W e hav e intro duced also the notations : ~ ρ − = ~ B − m − 1 , ~ τ − = ~ B + m − 1 , ~ ρ + = ~ b + m + 1 , ~ τ + = ~ b − m + 1 . There are some additional r elations which ensure that bo th T ( λ ) and its inv erse ˆ T ( λ ) b elong to the orthog onal group S O (2 r + 1) and that T ( λ ) ˆ T ( λ ) = 1 1. Next we intro duce the fundamen tal analy tic s olution (F AS) χ ± ( x, t, λ ) using the genera lized Gauss decom- po sition of T ( λ, t ), se e : 15, 19, 20 χ ± ( x, t, λ ) = φ ( x, t, λ ) S ± J ( t, λ ) = ψ ( x, t, λ ) T ∓ J ( t, λ ) D ± J ( λ ) , (13) This construction ensures that ξ ± ( x, λ ) = χ ± ( x, λ ) e iλJ x are analytic functions of λ for λ ∈ C ± . If Q ( x , t ) is a solution o f the MNLS eq. (7) then the matrix elements of T ( λ ) sa tis fy the line a r evolution equations 12 i d ~ b ± dt ± λ 2 ~ b ± ( t, λ ) = 0 , i d ~ B ± dt ± λ 2 ~ B ± ( t, λ ) = 0 , i dm ± 1 dt = 0 , i d m ± 2 dt = 0 . (14) Thu s the blo c k-diag onal matrices D ± ( λ ) c a n b e co nsidered a s g enerating functionals of the int egra ls of motion. The fact t hat all (2 r − 1) 2 matrix e lemen ts of m ± 2 ( λ ) f or λ ∈ C ± generate integrals of mo tion reflect the sup e rin tegrability of the mo del and are due to the degeneracy o f the dispe rsion law of (7). Note that D ± J ( λ ) allow ana lytic extension for λ ∈ C ± and that their zero es and p oles deter min e the discrete eig e n v alues 11 of L . 3.2 T he Riemann-Hilb ert Problem The F AS for real λ are linea rly related 12 χ + ( x, t, λ ) = χ − ( x, t, λ ) G 0 ,J ( λ, t ) , G 0 ,J ( λ, t ) = ˆ S − J ( λ, t ) S + J ( λ, t ) (15) Eq. (15) can be rewr iten in an equiv a len t form for the F AS ξ ± ( x, t, λ ) = χ ± ( x, t, λ ) e iλJ x : i dξ ± dx + Q ( x ) ξ ± ( x, λ ) − λ [ J, ξ ± ( x, λ )] = 0 , (16) and the relation lim λ →∞ ξ ± ( x, t, λ ) = 1 1 , (17) Then these F AS sa tisfy the RHP’s ξ + ( x, t, λ ) = ξ − ( x, t, λ ) G J ( x, λ, t ) , G J ( x, λ, t ) = e − iλJ ( x + λt ) G − J ( λ, t ) e iλJ ( x + λt ) , (18) Obviously the sewing function G J ( x, λ, t ) is uniquely deter mined by the Gauss facto rs S ± J ( λ, t ). In addition Zakharov-Shabat’s theorem 13, 14 states that G J ( x, λ, t ) depe nds on x and t in the wa y prescrib ed ab o ve then the corres p onding F AS sa tisf y the linear systems (16). If we hav e solved the RHP’s and know the F AS ξ + ( x, t, λ ) then the formula Q ( x, t ) = lim λ →∞ λ  J − ξ + ( x, t, λ ) J ˆ ξ + ( x, t, λ )  , (19) allows us to recov er the co rrespo nding p oten tial of L . 4. REDUCTIONS OF MNLS Along with the typical reduction Q = Q † men tioned ab o ve o ne can imp ose additional reductions us ing the reduction g roup prop osed by Mikhailov. 21 They are a utomatically compatible w ith the Lax representation of the co rrespo ndin g MNLS eq. Below we make use of t wo Z 2 -reductions: 22 1) C 1 U † ( x, t, λ ∗ ) C − 1 1 = U ( x, t, λ ) , C 1 V † ( x, t, λ ∗ ) C − 1 1 = V ( x, t, λ ) , 2) C 2 U T ( x, t, λ ) C − 1 2 = − U ( x, t, λ ) , C 2 V T ( x, t, λ ) C − 1 2 = − V ( x, t, λ ) , (20) where C 1 and C 2 are in volutions of the Lie alg e bra so (2 r + 1, i.e. C 2 i = 1 1. They can b e chosen to be either diagonal (i.e., elements of the Car tan s ubgroup o f S O (2 r + 1)) or elements o f the W eyl g roup. The t ypical reductions of the MNLS eqs. is a class 1) re duct ion obtained by s pecifying C 1 to b e the identit y automorphism of g ; b elo w we list s ev eral choices for C 1 leading to inequiv alen t r eductions: 1a) C 1 = 1 1 , ~ p ( x ) = ~ q ∗ ( x ) , 1b) C 1 = K 1 , ~ p ( x ) = K 01 ~ q ∗ ( x ) , 1c) C 1 = S e 2 , ~ p ( x ) = K 02 ~ q ∗ ( x ) , 1d) C 1 = S e 2 S e 3 , ~ p ( x ) = K 03 ~ q ∗ ( x ) , 2e) C 2 = K 4 , ~ q ( x ) = − K 04 s 0 ~ q ( x ) , ~ p ( x ) = − K 04 s 0 ~ p ( x ) , (21) where K j = blo c k-diag (1 , K 0 j , 1) , K 01 = diag ( ǫ 1 , . . . , ǫ r − 1 , 1 , ǫ r − 1 , . . . , ǫ 1 ) , j = 1 , 2 , 3 , (22) and ǫ j = ± 1. The matrices K 02 , K 03 and K 4 are no t diag onal and may take the for m: K 02 =   0 0 1 0 − 1 0 1 0 0   , K 4 =   0 0 1 0 K 04 0 1 0 0   , K 02 =       0 0 0 0 − 1 0 1 0 0 0 0 0 − 1 0 0 0 0 0 1 0 − 1 0 0 0 0       , K 03 =       0 0 0 0 − 1 0 0 0 1 0 0 0 − 1 0 0 0 1 0 0 0 − 1 0 0 0 0       , (23) Each of the above r eductions imp ose c o nstrain ts on the F AS, on the sca ttering matrix T ( λ ) and on its Gaus s factors S ± J ( λ ), T ± J ( λ ) a nd D ± J ( λ ). These hav e the form: ( S + ( λ ∗ )) † = K − 1 j ˆ S − ( λ ) K j ( T + ( λ ∗ )) † = K − 1 j ˆ T − ( λ ) K j ( D + ( λ ∗ )) † = K − 1 j ˆ D − ( λ ) K j ~ τ + = K 0 j ~ τ − , ∗ , ~ ρ + = K 0 j ~ ρ − , ∗ , (24) where the matrices K j are s pecific for each choice of the automor phisms C 1 , s e e eqs. (21), (22). In particular , fr om the las t line of (24) and (2 2 ) we g et: ( m + 1 ( λ ∗ )) ∗ = m − 1 ( λ ) , (25) and conse q uen tly , if m + 1 ( λ ) ha s zero es at the p oin ts λ + k , then m − 1 ( λ ) ha s zero es a t: λ − k = ( λ + k ) ∗ , k = 1 , . . . , N . (26) 5. SOLITON SO LUTIONS Let us now make us e of one of the versions of the dressing metho d 13, 14 which a llo ws one to co nstruct sing ular solutions of the RHP . In or der to obtain N - soliton solutions one has to apply dressing pro cedure with a 2 N -p oles dressing factor of the form u ( x, λ ) = 1 1 + N X k =1  A k ( x ) λ − λ + k + B k ( x ) λ − λ − k  . (27) The N -so liton solution itself ca n b e g enerated via the following formula Q N , s ( x ) = N X k =1 [ J, A k ( x ) + B k ( x )] . (28) The dr essing fa c tor u ( x, λ ) must satisfy the equa tion i∂ x u + Q N , s u − λ [ J, u ] = 0 (29) and the normalizatio n condition lim λ →∞ u ( x, λ ) = 1 1. The co nstruction of u ( x, λ ) ∈ S O ( n + 2) is ba sed on a n appropria te anzatz sp ecifying the form of its λ -dep endence 23, 24 The residues of u admit the following dec o mposition A k ( x ) = X k ( x ) F T k ( x ) , B k ( x ) = Y k ( x ) G T k ( x ) . where all matrices inv olv ed a re suppo sed to b e rec tangular and o f maximal rank s . By comparing the co efficien ts befo re the same p o wers of λ − λ ± k in (29) w e convince ourselves that the factors F k and G k can b e expressed by the fundamental a na lytic s olutions χ ± 0 ( x, λ ) a s fo llo ws F T k ( x ) = F T k, 0 [ χ + 0 ( x, λ + k )] − 1 , G T k ( x ) = G T k, 0 [ χ − 0 ( x, λ − k )] − 1 . The co nstan t rectangula r matrices F k, 0 and G k, 0 ob ey the alg ebraic r elations F T k, 0 S 0 F k, 0 = 0 , G T k, 0 S 0 G k, 0 = 0 . The other tw o types of factor s X k and Y k are s olutions to the algebra ic system S 0 F k = X k α k + X l 6 = k X l F T l S 0 F k λ + l − λ + k + X l Y l G T l S 0 F k λ − l − λ + k , S 0 G k = X l X l F T l S 0 G k λ + l − λ − k + Y k β k + X l 6 = k Y l G T l S 0 G k λ − l − λ − k . (30) The square s × s matrices α k ( x ) and β k ( x ) introduced a bov e dep end on χ + 0 and χ − 0 and their der iv a tiv es by λ as follows α k ( x ) = − F T 0 ,k [ χ + 0 ( x, λ + k )] − 1 ∂ λ χ + 0 ( x, λ + k ) S 0 F 0 ,k + α 0 ,k , β k ( x ) = − G T 0 ,k [ χ − 0 ( x, λ − k )] − 1 ∂ λ χ − 0 ( x, λ − k ) S 0 G 0 ,k + β 0 ,k . (31) Below for simplicity we will choos e F k and G k to b e 2 r + 1-comp onent vectors. Then o ne ca n show that α k = β k = 0 which simplifies the sys tem (30). W e also in tro duce the following more co n v enient parametriza tion for F k and G k , namely (see e q. (33)): F k ( x, t ) = S 0 | n k ( x, t ) i =   e − z k + iφ k − √ 2 s 0 ~ ν 0 k e z k − iφ k   , G k ( x, t ) = | n ∗ k ( x, t ) i =   e z k + iφ k √ 2 ~ ν 0 k ∗ e − z k − iφ k   , (32) where ~ ν 0 k are consta nt 2 r − 1-comp onent p olarization vectors and z j = ν j ( x + 2 µ j t ) + ξ 00 , φ j = µ j x + ( µ 2 j − ν 2 j ) t + δ 00 , h n T j ( x, t ) | S 0 | n j ( x, t ) i = 0 , or ( ~ ν 0 ,j s 0 ~ ν 0 ,j ) = 1 . (33) With this notations the p olarizatio n vectors automatically satisfy h n j ( x, t ) | S 0 | n j ( x, t ) i = 0. Thu s fo r N = 1 we get the system: | Y 1 i = − ( λ + 1 − λ − 1 ) | n 1 i h n † 1 | n 1 i , | X 1 i = ( λ + 1 − λ − 1 ) S 0 | n ∗ 1 i h n † 1 | n 1 i , (34) which is easily solved. As a r esult for the one-s o liton solutio n we get: ~ q 1s = − i √ 2( λ + 1 − λ − 1 ) e − iφ 1 ∆ 1  e − z 1 s 0 | ~ ν 01 i + e z 1 | ~ ν ∗ 01 i  , ∆ 1 = cosh(2 z 1 ) + h ~ ν † 01 | ~ ν 01 i . (35) F or n = 3 we put ν 0 k = | ν 0 k | e α 0 k get: Φ 1s; ± 1 = − p 2 | ν 01;1 ν 01;3 | ( λ + 1 − λ − 1 ) ∆ 1 e − iφ 1 ± iβ 13 (cosh( z 1 ∓ ζ 01 ) cos( α 13 ) − i sinh( z 1 ∓ ζ 01 ) sin( α 13 )) , Φ 1s;0 = − √ 2 | ν 01;2 | ( λ + 1 − λ − 1 ) ∆ 1 e − iφ 1 (sinh z 1 cos( α 02 ) + i cosh z 1 sin( α 02 )) , β 13 = 1 2 ( α 03 − α 01 ) , ζ 01 = 1 2 ln | ν 01;3 | | ν 01;1 | , α 13 = 1 2 ( α 03 + α 01 ) , (36) Note that the ‘center o f mass‘ of Φ 1s;1 (resp. o f Φ 1s; − 1 ) is shifted with res pect to the one o f Φ 1s;0 by ζ 01 to the right (resp to the left); b esides | Φ 1s;1 | = | Φ 1s; − 1 | , i.e. they have the same amplitudes. F or n = 5 we put ν 0 k = | ν 0 k | e α 0 k and get analogo usly: Φ 1s; ± 2 = − p 2 | ν 01;1 ν 01;5 | ( λ + 1 − λ − 1 ) ∆ 1 e − iφ 1 ± iβ 15 (cosh( z 1 ∓ ζ 01 ) cos( α 15 ) − i sinh( z 1 ∓ ζ 01 ) sin( α 15 )) , Φ 1s; ± 1 = p 2 | ν 01;2 ν 01;4 | ( λ + 1 − λ − 1 ) ∆ 1 e − iφ 1 ± iβ 24 (cosh( z 1 ∓ ζ 02 ) cos( α 24 ) − i sinh( z 1 ∓ ζ 01 ) sin( α 24 )) , Φ 1s;0 = − √ 2 | ν 01;3 | ( λ + 1 − λ − 1 ) ∆ 1 e − iφ 1 (cosh z 1 cos( α 03 ) − i sinh z 1 sin( α 03 )) , β 15 = 1 2 ( α 05 − α 01 ) , ζ 01 = 1 2 ln | ν 01;5 | | ν 01;1 | , α 15 = 1 2 ( α 05 + α 01 ) , β 24 = 1 2 ( α 04 − α 02 ) , ζ 02 = 1 2 ln | ν 01;4 | | ν 01;2 | , α 24 = 1 2 ( α 04 + α 02 ) , (37) Similarly the ‘cent er of mass‘ of Φ 1s;2 and Φ 1s;1 (resp. of Φ 1s; − 2 and Φ 1s; − 1 ) are shifted with r e s pect to the one of Φ 1s;0 by ζ 01 and ζ 02 to the right (resp to the left); b e sides | Φ 1s;2 | = | Φ 1s; − 2 | a nd | Φ 1s;1 | = | Φ 1s; − 1 | . F or N = 2 we g et: | n 1 ( x, t ) i = X 2 ( x, t ) f 21 λ + 2 − λ + 1 + Y 1 ( x, t ) κ 11 λ − 1 − λ + 1 + Y 2 ( x, t ) κ 21 λ − 2 − λ + 1 , | n 2 ( x, t ) i = X 1 ( x, t ) f 12 λ + 1 − λ + 2 + Y 1 ( x, t ) κ 12 λ − 1 − λ + 2 + Y 2 ( x, t ) κ 22 λ − 2 − λ + 2 , S 0 | n ∗ 1 ( x, t ) i = X 1 ( x, t ) κ 11 λ + 2 − λ + 1 + X 2 ( x, t ) κ 11 λ + 2 − λ − 1 + Y 2 ( x, t ) f ∗ 21 λ − 2 − λ − 1 , S 0 | n ∗ 2 ( x, t ) i = X 1 ( x, t ) κ 21 λ + 1 − λ − 2 + X 2 ( x, t ) κ 22 λ + 2 − λ − 2 + Y 1 ( x, t ) f ∗ 12 λ − 1 − λ − 2 , (38) where κ kj ( x, t ) = e z k + z j + i ( φ k − φ j ) + e − z k − z j − i ( φ k − φ j ) + 2  ~ ν † 0 k , ~ ν 0 j  , f kj ( x, t ) = e z k − z j − i ( φ k − φ j ) + e z j − z k + i ( φ k − φ j ) − 2  ~ ν T 0 k s 0 ~ ν 0 j  , (39) In other words: M ~ X ≡        0 f 21 λ + 2 − λ + 1 κ 11 λ − 1 − λ + 1 κ 21 λ − 2 − λ + 1 f 12 λ + 1 − λ + 2 0 κ 12 λ − 1 − λ + 2 κ 22 λ − 2 − λ + 2 κ 11 λ + 1 − λ − 1 κ 12 λ + 2 − λ − 1 0 f ∗ 21 λ − 2 − λ − 1 κ 21 λ + 1 − λ − 2 κ 22 λ + 2 − λ − 2 f ∗ 12 λ − 1 − λ − 2 0            X 1 X 2 Y 1 Y 2     =     | n 1 i | n 2 i S 0 | n ∗ 1 i S 0 | n ∗ 2 i     . (40) W e can rewrite M in blo c k-matrix form: M =  M 11 M 12 M 21 M 22  , M 22 = M ∗ 11 , M 21 = − M T 12 , M 11 = f 12 λ + 2 − λ + 1  0 1 − 1 0  , M 12 = κ 11 λ − 1 − λ + 1 κ 21 λ − 2 − λ + 1 κ 12 λ − 1 − λ + 2 κ 22 λ − 2 − λ + 2 ! . (41) The inverse o f M is given b y: M − 1 =  ( M 11 − M 12 ˆ M ∗ 11 M 21 ) − 1 − ( M 11 − M 12 ˆ M ∗ 11 M 21 ) − 1 M 12 ˆ M ∗ 11 − ( M ∗ 11 − M 21 ˆ M 11 M 12 ) − 1 M 21 ˆ M 11 ( M ∗ 11 − M 21 ˆ M 11 M 12 ) − 1  , (42) One can chec k by direct c alculation that: M 11 − M 12 ˆ M ∗ 11 M 21 = f ∗ 12 λ − 2 − λ − 1 Z  0 1 − 1 0  , M ∗ 11 − M 21 ˆ M 11 M 12 = f 12 λ + 2 − λ + 1 Z  0 1 − 1 0  , Z =  | f 12 | 2 | λ + 2 − λ + 1 | 2 − κ 12 κ 21 | λ + 2 − λ − 1 | 2 + κ 11 κ 22 4 ν 1 ν 2  , (43) Finally we g e t: M − 1 = 1 Z        0 f ∗ 12 λ − 1 − λ − 2 − κ 22 λ + 2 − λ − 2 κ 12 λ + 2 − λ − 1 − f ∗ 12 λ − 1 − λ − 2 0 κ 21 λ + 1 − λ − 2 − κ 11 λ + 1 − λ − 1 κ 22 λ + 2 − λ − 2 − κ 21 λ + 1 − λ − 2 0 − f 12 λ + 1 − λ + 2 − κ 12 λ + 2 − λ − 1 κ 11 λ + 1 − λ − 1 f 12 λ + 2 − λ + 1 0        , (44) F ro m eq s . (40) a nd (44) we obtain: | X 1 i = 1 Z  f ∗ 12 λ − 1 − λ − 2 | n 2 i − κ 22 λ + 2 − λ − 2 S 0 | n ∗ 1 i + κ 12 λ + 2 − λ − 1 S 0 | n ∗ 2 i  , | X 2 i = 1 Z  − f ∗ 12 λ − 1 − λ − 2 | n 1 i + κ 21 λ + 1 − λ − 2 S 0 | n ∗ 1 i − κ 11 λ + 1 − λ − 1 S 0 | n ∗ 2 i  , | Y 1 i = 1 Z  κ 22 λ + 2 − λ − 2 | n 1 i − κ 21 λ + 1 − λ − 2 | n 2 i − f 12 λ + 1 − λ + 2 S 0 | n ∗ 2 i  , | Y 2 i = 1 Z  − κ 12 λ + 2 − λ − 1 | n 1 i + κ 11 λ + 1 − λ − 1 | n 2 i + f 12 λ + 2 − λ + 1 S 0 | n ∗ 1 i  , (45) Inserting this result into eq. (28) w e o btain the following expre s sion for the 2-soliton so lution o f the MNLS: Q 2s ( x, t ) = [ J, A 1 + B 1 + A 2 + B 2 ] = 1 Z [ J, C ( x, t ) − S 0 C T ( x, t ) S 0 ] , C ( x, t ) = κ 22 λ + 2 − λ − 2 | n 1 ih n † 1 | − κ 12 λ + 2 − λ − 1 | n 1 ih n † 2 | − κ 21 λ + 1 − λ − 2 | n 2 ih n † 1 | + κ 11 λ + 1 − λ − 1 | n 2 ih n † 2 | − f ∗ 12 λ − 1 − λ − 2 | n 1 ih n 2 | S 0 − f 12 λ + 1 − λ + 2 S 0 | n ∗ 2 ih n † 1 | . (46) A t the end o f this section we note that the effect o f the reductions (20)–(21) co nsists in constra ining the po larization vectors. F or the reductio n 2e) we get ~ ν 0 k = K 04 ~ ν 0 k (47) In par ticular, for n = 3 and for K 04 = − 1 1 we hav e q 1 = q 3 , and q 2 arbitrar y . This reduction of eq . (1) is also impo rtan t for the BEC. 6 F ro m (47) we find ν 01 = ν 03 . The effect o f this cons tr ain t is that for the one-soliton solution we get Φ 1s;1 = Φ 1s;3 . Our next remark following 25 is that this reduction a pplied to the F = 1 MNLS (1) leads to a 2-comp onent MNLS which after the c hange o f v ariables q 1 = 1 2 ( w 1 + iw 2 ) , q 2 = 1 √ 2 ( w 1 − iw 2 ) , (48) leads to tw o disjoint NLS e q uations for w 1 and w 2 resp ectiv ely . It is o nly lo gical tha t applying the constraint ν 01 = ν 03 the ex plicit expression for the o ne-soliton s olution (36) simplifies and reduces to the standar d soliton solutions of the scala r NLS. 6. TW O SO LITON I N TERA C TIONS In this section we generalize the classic a l r esults of Za kharov and Shabat a b out s oliton interactions 26 to the class of MNLS eq ua tions r elated to BD.I symmetric spaces . F or detailed e xposition se e the monogra phs. 15 , 16 These results were generaliz e d for the vector nonlinear Schr¨ odinger equation by Ma nak ov, 17 see a lso. 27–29 The Zakharov Shabat appr oac h consisted in calculating the asymptotics of gener ic N -soliton so lution o f NLS for t → ±∞ and establishing the pure elastic character of the generic soliton int eractio ns. By generic here we mean N -soliton solution who se par a meters λ ± k = µ k ± iν k are such that µ k 6 = µ j for k 6 = j . The pur e elastic c hara c ter of the soliton interactions is demonstr ated by the fact that for t → ±∞ the gener ic N -soliton s o lution splits into sum of N o ne soliton solutions each pr eserving its amplitude 2 ν k and velocity µ k . The o nly effect of the interaction consists in shifting the center of mass and the initial phase of the solitons . These shifts ca n be expressed in ter ms of λ ± k only; for detailed exp osition see. 16 W e s tart with the simplest non-trivial case. Namely we use the 2 -soliton so lution derived ab ov e and calcula te its a symptotics along the tra jectory of the firs t so lito n. T o this end we keep z 1 ( x, t ) fixed a nd let τ = z 2 − z 1 tend to ±∞ . Therefore it will b e eno ugh to insert the asymptotic v alues of the matrix ele men ts of M for τ → ±∞ and keep only the leading terms . T ha t gives: κ 22 =  e 2 τ exp( ν 2 z 1 /ν 1 ) + 2 C 1 , for τ → ∞ , e − 2 τ exp( − ν 2 z 1 /ν 1 ) + 2 C 1 , for τ → − ∞ , κ 12 =  e τ exp((1 + ν 2 /ν 1 ) z 1 + i ( φ 1 − φ 2 )) + O (1) , for τ → ∞ , e − τ exp( − (1 + ν 2 /ν 1 ) z 1 − i ( φ 1 − φ 2 )) + O (1) , for τ → −∞ , κ 21 =  e τ exp((1 + ν 2 /ν 1 ) z 1 − i ( φ 1 − φ 2 )) + O (1) , for τ → ∞ , e − τ exp( − (1 + ν 2 /ν 1 ) z 1 + i ( φ 1 − φ 2 )) + O (1) , for τ → −∞ , f 12 =  e τ exp( − (1 − ν 2 /ν 1 ) z 1 + i ( φ 1 − φ 2 )) + O (1) , for τ → ∞ , e − τ exp((1 − ν 2 /ν 1 ) z 1 − i ( φ 1 − φ 2 )) + O (1) , for τ → −∞ , (49) After so mewhat lengthy ca lc ulations we get: lim τ →∞ ~ q 2s ( x, t ) = − i √ 2 ν 1 e − i ( φ 1 − α + ) ( e − z 1 − r + s 0 | ~ ν 01 i + e z 1 + r + | ~ ν ∗ 01 i ) cosh(2( z 1 + r + )) + ( ~ ν † 01 , ~ ν 01 ) , lim τ →−∞ ~ q 2s ( x, t ) = i √ 2 ν 1 e − i ( φ 1 + α + ) ( e − z 1 + r + s 0 | ~ ν 01 i + e z 1 − r + | ~ ν ∗ 01 i ) cosh(2( z 1 − r + )) + ( ~ ν † 01 , ~ ν 01 ) , (50) where r + = ln     λ + 1 − λ + 2 λ + 1 − λ − 2     , α + = arg λ + 1 − λ + 2 λ + 1 − λ − 2 . In other w ords the 2-solito n in teraction for the MNLS eqs. rela ted to the BD.I symmetric spaces is the same as the one o f the scala r NLS. Again we have that for larg e times the 2-s oliton solution splits into sum of 1-s oliton solutions with shifted center of masses and phas es and the v alue of these shifts r + and α + are indep enden t o n the n umber of comp onen ts of MNLS. It will b e interesting to chec k whether the N -soliton interactions co nsist of s equence of elementary 2-so liton interactions and the shifts ar e additive. 7. CONC LUSIONS AND DISC USSION Using the Zakhar o v-Shabat dressing metho d we have obtained the tw o-so lito n so lut ion and hav e used it to analyze the soliton interactions o f the MNLS equation. The co nclus ion is that after the interactions the s o litons recov er their po larization vectors ν 0 k , velo cities and fr equency velocities. The effect of the interaction is, like in for the s c alar NLS equation, shift of the cent er of mass z 1 → z 1 + r + and shift of the pha se φ 1 → φ 1 + α + . Both shifts a re expressed throug h the rela ted eigenv alues λ ± j only . The next step would be to a nalyze multi-soliton interactions. Our h yp othesis is that each so liton will a cquire a total shift of the cent er of mass that is sum o f all elemen tary shifts fro m e a c h tw o soliton in teractio ns. Similar result is exp ected for the total phase shift o f the so liton. Pro ofs of these facts will b e published elsewhere. REFERENCES [1 ] Ieda, J ., Miyak aw a, T. a nd W adati, M., ”Exa ct Analysis of Soliton Dynamics in Spinor Bose-Eins tein Condensates”, Phys. Rev Lett. 93 , 19410 2 (2004). [2 ] Ieda, J., Miyak awa, T. and W adati, M., ”Matter-wav e solito ns in an F = 1 spinor Bos e-Einstein con- densate”, J . Phys. So c. J pn. 73 , 2996 (200 4 ). [3 ] Li, L ., Li, Z., Malomed, B. A., Mihala che, D. and Liu, W. M., ”Exa c t Solito n Solutions and Nonlinear Mo dulation Instability in Spinor Bose - Einstein Condens ates”, P h ys. Rev. A 7 2 , 0 33611 (2005). [4 ] Uchiy ama, M., Ieda, J. and W adati, M., ” D ark solitons in F = 1 spinor Bo se–Einstein condensate” J. Phys. So c. Jpn. 75 0640 0 2 (2006 ). [5 ] Dokto r o v, E . V., W a ng, J. and Y ang, J., ”Perturbation theory for br igh t spinor Bose-E instein condensa te solitons”, Phys. Rev. A 77 , 0 43617 (2008). [6 ] Nistaza kis, H. E., F ra n tzesk akis , D. J., Kevre k idis, P . G., Malomed, B . A. and Carre tero-Gonzalez R., ”Bright-Dark Soliton Co mplexes in Spinor B o se-Einstein Condens a tes” Phys. Rev. A 77 , 03 3612 (2008). [7 ] Uchiy ama, M., Ie da, J. and W adati, M., ”Multicomp onen t Bright Solitons in F = 2 Spinor Bose-Einstein Condensates”, J. P hys. So c. Japan, 7 6 , 740 05 (2 0 07). [8 ] F ordy , A.P . a nd Kulish, P . P ., ” No nlinear Schrodinger Eq uations and Simple L ie Algebras” , Commun. Math. P h ys. 89 4 27–443 (1 983). [9 ] Tsuchida, T. and W adati, M., ”The co upled mo dified Korteweg-de V ries equations ”, J. Phys. So c. Jpn. 67 11 75 (19 98). [1 0 ] Kostov, N. A., Atanasov, V. A., Gerdjikov, V. S. and Grahovski, G. G., ”On the soliton s o lutions o f the spinor B ose-Einstein condensate”, Pro ceedings of SPIE 6604 , 6 6041T (2007). [1 1 ] Gerdjiko v, V. S., Kaup, D. J., Kos to v, N. A. and V alchev, T. I., ” On classifica tion of s o liton so lutions of multicompo nen t nonlinear evolution equations” J. Phys. A: Math. Theor . 41 3152 13 (200 8). [1 2 ] Gerdjiko v, V. S., Kostov, N. A. and V alchev T. I., ”Solutions of multi-component NLS mo dels and Spinor Bo se-Einstein condensates” , Physica D 2 3 8 , 1 306-1310 (2 0 09). arXiv:0802.439 8 [nl in.SI] . [1 3 ] Zakharov, V. E. and A. B. Sha ba t A. B., ”A scheme for in tegrating the nonlinear equations of ma themati- cal ph ysics by the method of the in verse scattering problem” . I. F unctional Analys is and Its Applica tions, 8 , 2 26–235 (1 974). [1 4 ] Zakharov, V. E . and Shabat, A. B., ”Integration of no nlinear equations of mathematical physics by the metho d of inverse sca ttering”. II. F unctional Analysis and Its Applications, 13 , 1 66–174 (1 979). [1 5 ] Zakharov, V. E ., Mana k ov S V., Novik ov S P . a nd Pitaevs kii L. I., [Theory of solitons. The inv erse scattering metho d], Plenum, N.Y. (19 84). [1 6 ] F addee v , L. D. and T akhtadjan, L. A., [Ha milt onian Appro ac h in the Theo ry of So lito ns], Spring er V erla g, Berlin, (19 8 7). [1 7 ] Manako v, S. V., ”O n the theo ry of tw o-dimensiona l stationa r y self-fo cusing of electromag netic wa ves”, Zh. E ksp. T eor . Fiz [Sov.Ph ys. JETP ], 6 5 [ 3 8 ], 5 05–516 [2 48–253], 1973 [1 974]. [1 8 ] Gerdjiko v, V. S., ”The Za kharov-Shabat dr essing metho d and the represe ntation theory of the semisim- ple Lie alg ebras”, P h ys. Lett. A, 126A, 184–1 88, (1987). [1 9 ] Gerdjiko v, V. S., ” Generalised F our ier tra nsforms for the soliton equations. Gauge cov ariant for m ula- tion”, Inv erse Problems, 2 , 51–7 4, (19 86). [2 0 ] Gerdjiko v V. S., ”The Gener alized Zakha ro v–Shabat Sys tem and the Soliton Perturbations” Theor . Math. P h ys. 99 , 2 92–299 (1 994). [2 1 ] Mikhailov, A. V., ”The reductio n problem a nd the in verse scattering method” , Physica D: Nonlinear Phenomena, 3 , 73– 117 (19 8 1). [2 2 ] Gerdjiko v, V. S., ”On Reductions of Soliton So lut ions of multi-component NLS mo dels a nd Spinor Bose- Einstein condensates” , In AIP Conference pro ceedings of Fir st Conference on Application of Ma themat- ics in T echnical and Natur al Sciences, Sozo p ol, Bulga ria, J une 2 2-27, 2009 (In press). [2 3 ] Zakharov, V. E. a nd Mikhailov, A. V., ” O n The Integrability of Classica l Spinor Mo dels in Two- dimensional Spa c e-time”, Comm. Ma th . Phys. 7 4 , 2 1–40 (1980). [2 4 ] Gerdjiko v, V. S., Grahovski, G. G. and Ko sto v, N. A., ”On the multi-component NLS type equa tions on symmetric spa ces and their reductions” , Theo r. Math. P h ys. 144 , 11 4 7–1156 (200 5). [2 5 ] Park, Q-Han and Shin, H. J., ” P ainlev´ e analysis of the coupled nonlinear Schr¨ o dinger eq ua tion for po larized optical wav es in a n iso tropic medium”, Phys. Rev. E 59 , 23 73–2379 (199 9). [2 6 ] Zakharov, V. E. and Shabat, A. B., ”Exact theory of t wo-dimensional self-fo cusing and one-dimensio nal self-mo dulation of w av es in nonlinear media” , Soviet Physics-JETP , 34 , 6 2–69 (1972). [2 7 ] M. Ablowitz, J., Pr inari B. and T rubatch A. D., [Discrete and contin uous no nlinear Schr¨ oding er systems] Cambridge Univ. P ress, Cam bridge, (20 04). [2 8 ] Kanna T. and La kshmanan, M., ”Exact so liton s o lutions of coupled nonlinea r Sch r¨ odinger equations: Shap e-c hanging collisions , logic gates, a nd pa rtially coher en t solitons”, Ph ys. Rev. E 67 , 0466 17 (20 03). [2 9 ] Tsuchida, T., ”N-s oliton co llis ion in the Manakov mo del”, Pro g . Theor . Phys. 111 151-18 2 (2 0 04).