Bose-Einstein Condensates and spectral properties of multicomponent nonlinear Schrodinger equations

Manuscript submitted to W ebsite: http://AIMsciences.org AIMS’ Journals V olume X , Number 0X , XX 2 00X pp. X–XX BOSE-EINSTEIN CONDENSA TES AND SPECTRAL PR OPER TIES OF MUL TI COMPONENT N ONLINEAR SCHR ¨ ODINGER EQUA TIONS Vladimir S. Gerdjiko v Institute for Nuclear Research and N uclear Energy , Bulgarian academ y of sciences 72 Tsarigradsko ch aussee, 1784 Sofia, Bul gari a Abstract. W e analyze the prop erties of the soliton solutions of a class of mo d- els describing one-dimensional BEC wi th spin F . W e describe the minimal sets of scattering data which determine uniquely b oth the corresponding p otent ial of the Lax operator and its scatte ring matrix. Next we give s everal reduc- tions of these MNLS, derive their N -soliton solutions and analyze the soli ton int eractions. Finally w e prov e an imp ortant theorem proving that if the initial conditions satisfy the reduction then one gets a solution of th e reduced MNLS. 1. INTR O DUCTION. It is w ell known tha t Bos e-Einstein condensate (BEC) of alk ali atoms in the F = 1 hype r fine state, elongated in x direction and con- fined in the transverse directions y , z b y purely optical means are descr ib ed by a 3-comp onent normalized spinor wav e vector Φ ( x, t ) = (Φ 1 , Φ 0 , Φ − 1 ) T ( x, t ). Con- sidering dimensio nless units and using sp e cial c hoices for the s cattering le ng ths o ne can show that Φ ( x, t ) satisfies the multicompo nent nonlinear Sc hr¨ odinger (MNLS) equation [ 14 ], see also [ 15 , 1 9 , 26 , 2 , 2 2 ]: 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 . Similarly spinor BE C with F = 2 is describ ed by a 5 -comp onent normalize d spinor wa ve vector Φ ( x, t ) = (Φ 2 , Φ 1 , Φ 0 , Φ − 1 , Φ − 2 ) T ( x, t ). F or specific choices of the scattering leng ths in dimensionles s co o rdinates the cor resp onding s et of equatio ns for Φ ( x, t ) take the form [ 27 ]: 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 = 0 , 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 = 0 , (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 = 0 . Both mo dels hav e natur a l Lie algebraic in terpretation and are related to the sym- metric spaces BD . I ≃ SO(n + 2) / SO(n) × SO(2) with n = 3 and n = 5 res pe c tively . They are integrable by means of inv er s e scattering transform metho d [ 4 , 25 , 12 ]. 2000 Mathematics Subje ct Classific ation. Primar y: 35Q51, 37K40; Secondary: 34K17 . Key wor ds and phr ases. Bose-Einstein condensates, Multicomp onen t nonlinear Schr¨ odinger equations, Soli ton solutions, Soliton interact ions, Reductions of MNLS. 1 2 VLADIMIR S. GERDJIKO V Using a mo difica tion of the Zakharov-Shabat ‘dressing metho d’ w e describ e the soliton solutions [ 14 , 17 ] and the e ffects of the reductions on them. Sections 2 contains the basic details on the direct and inv erse scattering problems for the La x opera tor. Section 3 is devoted to the constr uction of their soliton solutions. In Section 4 we formulate the minimal sets of sca ttering data L which determine uniquely b oth the scattering ma trix and the p otential Q ( x, t ). Section 5 gives a few imp or tant exa mples of alge br aic reductions of the MNLS. In Section 6 we analyze the so lito n interactions of the MNLS. T o this end we ev a luate the limits of the generic t w o-soliton solution for t → ±∞ . As a res ult we establish that the effect of the interactions on the soliton pa r ameters is ana logous to the one for the scalar NLS equation and consists in shifts of the ‘ce nter of mass’ and shift in the phase. In Section 7 we prov e a n imp orta nt theorem proving that if the initial conditions satisfy the reduction then one gets a s o lution of the r educed MNLS. 2. The metho d for solving MNLS for an y F . 2.1. The Lax represen tation. The ab ov e MNLS equa tions ( 1 ) and ( 2 ) are the first tw o mem bers of a series o f MNLS equations related to the BD.I -type symmetric spaces. They allow Lax representation as follows [ 4 , 10 , 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 , (3) where 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 )  . (4) F or those familiar with Lie algebras I remind that, a s usual, Q ( x, t ) and J are elements o f the cor resp onding Lie algebra , which in our ca se is g ≃ so ( n + 2). The choice of the Ca r tan subalgebra element J deter mines the co-a djoint orbit of g ; in our ca se J is dual to e 1 , see [ 13 ]. It intro duces g rading in g = g (0) ⊕ g (1) where g (0) ≃ so ( n ). The ro ot system of g (0) consists of all ro ots of so ( n + 2 ) whic h are orthogo nal to e 1 ; the linear subspace g (1) is spanned by the W eyl g enerator s E α and E − α for which the ro ots α ∈ ∆ + 1 are such that their sca la r pro ducts ( α, e 1 ) = 1. Thu s the p otential Q ( x, t ) = X α ∈ ∆ + 1 ( q α ( x, t ) E α + p α ( x, t ) E − α ) (5) may be vie wed as lo cal co or dinate of the ab ov e men tioned s y mmetric space. The linear op er a tor ad J X = [ J, X ] and ad − 1 J is well defined on the imag e o f ad J in g . In wha t follows we will use the t ypical r epresentation of so ( n + 2) with n = 2 r − 1 in which Q and J take the following blo ck-matrix structure: Q ( x, t ) =   0 ~ q T 0 ~ p 0 s 0 ~ q 0 ~ p T s 0 0   , J = diag(1 , 0 , . . . 0 , − 1) . (6) F or ph ysical applications one uses mostly p otentials satisfying the typical reduction, i.e. ~ p ( x , t ) = ~ q ∗ ( x, t ). The vector ~ q ( x, t ) for integer F = r has 2 r + 1 comp onents ~ q ( x, t ) = (Φ r − 1 , . . . , Φ 0 , . . . , Φ − r +1 ) T ( x, t ) , (7) BEC AND SPE CTRAL PR OPER TIES OF MNLS 3 and the cor resp onding matrices s 0 ent er in the definition of the orthogo nal a lg ebras so (2 r − 1); namely X ∈ so (2 r + 1) if 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   . (8) By E sp ab ov e we mean 2 r + 1 × 2 r + 1 matrix with matrix elements ( E sp ) ij = δ si δ pj . With the definition of orthogo nality used in ( 8 ) the Carta n g enerator s H k = E k,k − E 2 r + 2 − k, 2 r +2 − k are represented by diago nal matr ices. If we make use o f the t ypical r eduction Q = Q † (or ~ p ∗ = ~ q ) the g eneric MNLS t ype equations related to BD . I . acquir e the for m: i~ q t + ~ q xx + 2( ~ q † , ~ q ) ~ q − ( ~ q , s 0 ~ q ) s 0 ~ q ∗ = 0 . (9) and for r = 2 (resp. r = 3) coincides with the MNLS eq. ( 1 ) (r esp. with eq. ( 3 )). The Hamiltonians for the MNLS equations ( 9 ) are g iven by H MNLS = Z ∞ −∞ dx  ( ∂ x ~ q † , ∂ x ~ q ) − ( ~ q † , ~ q ) 2 + 1 2 | ( ~ q T , s 0 ~ q ) | 2  . (10) 2.2. The Direct and the Inv erse scattering proble m for L . W e remind s o me basic features of the scatter ing theory for the Lax o pe r ators L , see [ 10 , 12 ]. There we hav e made use of the gener al theory developed in [ 32 , 33 , 29 , 3 , 6 ] and the references ther ein. The J ost solutions of L ar e defined by: lim x →−∞ φ ( x, t, λ ) e iλJ x = 1 1 , lim x →∞ ψ ( x, t, λ ) e iλJ x = 1 1 (11) and the scattering matrix T ( λ, t ) ≡ ψ − 1 φ ( x, t, λ ). The sp ecial c hoic e of J and the fact that the Jost solutions and the scattering matrix take v alues in the g roup S O (2 r + 1) w e can use the following blo ck-matrix structure o f 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    , (12) where ~ b ± ( λ, t ) a nd ~ B ± ( λ, t ) a r e 2 r − 1 - comp onent vectors, T 22 ( λ ) is 2 r − 1 × 2 r − 1 blo ck matrix, and m ± 1 ( λ ), and c ± 1 ( λ ) are scalar functions. Below we often use ˆ X to denote the matrix in verse to X . Remark 1. The t y pic a l reduction ~ p ( x, t ) = ~ q ∗ ( x, t ) mentioned a bove imp os es on T ( λ, ) the constraint T † ( λ, t ) = ˆ T ( λ, t ) for real v a lues o f λ ∈ R , i.e. m + 1 ( λ ) = m − , ∗ 1 ( λ ) , ~ B 1 − ( λ ) = ~ b 1 + , ∗ ( λ ) , c + 1 ( λ ) = c − , ∗ 1 ( λ ) , ~ B 1 + ( λ ) = ~ b 1 − , ∗ ( λ ) . (13) The Lax representation ( 1 ) allows one to prov e that if ~ q ( x, t ) sa tisfie s the MNLS ( 9 ) then the scattering matrix T ( λ, t ) satisfies the linear evolution equation [ 12 ]: i dT dt − λ 2 [ J, T ( λ, t )] = 0 , (14) 4 VLADIMIR S. GERDJIKO V or in comp onents: 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 . (15) Thu s the blo ck-diagonal matric es D ± ( λ ) can b e considered as generating functionals of the in tegrals of motion. Thus the problem of solving the MNLS eq. is based on the effective analysis of the mapping b etw ee n the potential Q ( x, t ) of L and the scattering matr ix T ( λ, t ). 2.3. The fundamen tal ana lytic solutio n and the Riemann-Hilb e rt prob- lem. The most effectiv e metho d for the ab ov e mentioned analysis consists in con- structing the fundamental analytic solutio n (F AS) of L - op erator s of type ( 3 ) and reducing the in v erse scatter ing pr o blem to an equiv alent Riemann-Hilb ert pr oblem (RHP). Sk ipping the details (see [ 11 ]) we just outline the construction of F AS for L . Obviously the F AS, like any other fundamental solutions of L m us t be linearly related to the Jost s olutions. F or the class of p o tentials Q ( x, t ) with v anishing bo undary co nditions there exist tw o F AS χ ± ( x, t, λ ) whic h allow analytic extension for λ ∈ C ± resp ectively . F or real λ they are rela ted to the Jost solutions by χ ± ( x, t, λ ) = φ ( x, t, λ ) S ± J ( t, λ ) = ψ ( x, t , λ ) T ∓ J ( t, λ ) D ± J ( λ ) , (16 ) where T ∓ J ( t, λ ), D ± J ( λ ) a nd T ∓ J ( t, λ ) are the g eneralized Gaus s factors o f T ( λ, t ), see [ 29 , 5 , 7 ]: T ( λ, t ) = T − J D + J ˆ S + J , T ( λ, t ) = T + J D − J ˆ S − J , T ∓ J ( λ, t ) = e ± ( ~ ρ ± , ~ E ∓ 1 ) , S ± J ( λ, t ) = e ± ( ~ τ ± , ~ E ± 1 ) , D ± J ( λ ) = diag  ( m ± 1 ) ± 1 , m ± 2 , ( m ± 1 ) ∓ 1  , (17) Here ~ τ ± ( λ, t ) =  τ ± r − 1 , . . . , τ ± 0 , . . . , τ ± − r +1  T ( λ, t ) ,  ~ τ + , ~ E + 1  = r − 1 X k =1 ( τ + k E e 1 − e k +1 + τ + − k E e 1 + e k +1 ) + τ + 0 E e 1 ,  ~ τ − , ~ E − 1  = r − 1 X k =1 ( τ − k E − e 1 + e k +1 + τ − − k E − e 1 − e k +1 ) + τ − 0 E − e 1 , (18) and similar express ions for  ~ ρ ± , ~ E ∓ 1  . Above we have made use o f the fac t that ∆ + 1 consists of the r o ots { e 1 − e k , e 1 , e 1 + e k } r − 1 k =1 . The functions m ± 1 and n × n matrix - v alued functions m ± 2 are ana lytic for λ ∈ C ± . One can chec k, that the analo gs of the r eflection c o efficients ~ ρ ± and ~ τ ± are expressed by: ~ ρ − = ~ B − m − 1 , ~ τ − = ~ B + m − 1 , ~ ρ + = ~ b + m + 1 , ~ τ + = ~ b − m + 1 . Remark 2. The typical reduction means that for λ ∈ R the re flection co efficients are cons trained by (see remark 1 a b ove): ~ ρ + ( λ ) = ~ ρ − , ∗ ( λ ) , ~ τ + ( λ ) = ~ τ − , ∗ ( λ ) , λ ∈ R . (19) BEC AND SPE CTRAL PR OPER TIES OF MNLS 5 There are some additional relations whic h ensure that b o th T ( λ ) and its in verse ˆ T ( λ ) b elong to the or tho gonal group S O (2 r + 1) and that T ( λ ) ˆ T ( λ ) = 1 1. The F AS χ ± ( x, t, λ ) ar e related b y: 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 ) (20) Below for conv enience w e introduce ξ ± ( x, λ ) = χ ± ( x, λ ) e iλJ x which sa tis fy the equation: i dξ ± dx + Q ( x ) ξ ± ( x, λ ) − λ [ J, ξ ± ( x, λ )] = 0 , (21) and the relation lim λ →∞ ξ ± ( x, t, λ ) = 1 1 , (22) Then ξ ± ( x, λ ) satis fy the RHP’s ξ + ( x, t, λ ) = ξ − ( x, t, λ ) G J ( x, t, λ ) , G J ( x, t, λ ) = e − iλJ ( x + λt ) G 0 ,J ( λ ) e iλJ ( x + λt ) , (23) with sewing function G J ( x, t, λ ) uniquely deter mined by the Gauss factor s S ± J ( t, λ ) taken for t = 0: G 0 ,J ( λ ) = ˆ S − J (0 , λ ) S + J (0 , λ ) . The analy ticity prop erties of these F AS follow from the equiv alent set of int egral equations: ξ + 1 j ( x, λ ) = δ 1 j + i Z x ∞ dy e − iλ ( x − y ) 2 r − 1 X p =1 q r − p ( y ) ξ + p +1 ,j ( y , λ ) , ξ + kj ( x, λ ) = δ kj + i Z x −∞ dy ( q ∗ r − k +1 ( y ) ξ + 1 ,j ( y , λ ) − ( − 1) r + k q − r + k − 1 ( y ) ξ + 2 r + 1 , j ( y , λ )) , 2 ≤ k , j ≤ 2 r ; ξ + 2 r + 1 , j ( x, λ ) = δ 2 r + 1 , j + i Z x −∞ dy e iλ ( x − y ) 2 r − 1 X p =1 ( − 1) p +1 q ∗ − r + p ( y ) ξ + p +1 ,j ( y , λ ) , (24) and a similar set o f integral eq ua tions for ξ − 1 j ( x, λ ) = δ 1 j + i Z x −∞ dy e − iλ ( x − y ) 2 r − 1 X p =1 q r − p ( y ) ξ − p +1 ,j ( y , λ ) , ξ − kj ( x, λ ) = δ kj + i Z x −∞ dy ( q ∗ r − k +1 ( y ) ξ − 1 ,j ( y , λ ) − ( − 1) r + k q − r + k − 1 ( y ) ξ − 2 r + 1 , j ( y , λ )) , 2 ≤ k , j ≤ 2 r ; ξ − 2 r + 1 , j ( x, λ ) = δ 2 r + 1 , j + i Z x ∞ dy e iλ ( x − y ) 2 r − 1 X p =1 ( − 1) p +1 q ∗ − r + p ( y ) ξ − p +1 ,j ( y , λ ) , (25) The RHP ( 23 ) with the a dditional condition ( 22 ) is kno wn as an RHP with canonical no r malization. Remark 3. An immediate conseq uence of the a nalyticity of ξ ± ( x, t, λ ) is that D ± ( λ ) are analytic functions for λ ∈ C ± . This fact follo ws from the relation lim x →∞ ξ ± ( x, λ ) = D ± ( λ ). 6 VLADIMIR S. GERDJIKO V Zakharov and Shaba t proved a theorem [ 32 , 33 ] which states that if G J ( x, λ, t ) satisfies: i dG dx − λ [ J, G ( x, λ, t )] = 0 , i dG dt − λ 2 [ J, G ( x, λ, t )] = 0 , (26) then the cor resp onding s olutions of the RHP allow one to co nstruct χ ± ( x, λ ) = ξ ± ( x, λ ) e − iλJ x as a fundamental solution of the La x pair eq. ( 1 ). W e will say that ξ ± 0 ( x, λ ) is a reg ular so lutio n to the RHP ( 23 ) if the block- diagonal pa rt o f it ha s neither zer o es nor po les in its whole r egion of analy ticit y . 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 )  , (27) allows us to recov er the cor resp onding p otential of L . 3. Singular solutions of RH P and sol iton solutions of MNLS. Zak harov- Shabat’s theorem ensur es that if a given RHP allows r egular solution, then this solution is unique. How ever the RHP may hav e many singula r solutio ns. The construction o f s uch singular solutions star ting from a g iven regular o ne is known as the dr essing Zakha r ov-Shabat metho d [ 32 , 33 ]. Indeed, if ξ ± 0 ( x, t, λ ) are reg ular solutions to the RHP , then ξ ± ( x, t, λ ) = u ( x, t, λ ) ξ ± 0 ( x, t, λ ) (28) with conv enien tly c hosen dressing factor u ( x, t, λ ) may again b e a s o lution of the RHP [ 32 , 3 3 ]. Obviously this factor m ust be analytic (with the exception of finite nu m ber of singular p o ints) in the whole complex λ -plane a nd can explicitly be constructed using only the s olution o f the reg ular RHP . In order to obtain N -soliton solutions one has to apply the dressing pro cedure to the trivial so lution of the RHP ξ 0 ( x, t, λ ) = 1 1. W e cho o se a dressing factor with 2 N -p oles [ 11 ]: u ( x, t, λ ) = 1 1 + N X k =1  A k ( x, t ) λ − λ + k + B k ( x, t ) λ − λ − k  . (29) The N -so liton so lution itself can be genera ted via the follo wing formula Q N , s ( x, t ) = N X k =1 [ J, A k ( x, t ) + B k ( x, t )] . (30) The dres sing fac to r u ( x, λ ) must satisfy the equa tion i ∂ u ∂ x + Q N , s ( x, t ) u ( x, t, λ ) − λ [ J, u ( x, t, λ )] = 0 (31) and the normaliza tion condition lim λ →∞ u ( x, λ ) = 1 1. The residues o f u admit the following decomp os ition A k ( x, t ) = X k ( x, t ) F T k ( x, t ) , B k ( x, t ) = Y k ( x, t ) G T k ( x, t ) , where all matrices in volved are supp osed to b e rectangular and of max imal rank s [ 30 , 9 ]. By comparing the co efficients b efor e the same powers o f λ − λ ± k in ( 31 ) we convince ours elves that the factors F k and G k can b e expr e ssed by the fundamental analytic solutions χ ± 0 ( x, t, λ ) = e − iλ ( x + λt ) J as fo llows F T k ( x, t ) = F T k, 0 [ χ + 0 ( x, t, λ + k )] − 1 , G T k ( x, t ) = G T k, 0 [ χ − 0 ( x, t, λ − k )] − 1 . BEC AND SPE CTRAL PR OPER TIES OF MNLS 7 The cons ta nt re c ta ngular matr ices F k, 0 and G k, 0 ob ey the a lgebraic rela tions 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 fac to rs X k ( x, t ) a nd Y k ( x, t ) are solutions to the alg ebraic 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 . (32) The squar e s × s matr ices α k ( x, t ) and β k ( x, t ) intro duced ab ov e dep end on χ + 0 and χ − 0 and their deriv a tives b y λ as fo llows α k ( x, t ) = − F T 0 ,k [ χ + 0 ( x, t, λ + k )] − 1 ∂ λ χ + 0 ( x, t, λ + k ) S 0 F 0 ,k + α 0 ,k , β k ( x, t ) = − G T 0 ,k [ χ − 0 ( x, t, λ − k )] − 1 ∂ λ χ − 0 ( x, t, λ − k ) S 0 G 0 ,k + β 0 ,k . (33) Below for s implicit y we will c ho ose F k and G k to b e 2 r + 1-comp onent vectors. Then one can show that α k = β k = 0 which simplifies the system ( 32 ). W e also int ro duce the following more conv enient para metrization for F k and G k , namely (see eq. ( 35 )): 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   , (34) where ~ ν 0 k are constant 2 r − 1-co mpo nent p olariza tion vectors and z j = ν j ( x + 2 µ j t ) + ξ 0 j , φ j = µ j x + ( µ 2 j − ν 2 j ) t + δ 0 j , h n T j ( x, t ) | S 0 | n j ( x, t ) i = 0 , or ( ~ ν 0 ,j s 0 ~ ν 0 ,j ) = 1 . (35) With this nota tio ns the p olar ization vectors automatically s a tisfy the condition h n j ( x, t ) | S 0 | n j ( x, t ) i = 0 . Thus for 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 , (36) which is eas ily solved. As a result for the one-so liton so lution 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 = cos h(2 z 1 ) + h ~ ν † 01 | ~ ν 01 i . (37) 8 VLADIMIR S. GERDJIKO V F or n = 3 we put ν 0 k = | ν 0 k | e α 0 k and 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 ) c o s( α 13 ) − i sinh( z 1 ∓ ζ 01 ) s in( 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 ) , (38) Note that the ‘center o f mass‘ o f Φ 1s;1 (resp. of Φ 1s; − 1 ) is shifted with res pe ct to the o ne of Φ 1s;0 by ζ 01 to the r ig ht (resp to the left); besides | 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 ana lo gously: 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 ) c o s( α 15 ) − i sinh( z 1 ∓ ζ 01 ) s in( 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 ) c o s( α 24 ) − i sinh( z 1 ∓ ζ 01 ) s in( 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 ) . (39) Similarly the ‘center of mass‘ of Φ 1s;2 and Φ 1s;1 (resp. of Φ 1s; − 2 and Φ 1s; − 1 ) are shifted with resp ect to the one of Φ 1s;0 by ζ 01 and ζ 02 to the right (res p to the left); bes ides | Φ 1s;2 | = | Φ 1s; − 2 | and | Φ 1s;1 | = | Φ 1s; − 1 | . F or N = 2 we get: | 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 , (40) 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  , (41) BEC AND SPE CTRAL PR OPER TIES OF MNLS 9 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     . (42) W e c a n rew r ite M in block-matrix for m: 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 ! . (43) The inv erse of M is given by: M − 1 =  N − 1 1 −N − 1 1 M 12 ˆ M ∗ 11 −N − 1 2 M 21 ˆ M 11 N − 1 2  , N 1 = M 11 − M 12 ˆ M ∗ 11 M 21 , N 2 = M ∗ 11 − M 21 ˆ M 11 M 12 (44) F rom eq s. ( 42 ) a nd ( 44 ) we obtain [ 12 ]: | 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) where 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  . (46) Inserting this result into eq. ( 30 ) w e o btain the follo wing express ion for the 2-soliton s olution 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 | . (47) 4. The minimal sets of scattering data. It is well known that the lo cations of the singularities of the RHP λ ± k ∈ C ± are zero es of the functions m ± 1 ( λ ) and discrete eigenv alues of the Lax op er ator L . W e will say that these eigenv alues are simple if the corr esp onding eigensubspaces are o ne dimensional. This cor resp onds to our choice of F k ( x, t ) and G k ( x, t ) as vectors. Eigensubspace s of higher multiplicities s > 1 can be obta ined choo sing F k ( x, t ) a nd G k ( x, t ) as s × (2 r + 1) matrices of rank s . 10 VLADIMIR S. GERDJIKO V Theorem 4.1 . L et the p otential Q ( x, t ) b e such that the c orr esp onding L ax op er ator L has finite nu m b er of simple discr ete eigenvalues lo c ate d at the p oints λ ± k ∈ C ± r esp e ct ively, k = 1 , . . . , N . Then as minimal sets of sc att ering data uniquely de- termining b oth t he sc attering matrix T ( λ, t ) and the c orr esp onding p otential Q ( x, t ) one c an c onsider the sets T 1 ≡ { ~ τ + ( λ, 0 ) , λ ∈ R ; ~ τ + k , λ + k ∈ C + , k = 1 , . . . N } , T 2 ≡ { ~ ρ ± ( λ, 0 ) , λ ∈ R ; ~ ρ + k , λ + k ∈ C + , k = 1 , . . . N } , (48) wher e the c onst ant ve ctors ~ τ + 0 k and ~ ρ ± 0 k ~ τ + 0 k =   e ξ 0 k − iδ 0 k √ 2 ~ ν 0 k e − ξ 0 k + iδ 0 k   , ~ ρ + 0 k =   e η 0 k − iθ 0 k √ 2 ~ µ 0 k e − η 0 k + iθ 0 k   , (49) and the ve ctors ~ ν 0 k and ~ µ 0 k satisfy the normalization c ondition ( ~ ν T 0 k s 0 ~ ν 0 k ) = 1 and ( ~ µ T 0 k s 0 ~ µ 0 k ) = 1 . Remark 4. The da ta λ + k and λ − k = ( λ + k ) ∗ characterize the disc r ete eigenv alues of L . The vectors ~ τ + 0 k and ~ τ − 0 k = ( ~ τ + 0 k ) ∗ (resp. ~ ρ + 0 k and ~ ρ − 0 k = ( ~ ρ + 0 k ) ∗ ) determine the corres p o nding eigenfunction of L . No te also that by definition these vectors satisfy ( ~ τ + ,T 0 k s 0 ~ τ + 0 k ) = 0 and ( ~ ρ + ,T 0 k s 0 ~ ρ + 0 k ) = 0 . Outline of the pr o of. Let us b e given T 1 . Using ~ τ + ( λ, t ) and ~ τ − ( λ, t ) = ( ~ τ + ( λ, t )) ∗ we construct S + 0 J ( λ, t ) and S − 0 J ( λ, t ) and therefore obtain also the sewing function G 0 ( λ, t ) = ˆ S − 0 J ( λ, t ) S + 0 J ( λ, t ) for a reg ular RHP . According to the Z a kharov-Shabat theorem it has unique solution ξ ± 0 ( x, t, λ ). The cor resp onding r egular potential is obtained by: Q 0 ( x, t ) = lim λ →∞ λ  J − ξ ± 0 ( x, t, λ ) J ˆ ξ ± 0 ( x, t, λ )  = [ J, ξ + 01 ( x, t )] , (50) where ξ + 01 ( x, t ) = lim λ →∞ λ ( ξ + 0 ( x, t, λ ) − 1 1). Next we use the dress ing metho d to dress the regula r solution ξ ± 0 ( x, t, λ ) with the dr essing factor u ( x, t, λ ) of the form ( 29 ). In order to do it we make use of the set of eig env alues λ + k and λ − k = ( λ + k ) ∗ and instea d of the pola rization vectors ( 34 ) we use : | n k ( x, t ) i = ξ + 0 ( x, t, λ + k ) e − iλ + k ( x + λ + k t ) J ~ τ 0 k + , | n ∗ k ( x, t ) i = ξ − 0 ( x, t, λ − k ) e − iλ − k ( x + λ − k t ) J ~ τ 0 k − . (51) After s olving the algebraic e q uations for | X k ( x, t ) i and | Y k ( x, t ) i we find ex plicitly the dr essed p otential Q ( x, t ) = Q 0 ( x, t ) + N X k =1 [ J, A k ( x, t ) + B k ( x, t )] , (52) which pr oves the first part o f the theorem. Let us now s how ho w o ne can recover T ( λ, t ) fr om T 1 . Given the r egular solution ξ ± 0 ( x, t, λ ) we can find D ± 0 ,J ( λ ) = lim x →∞ ξ ± 0 ( x, t, λ ) , (53) and als o T ∓ 0 ,J ( λ ) D ± 0 ,J ( λ ) = lim x →∞ e i ( λx + λ 2 t ) J ξ ± 0 ( x, t, λ ) e − i ( λx + λ 2 t ) J . (54) BEC AND SPE CTRAL PR OPER TIES OF MNLS 11 Thu s we have recovered all Gauss factors T ∓ 0 ,J ( λ ), D ± 0 ,J ( λ ) and S ± 0 ,J ( λ ) of the ‘un- dressed’ sca ttering ma trix T 0 ( λ, t ), so T 0 ,J ( λ, t ) = T ∓ 0 ,J ( λ, t ) D ± 0 ,J ( λ ) ˆ S ± 0 ,J ( λ, t ) . (55) In order to ta ke into a ccount the effect of dress ing we make use of the relations betw een the dressed and undres sed Jo st so lutions: ψ ( x, t, λ ) = u ( x, t, λ ) ψ 0 ( x, t, λ ) ˆ u + ( λ ) , φ ( x, t, λ ) = u ( x, t, λ ) φ 0 ( x, t, λ ) ˆ u − ( λ ) , (56) where u ± ( λ ) = lim x →±∞ u ( x, t, λ ). As a result w e get: T ( λ, t ) = ˆ ψ ( x, t, λ ) φ ( x, t, λ ) = u + ( λ ) ˆ ψ 0 ( x, t, λ ) φ 0 ( x, t, λ ) ˆ u − ( λ ) = u + ( λ ) T 0 ( λ, t ) ˆ u − ( λ ) . (57) Skipping the details we state the result: u + ( λ ) =   c ( λ ) 0 0 0 1 1 0 0 0 1 /c ( λ )   , u − ( λ ) =   1 /c ( λ ) 0 0 0 1 1 0 0 0 c ( λ )   , (58) where c ( λ ) = Q N j =1 λ − λ + j λ − λ − j . The fact that the set T 2 is als o a minimal set o f sca tter ing data is prov e d ana lo- gously . 5. Reductions of MNLS. Along with the typical r eduction Q = Q † men tioned ab ov e one can imp ose additiona l reductio ns using the r eduction g roup prop osed by Mikhailov [ 21 ]. They are automa tically compatible with the Lax representation of the c o rresp onding MNLS eq. Below we make us e of tw o t yp es of Z 2 -reductions[ 8 ]: 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 ) , (59) where C 1 and C 2 are in volutions of the Lie algebra so (2 r + 1, i.e. C 2 i = 1 1. They can be chosen to b e either diagonal (i.e., elements of the Cartan subgroup of S O (2 r + 1)) or elements o f the W eyl gr oup. The t ypical r e ductio ns o f the MNLS eqs . is a cla ss 1) reduction obtained by sp ecifying C 1 to be the identit y automorphism of g ; below we list several choices for C 1 leading to inequiv alent re ductio ns: 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 ) . (60) W e a lso make use of type 2) reductions: 2e) C 2 = K 4 , ~ q ( x ) = − K 04 s 0 ~ q ( x ) , ~ p ( x ) = − K 04 s 0 ~ p ( x ) , 2f ) C 2 = K 5 , ~ q ( x ) = K 05 ~ q ( x ) , ~ p ( x ) = K 05 ~ p ( x ) , (61) where K j = blo ck-diag (1 , K 0 j , 1) , K 01 = diag ( ǫ 1 , . . . , ǫ r − 1 , 1 , ǫ r − 1 , . . . , ǫ 1 ) , (62) 12 VLADIMIR S. GERDJIKO V for j = 1 , 2 , 3 , 5 and ǫ j = ± 1 . The matrices K 02 , K 03 and K 4 are not diago nal and may take the form: 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       . (63) Each of the ab ove reductions impose constraints on the F AS, on the scattering matrix T ( λ ) and on its Gauss factors S ± J ( λ ), T ± J ( λ ) a nd D ± J ( λ ). F or the type 1 reductions (cases 1a) – 1d)) 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 − , ∗ , j = 1 , 2 , 3 (64) where the ma tr ices K j are s pe c ific for ea ch choice of the a utomorphisms C 1 , see eqs. ( 60 ). In par ticular, from the last line of ( 64 ) and ( 61 ) we get: ( m + 1 ( λ ∗ )) ∗ = m − 1 ( λ ) , (65) and consequently , if m + 1 ( λ ) has zero es at the p oints λ + k , then m − 1 ( λ ) has zero es at: λ − k = ( λ + k ) ∗ , k = 1 , . . . , N . (66) F or the type 2) r eductions we obtain: 2e) ( S ± ( λ )) T = K − 1 4 ˆ S ± ( λ ) K 4 ( T ± ( λ )) T = K − 1 4 ˆ T ± ( λ ) K 4 ( D ± ( λ )) T = K − 1 4 ˆ D ± ( λ ) K 4 ~ τ ± = − K 04 s 0 ~ τ ± , ~ ρ ± = − K 04 s 0 ~ ρ ± , (67) and 2f ) ( S + ( λ )) T = K − 1 5 ˆ S − ( − λ ) K 5 ( T + ( λ )) T = K − 1 5 ˆ T − ( − λ ) K 5 ( D + ( λ )) T = K − 1 5 ˆ D − ( − λ ) K 5 ~ τ + ( λ ) = K 05 ~ τ − ( − λ ) , ~ ρ + ( λ ) = − K 05 ~ ρ − ( − λ ) , (68) F or the 2e) reduction with n = 3 we ma y c ho ose K 4 to cor resp onds to the W eyl group element S e 1 , so K 04 = 1 1. As a result w e get: Φ 1 = − Φ − 1 (69) and Φ 0 arbitrar y . This re ductio n of eq. ( 1 ) is also impo rtant for the BEC [ 22 ]. F rom ( 67 ) we find ν 01 = ν 03 . The effect of this constra in t is that for the o ne-soliton solution we get Φ 1s;1 = − Φ 1s; − 1 . Our next remark fo llowing [ 23 ] is that this r eduction applied to the F = 1 MNLS ( 1 ) leads to a 2- comp onent MNLS which after the change o f v ar iables Φ 1 = 1 2 ( w 1 + iw 2 ) , Φ 0 = i √ 2 ( w 1 − iw 2 ) , (70) leads to tw o disjoint NLS equations for w 1 and w 2 resp ectively . BEC AND SPE CTRAL PR OPER TIES OF MNLS 13 It is only logical that applying the constraint ν 01 = ν 03 the ex plicit express ion for the o ne-soliton solution ( 38 ) simplifies and reduces to the standard soliton solutions of the scalar NLS. F or the other tw o ex amples of type 2) reductio ns w e choose n = 5 and K 4 , and K 5 corres p o nd to the W eyl g roup elemen ts S e 2 S e 3 and S e 2 − e 3 resp ectively . Then K 04 = − s 0 and K 05 =       0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 − 1 0 0 0 − 1 0       , (71) F or thes e choices of K 4 , K 5 we o btain: 2e) Φ 2 = Φ − 2 , Φ 1 = Φ − 1 , 2f ) Φ ± 2 = ± c √ 1 + c 2 Φ ′ ± 1 , Φ ± 1 = 1 √ 1 + c 2 Φ ′ ± 1 , (72) It r educes the F = 2 s pin BEC mo del in to the F = 1 mo del. The corres po nding relations for the Gauss factors and for the p o larizatio n vectors are given by: Φ ± 2 = ± c √ 1 + c 2 Φ ′ ± 1 , Φ ± 1 = 1 √ 1 + c 2 Φ ′ ± 1 , (73) 6. Tw o Soliton i n teractions. In this s ection we g e neralize the cla ssical results of Zakharov and Shabat ab out soliton in teractions [ 31 ] to the cla ss of MNLS e q uations related to BD.I symmetric spaces. F or detailed expositio n see the monog raphs [ 29 , 3 ]. Thes e results were genera lized for the vector nonlinear Schr¨ o dinger equation by Ma nako v [ 20 ], see also [ 1 , 1 6 , 28 ]. T he Za kharov Shabat appro ach consisted in calculating the asy mpto tics of gener ic N - soliton solution of NLS for t → ± ∞ and establishing the pur e elastic character of the generic so liton in teractions. By generic her e w e mean N -so liton solution who s e par ameters λ ± k = µ k ± iν k are such that µ k 6 = µ j for k 6 = j . The pure elastic character of the s o liton in teractions is demonstrated by the fact that for t → ± ∞ the g e neric N -solito n s o lution splits int o sum of N one soliton s olutions each pre s erving its a mplitude 2 ν k and v elo city µ k . The only effect of the interaction consists in shifting the cen ter of ma ss and the initial phase of the s o litons. These s hifts ca n b e expresse d in terms o f λ ± k only; for detailed exp o s ition see [ 3 ]. Let us apply these ideas to the MNLS equations studied a bove. Namely we use the 2-soliton so lutio n ( 47 ) derived a bove and calculate its as y mptotics along the tra jector y of the first soliton. T o this end we keep z 1 ( x, t ) fixed and let τ = z 2 − z 1 tend to ± ∞ . Ther efore it will be enough to insert the asymptotic v alues of the matrix elements o f M for τ → ±∞ and k eep only the leading terms. F or τ → ∞ that gives: κ 22 ≃ e 2 τ exp( ν 2 z 1 /ν 1 ) + 2 C 1 , κ 12 = e τ exp((1 + ν 2 /ν 1 ) z 1 + i ( φ 1 − φ 2 )) + O (1) , κ 21 = e τ exp((1 + ν 2 /ν 1 ) z 1 − i ( φ 1 − φ 2 )) + O (1) , f 12 = e τ exp( − (1 − ν 2 /ν 1 ) z 1 + i ( φ 1 − φ 2 )) + O (1) , (74) 14 VLADIMIR S. GERDJIKO V while for τ → −∞ we g et: κ 22 ≃ e − 2 τ exp( − ν 2 z 1 /ν 1 ) + 2 C 1 , κ 12 = e − τ exp( − (1 + ν 2 /ν 1 ) z 1 − i ( φ 1 − φ 2 )) + O (1) , κ 21 = e − τ exp( − (1 + ν 2 /ν 1 ) z 1 + i ( φ 1 − φ 2 )) + O (1) , f 12 = e − τ exp((1 − ν 2 /ν 1 ) z 1 − i ( φ 1 − φ 2 )) + O (1) , (75) After s omewhat le ng thy ca lculations 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 ) , (76) where r + = ln     λ + 1 − λ + 2 λ + 1 − λ − 2     , α + = ar g λ + 1 − λ + 2 λ + 1 − λ − 2 . F or n = 3 and n = 5 the r ight hand sides o f ( 76 ) coincide with the one-soliton solutions ( 38 ) a nd ( 39 ) resp ectively . This means that the 2 - soliton interaction fo r the ab ov e MNLS eq s . is purely elastic. The solito ns preser ve their shap es and velocities and the only effect of the interaction cons ist in shifts o f the center of mass and the phase. F r o m this p oint of view the interaction is the s ame like for the s calar NLS eq . It is important to check whether the N -so liton in teractions consist o f sequence of elementary 2 -soliton interactions and the shifts a re additive. 7. Effects of reductions and initial conditions on M NLS. Theorem 7.1. L et the m inimal set of sc att ering data T j , j = 1 , 2 for t = 0 satisfy the r e duction c onditions ( 67 ). The n the solution ~ q ( x, t ) of the MNLS with such initial data wil l satisfy the c orr esp onding re duction 2e) ( 61 ). Pr o of. Let the minimal se ts of scattering data, say T 1 satisfy the reduction condi- tions ( 67 ) for t = 0. It is easy to chec k that their evolution law ( 15 ) is compatible with the reductio n, s o ( 67 ) will hold for all t > 0. As a result the corres po nding Gauss fa ctors S ± , T ± and D ± , and consequently , the sewing function in the RHP G ( x, t, λ ) will satisfy G ( x, t, λ ) = K − 1 4 ˆ G T ( x, t, λ ) K 4 . (77) The nex t consequence is that b o th ξ ± and K − 1 4 ˆ ξ ± ,T K 4 are solutions of the RHP ( 23 ) with the same s e wing function and the same canonica l norma lization. Therefor e from the uniquenes s of the so lution of RHP we get that the r egular solutions of this RHP sa tis fy: ξ ± 0 ( x, t, λ ) = K − 1 4 ˆ ξ ± ,T 0 ( x, t, λ ) K 4 . (78) Next we note tha t the scattering data rela ted to the discrete sp ectrum also satisfy the reduction co nditio ns . This means that the dre s sing factor u ( x, t, λ ) and the singular s o lutions ξ ± ( x, t, λ ) = u ( x, t, λ ) ξ ± 0 ( x, t, λ ) ˆ u − ( λ ) als o satisfy: u ( x, t, λ ) = K − 1 4 ˆ u T ( x, t, λ ) K 4 , ξ ± ( x, t, λ ) = K − 1 4 ˆ ξ ± ,T ( x, t, λ ) K 4 . (79) It r emains to chec k that fro m e q uations ( 27 ) and ( 79 ) there follows: Q ( x, t ) = − K − 1 4 Q T ( x, t ) K 4 . (80) BEC AND SPE CTRAL PR OPER TIES OF MNLS 15 Remark 5. Note that the a bove a rguments are no t specific for the ch oice o f K 4 . The ab ove theorem can b e proved along the sa me lines for a ny re ductio n of t ype 1 and type 2. A simple consequence of the ab ove theorem is the following. C o nsider n = 3 and choose τ + 1 = τ + 3 for t = 0 . Then the corresp o nding solution of F = 1 BE C ( 1 ) will also sa tisfy Φ 1 = − Φ − 1 for a ll t > 0, i.e. will b e a solution to 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(2 | Φ 1 | 2 + | Φ 0 | 2 )Φ 0 − 2Φ ∗ 0 Φ 2 1 = 0 , (81) If we inser t eq. ( 70 ) into ( 81 ) we obtain i∂ t w 1 + ∂ 2 x w 1 + 2 | w 1 | 2 w 1 = 0 , i∂ t w 2 + ∂ 2 x w 2 + 2 | w 2 | 2 w 2 = 0 , (82) Therefore, if we wan t to ana lyze the spe cific features of F = 1 BEC we hav e to av oid suc h initial conditions. Similarly , if for n = 5 we cho ose in ( 39 ) ν 01;1 = ν 01 , 5 , ν 01;2 = − ν 01 , 4 we will obtain in fact a s olution to F = 1 BEC. 8. Conclusions and discussi o n. Using the Zakharov-Shabat dressing metho d we hav e obtained the tw o -soliton solution and ha ve used it to analyze the soliton int eractions of the MNLS eq ua tion. The conclusio n is that after the interactions the solitons recov e r their pola rization vectors ν 0 k , velo cities and fre quency velo cities. The effect o f the in tera c tio n is, lik e in for the scala r NLS equatio n, shift o f the center of mass z 1 → z 1 + r + and shift of the phase φ 1 → φ 1 + α + . Both shifts are expressed thro ugh the rela ted eigenv alues λ ± j only . 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