Permutability of Backlund Transformations for N=2 Supersymmetric Sine-Gordon

P erm utability of Bac klund T ransformations for N = 2 Sup ersymmetric Sine-Gordo n J.F. Gomes, L.H. Ymai a nd A.H. Zimerman Instituto de F sica T erica, IFT-UNESP Univ ersidade Estadual Paulista Caixa Postal 7053 2-2 01156 -970 So Paulo, SP , Brazil ABSTRAC T The p erm utability of t w o Bac klund tr a nsformations is emplo ye d to construct a non linear sup erp osition fo rm ula and to generate a class of solutions for the N = 2 super sine-Gordon mo del. W e presen t explicitly the one and t wo soliton solutions. 1 In tro ductio n Bac klund transformations reduce the order o f the non-linear differen tia l equations making the system sometimes effec tively more tractable. Starting with a sim ple input solution, w e may b e able to solve for a more complicated one. In many cases, this ma y b e v ery difficult to accomplish. A conv enien t a nd p ow erful wa y is to use the p ermutability the or em whic h pro vides a closed algebraic non- linear sup erp osition form ula for the solutions. The Bac klund transformation and the P erm utabilit y theorem are employ ed to deriv e a se ries of consistency conditions whic h are satisfied by soliton solutions of certain class of in tegra ble mo dels. Within suc h class, we encoun ter the sine-Gor do n [1] and KdV [2] equations. This framew ork was a lso applied to the N = 1 sup er KdV [3] and sup er sinh-Gordon [4] in order to deriv e its soliton solutions. The N = 2 sup er sine-G o rdon mo del w as pro p osed in [5 ] and la ter in [6] its algebraic structure w as unco ve r ed. Certain solutions of this mo del hav e already b een constructed [7], ho we ver they we re suc h that inv olv e a single Grassmann parameter. In this pap er we extend the non-linear sup erp osition f orm ulae for soliton solutions o f the N = 2 sup er sine-G o rdon mo del. These for mulae are deriv ed from the Ba c klund transformation prop osed in [8] and the p erm utability condition whic h implies that the order of tw o success ive Bac klund transformations is irrelev an t. As examples, w e presen t explicitly the 1-and 2-soliton solutions with distinct Grassmann parameters. Recen tly the P ohlmey er reduction of Ad S n xS n sup erstring mo dels ha v e b een considered [9] whic h in the simple case of n = 2 w a s shown [10] to b e equiv alen t to the N = 2 sup ersymmetric sine-Gordon. This pap er is orga nized as follows . In Section 2 w e discuss the N = 2 sup er sin e- Gordon and its Bac klund T ransformation. In Section 3 w e apply the p erm utability condition to deriv e a closed algebraic non- linear sup erp o sition form ulae in v o lving solutions of the mo del. Finally in Section 4 and 5 w e presen t the 1- and 2 -soliton solutions resp ectiv ely . In the app endix A w e presen t the Bac klund transformation in comp onents. In app endices B and C we give details fo r the deriv ation of the sup erp osition formulae. 2 N = 2 sup er sine-Gordon - Bac klund T ransformation Let us start by in tro ducing the N = 2 superfields [5] φ ± = ϕ ± ( z ± , ¯ z ± ) + θ ± ψ ∓ ( z ± , ¯ z ± ) + ¯ θ ± ¯ ψ ∓ ( z ± , ¯ z ± ) + θ ± ¯ θ ± F ± ( z ± , ¯ z ± ) , where z ± = z ± 1 2 θ + θ − , ¯ z ± = ¯ z ± 1 2 ¯ θ + ¯ θ − . The sup erfield comp onen ts φ ± can b e expanded in Grassmann v ariables θ ± and ¯ θ ± . F or instance, the comp onent ϕ ± ( z ± , ¯ z ± ) giv es r ise to ϕ ± ( z ± , ¯ z ± ) = ϕ ± ± 1 2 θ + θ − ∂ z ϕ ± ± 1 2 ¯ θ + ¯ θ − ∂ ¯ z ϕ ± + 1 4 θ + θ − ¯ θ + ¯ θ − ∂ z ∂ ¯ z ϕ ± . 1 By expanding all comp o nen ts of φ ± ,w e obtain φ ± = ϕ ± + θ ± ψ ∓ + ¯ θ ± ¯ ψ ∓ ± 1 2 θ + θ − ∂ z ϕ ± ± 1 2 ¯ θ + ¯ θ − ∂ ¯ z ϕ ± + θ ± ¯ θ ± F ± ± θ ± ¯ θ + ¯ θ − 1 2 ∂ ¯ z ψ ∓ ± ¯ θ ± θ + θ − 1 2 ∂ z ¯ ψ ∓ + 1 4 θ + θ − ¯ θ + ¯ θ − ∂ z ∂ ¯ z ϕ ± . W e next in tro duce the sup er deriv ativ es D ± = ∂ ∂ θ ± + 1 2 θ ∓ ∂ z , ¯ D ± = ∂ ∂ ¯ θ ± + 1 2 ¯ θ ∓ ∂ ¯ z , satisfying the fo llo wing conditio ns D 2 ± = 0 , ¯ D 2 ± = 0 , { ¯ D ± , D ± } = 0 , { ¯ D ± , D ∓ } = 0 , { D + , D − } = ∂ z , { ¯ D + , ¯ D − } = ∂ ¯ z . The equations of motion for the sup ersymmetric sine-Gordon mo del with N = 2 a r e g iv en by [5] ¯ D ± D ± φ ± = g sin  β φ ∓  , (1) where g is a mass pa r ameter a nd β is the coupling constan t. F rom now on w e assume β = 1 whic h ma y b e re-inserted by a con v enien t field reparametrization. In components, the equations of motion for the N = 2 sup er sine-Gordon reads, F ± = g sin ϕ ∓ , ∂ ¯ z ψ ∓ = g cos ϕ ∓ ¯ ψ ± , ∂ z ¯ ψ ∓ = − g cos ϕ ∓ ψ ± , ∂ z ∂ ¯ z ϕ ± = − g cos ϕ ∓ F ∓ − g sin ϕ ∓ ψ ± ¯ ψ ± . Moreo v er the the chiral, φ + and the an ti- c hiral, φ − sup erfields satisfy the conditions ¯ D ± φ ∓ = D ± φ ∓ = 0 . (2) Let us now recall the Backlund transformation for the N = 2 sup er sine-Gordon mo del [8]. F or this purp ose, consider the pair of first order differen tial equations D + φ + 1 = D + φ + 2 − 8 κ F cos φ − 1 + φ − 2 2 ! , (3) ¯ D + φ + 1 = − ¯ D + φ + 2 + κ G cos φ − 1 − φ − 2 2 ! , (4) where F and G are f ermionic auxiliary sup erfields and κ is an a rbitrary constant. The ab ov e equation and the condition ( ¯ D + D + + D + ¯ D + ) φ + 1 = 0 , 2 leads to the equations o f motion ¯ D + D + φ + 2 = g sin φ − 2 , pro vided the sup erfields F and G s a tisfy ¯ D + F = − κg 4 sin φ − 1 − φ − 2 2 ! , D + G = − 2 g κ sin φ − 1 + φ − 2 2 ! . (5) In a similar w ay , D − φ − 1 = D − φ − 2 + λ G cos φ + 1 + φ + 2 2 ! , (6) ¯ D − φ − 1 = − ¯ D − φ − 2 − 8 λ F cos φ + 1 − φ + 2 2 ! , (7) where λ is another a rbitrary constan t. T ogether with the condition ( ¯ D − D − + D − ¯ D − ) φ − 1 = 0 , yields ¯ D − D − φ − 2 = g sin φ + 2 , pro vided G and F satisfy ¯ D − G = 2 g λ sin φ + 1 − φ + 2 2 ! , D − F = λg 4 sin φ + 1 + φ + 2 2 ! . (8) Acting with D + in eqn. (3), ¯ D + in (4), D − in (6) and ¯ D − in (7) w e find D + F = 0 , ¯ D + G = 0 , D − G = 0 , ¯ D − F = 0 . (9) These last conditions allow s us to rewrite the fermionic sup erfields into tw o distinct manners, i.e., F = D + Φ + 1 = ¯ D − Φ − 2 , (10) G = D − Φ − 1 = ¯ D + Φ + 2 , (11) where the c hira l Φ + p and anti-c hiral Φ − p , p = 1 , 2 sup erfields are defined as Φ ± 1 = q ± 1 ( z ± , ¯ z ± ) + θ ± ζ ± 1 ( z ± , ¯ z ± ) + ¯ θ ± ζ ± 2 ( z ± , ¯ z ± ) + θ ± ¯ θ ± q ± 2 ( z ± , ¯ z ± ) , Φ ± 2 = p ± 1 ( z ± , ¯ z ± ) + θ ± ξ ± 1 ( z ± , ¯ z ± ) + ¯ θ ± ξ ± 2 ( z ± , ¯ z ± ) + θ ± ¯ θ ± p ± 2 ( z ± , ¯ z ± ) . The second equalit y in (10) implies ζ + 1 = ξ − 2 , q + 2 = ∂ ¯ z p − 1 , p − 2 = − ∂ z q + 1 , ∂ z ζ + 2 = − ∂ ¯ z ξ − 1 , (12) whilst the second equalit y in (11) implies ζ − 1 = ξ + 2 , p + 2 = − ∂ z q − 1 , q − 2 = ∂ ¯ z p + 1 , ∂ z ζ − 2 = − ∂ ¯ z ξ + 1 . (13) Eqns. (3)- (9) describ e the Ba c klund transformation for the N = 2 sup er sine-Gordon system. In app endix A w e presen t these equations in comp onen ts. 3 3 The P erm utabilit y condition A Bac klund transformation from φ ± 0 to φ ± 1 is describ ed by D + ( φ + 0 − φ + 1 ) = − 8 κ 1 F (0 , 1) cos φ − 0 + φ − 1 2 ! , (14) ¯ D + ( φ + 0 + φ + 1 ) = κ 1 G (0 , 1) cos φ − 0 − φ − 1 2 ! , (15) D − ( φ − 0 − φ − 1 ) = λ 1 G (0 , 1) cos φ + 0 + φ + 1 2 ! , (16) ¯ D − ( φ − 0 + φ − 1 ) = − 8 λ 1 F (0 , 1) cos φ + 0 − φ + 1 2 ! , (17) where w e ha ve in tro duced the sup erscript indices (0 , 1) for the auxiliary fermionic sup erfields denoting its dependence in φ ± 0 and φ ± 1 . The later, in turn satisfy the following condition (as in (5) and (8)) ¯ D + F (0 , 1) = − g κ 1 4 sin φ − 0 − φ − 1 2 ! , (18) D + G (0 , 1) = − g 2 κ 1 sin φ − 0 + φ − 1 2 ! , (19) ¯ D − G (0 , 1) = g 2 λ 1 sin φ + 0 − φ + 1 2 ! , (20) D − F (0 , 1) = g λ 1 4 sin φ + 0 + φ + 1 2 ! . ( 2 1) The c hiral conditions (9), i.e., ¯ D − F (0 , 1) = 0 , D + F (0 , 1) = 0 , ¯ D + G (0 , 1) = 0 , D − G (0 , 1) = 0 , are a ut o matically satisfied using eqn. (10), i.e., express ing the fermionic sup erfields as deriv a- tiv es o f chiral sup erfields, F (0 , 1) = D + Φ +(0 , 1) 1 = ¯ D − Φ − (0 , 1) 2 , G (0 , 1) = D − Φ − (0 , 1) 1 = ¯ D + Φ +(0 , 1) 2 , where the superscript indices indicate whether the sup erfield Φ ± 1 and Φ ± 2 dep end up on φ ± 0 and φ ± 1 . Acting with sup er deriv ativ es D − , ¯ D − , D + and ¯ D + on eqns.(14), (15), (16) and (17) resp ectiv ely , w e find ∂ z ( φ + 0 − φ + 1 ) = − 2 γ 1 s + 0 , 1 c − 0 , 1 + 8 κ 1 F (0 , 1) D − c − 0 , 1 , (22) ∂ ¯ z ( φ + 0 + φ + 1 ) = 2 g 2 γ 1 ¯ s + 0 , 1 ¯ c − 0 , 1 − κ 1 G (0 , 1) ¯ D − ¯ c − 0 , 1 , (23) ∂ z ( φ − 0 − φ − 1 ) = − 2 γ 1 s − 0 , 1 c + 0 , 1 − λ 1 G (0 , 1) D + c + 0 , 1 , (24) ∂ ¯ z ( φ − 0 + φ − 1 ) = 2 g 2 γ 1 ¯ s − 0 , 1 ¯ c + 0 , 1 + 8 λ 1 F (0 , 1) ¯ D + ¯ c + 0 , 1 , (25) 4 where γ 1 = g λ 1 κ 1 and c ± j,k = cos φ ± j + φ ± k 2 ! , s ± j,k = sin φ ± j + φ ± k 2 ! , (26) ¯ c ± j,k = cos φ ± j − φ ± k 2 ! , ¯ s ± j,k = sin φ ± j − φ ± k 2 ! , (27) W e now assume tha t the order of t wo successiv e Back lund tr a nsformations is irrelev an t leading to the same final result. Such condition is kno wn as the p erm utability theorem, i.e., φ ± 0 γ 1 / / φ ± 1 γ 2 / / φ ± 12 and in the inv erse order, φ ± 0 γ 2 / / φ ± 2 γ 1 / / φ ± 21 , do es not change the final result, φ ± 12 = φ ± 21 ≡ φ ± 3 . The p ermutabilit y theorem applied to the Back lund equation (14) leads to D + ( φ + 0 − φ + 1 ) = − 8 κ 1 F (0 , 1) c − 0 , 1 , D + ( φ + 1 − φ + 3 ) = − 8 κ 2 F (1 , 3) c − 1 , 3 , D + ( φ + 0 − φ + 2 ) = − 8 κ 2 F (0 , 2) c − 0 , 2 , D + ( φ + 2 − φ + 3 ) = − 8 κ 1 F (2 , 3) c − 2 , 3 . (28) T aking into accoun t that the sum of the first t wo and the last tw o equations a re the same, w e obtain, 1 κ 1 F (0 , 1) c − 0 , 1 + 1 κ 2 F (1 , 3) c − 1 , 3 = 1 κ 2 F (0 , 2) c − 0 , 2 + 1 κ 1 F (2 , 3) c − 2 , 3 . (29) Similarly , fro m (17), w e o bt a in ¯ D − ( φ − 0 + φ − 1 ) = − 8 λ 1 F (0 , 1) ¯ c + 0 , 1 , ¯ D − ( φ − 1 + φ − 3 ) = − 8 λ 2 F (1 , 3) ¯ c + 1 , 3 , ¯ D − ( φ − 0 + φ − 2 ) = − 8 λ 2 F (0 , 2) ¯ c + 0 , 2 , ¯ D − ( φ − 2 + φ − 3 ) = − 8 λ 1 F (2 , 3) ¯ c + 2 , 3 , (30) leading to 1 λ 1 F (0 , 1) ¯ c + 0 , 1 − 1 λ 2 F (1 , 3) ¯ c + 1 , 3 = 1 λ 2 F (0 , 2) ¯ c + 0 , 2 − 1 λ 1 F (2 , 3) ¯ c + 2 , 3 . (31) W e prop ose as solution for the non-linear sup erp osition formula φ ± 12 = φ ± 21 = φ ± 3 , φ ± 3 = φ ± 0 + Γ ± + ∆ ± , (32) 5 with Γ ± ( x, y ) = 2 arctan  δ tan  x + y 4  ± 2 arctan  δ tan  x − y 4  , x = φ + 1 − φ + 2 , y = φ − 1 − φ − 2 , (33) δ = γ 1 + γ 2 γ 1 − γ 2 , γ k = g λ k κ k . Notice that t he solution φ ± 3 when the fermionic sup erfields a re neglected is deriv ed in the app endix B to b e φ ± 3 = φ ± 0 + Γ ± . The term ∆ ± comes from the contribution of the fermionic sup erfields and has the follo wing form ∆ ± = 2 X j,k =1 Λ ± j,k f j,k + Λ ± 0 f 0 , f j,k = F (0 ,j ) G (0 ,k ) , f 0 = F (0 , 1) F (0 , 2) G (0 , 1) G (0 , 2) , where we hav e assumed the co efficien ts Λ ± to b e functionals of x = ( φ + 1 − φ + 2 ) and y = ( φ − 1 − φ − 2 ),i.e., Λ ± j,k = Λ ± j,k ( x, y ) , Λ ± 0 = Λ ± 0 ( x, y ) . (34) Observ e that there are no terms lik e Λ ± 1 F (0 , 1) F (0 , 2) nor Λ ± 2 G (0 , 1) G (0 , 2) due to c hiral equations (2). Λ ± are determined in app endix C where, Λ + 1 , 1 = Λ + 2 , 2 = − 8 µ − g η + η − cos  x 2  sin  y 2  , Λ + 1 , 2 = 8 µ − g η + η − λ 2 λ 1 ! sin  y 2  , Λ + 2 , 1 = 8 µ − g η + η − λ 1 λ 2 ! sin  y 2  , Λ + 0 = − 32 µ − ( g η + η − ) 2 sin  x 2   cos  y 2  ( a + cos x − cos y ) − 2 µ + cos  x 2  , Λ − 1 , 1 = Λ − 2 , 2 = 8 µ − g η + η − cos  y 2  sin  x 2  , Λ − 1 , 2 = − 8 µ − g η + η −  κ 2 κ 1  sin  x 2  , Λ − 2 , 1 = − 8 µ − g η + η −  κ 1 κ 2  sin  x 2  , Λ − 0 = − 32 µ − ( g η + η − ) 2 sin  y 2   cos  x 2  ( a − cos x + cos y ) − 2 µ + cos  y 2  , µ ± = γ 1 γ 2 ± γ 2 γ 1 , a = 1 2 γ 2 1 γ 2 2 + γ 2 2 γ 2 1 ! + 3 , η ± = µ + − 2 cos  x ± y 2  . 6 3.1 Solution in Comp o nen ts In comp onen ts the non-linear sup erp osition for m ula ( 3 2) yields the following express ions, ϕ + 3 = ϕ + 0 + ˜ Γ + − 8 µ − g ˜ η + ˜ η − A + 1 − B + 1 + 4 g ˜ η + ˜ η − C + 1 ! , ψ − 3 = ψ − 0 + F 1 , 2 ψ − 1 , 2 + 8 µ − g ˜ η + ˜ η − A + 2 − B + 2 + 4 g ˜ η + ˜ η − C + 2 ! , ¯ ψ − 3 = ¯ ψ − 0 + F 1 , 2 ¯ ψ − 1 , 2 + 8 µ − g ˜ η + ˜ η − A + 3 − B + 3 + 4 g ˜ η + ˜ η − C + 3 ! , ϕ − 3 = ϕ − 0 + ˜ Γ − + 8 µ − g ˜ η + ˜ η − A − 1 − B − 1 − 4 g ˜ η + ˜ η − C − 1 ! , ψ + 3 = ψ + 0 + F 1 , 2 ψ + 1 , 2 − 8 µ − g ˜ η + ˜ η − A − 2 − B − 2 − 4 g ˜ η + ˜ η − C − 2 ! , ¯ ψ + 3 = ¯ ψ + 0 + F 1 , 2 ¯ ψ + 1 , 2 − 8 µ − g ˜ η + ˜ η − A − 3 − B − 3 − 4 g ˜ η + ˜ η − C − 3 ! , where ˜ Γ ± = 2 ar ctan " δ tan ϕ + 1 , 2 + ϕ − 1 , 2 4 !# ± 2 arctan " δ tan ϕ + 1 , 2 − ϕ − 1 , 2 4 !# , ˜ η ± = µ + − 2 cos ϕ + 1 , 2 ± ϕ − 1 , 2 2 ! , F 1 , 2 = δ 2     sec 2  ϕ + 1 , 2 + ϕ − 1 , 2 4  1 + δ 2 tan 2  ϕ + 1 , 2 + ϕ − 1 , 2 4  + sec 2  ϕ + 1 , 2 − ϕ − 1 , 2 4  1 + δ 2 tan 2  ϕ + 1 , 2 − ϕ − 1 , 2 4      , 7 A + 1 = cos ϕ + 1 , 2 2 ! sin ϕ − 1 , 2 2 !  ζ +(0 , 1) 1 ξ +(0 , 1) 2 + ζ +(0 , 2) 1 ξ +(0 , 2) 2  , A + 2 = − cos ϕ + 1 , 2 2 ! sin ϕ − 1 , 2 2 !  ζ +(0 , 1) 1 p +(0 , 1) 2 + ζ +(0 , 2) 1 p +(0 , 2) 2  − Σ +  ζ +(0 , 1) 1 ξ +(0 , 1) 2 + ζ +(0 , 2) 1 ξ +(0 , 2) 2  ψ − 1 , 2 , A + 3 = − cos ϕ + 1 , 2 2 ! sin ϕ − 1 , 2 2 !  q +(0 , 1) 2 ξ +(0 , 1) 2 + q +(0 , 2) 2 ξ +(0 , 2) 2  − Σ +  ζ +(0 , 1) 1 ξ +(0 , 1) 2 + ζ +(0 , 2) 1 ξ +(0 , 2) 2  ¯ ψ − 1 , 2 , B + 1 = sin ϕ − 1 , 2 2 ! λ 2 λ 1 ζ +(0 , 1) 1 ξ +(0 , 2) 2 + λ 1 λ 2 ζ +(0 , 2) 1 ξ +(0 , 1) 2 ! , B + 2 = − sin ϕ − 1 , 2 2 ! λ 2 λ 1 ζ +(0 , 1) 1 p +(0 , 2) 2 + λ 1 λ 2 ζ +(0 , 2) 1 p +(0 , 1) 2 ! − Ω + λ 2 λ 1 ζ +(0 , 1) 1 ξ +(0 , 2) 2 + λ 1 λ 2 ζ +(0 , 2) 1 ξ +(0 , 1) 2 ! ψ − 1 , 2 , B + 3 = − sin ϕ − 1 , 2 2 ! λ 2 λ 1 q +(0 , 1) 2 ξ +(0 , 2) 2 + λ 1 λ 2 q +(0 , 2) 2 ξ +(0 , 1) 2 ! − Ω + λ 2 λ 1 ζ +(0 , 1) 1 ξ +(0 , 2) 2 + λ 1 λ 2 ζ +(0 , 2) 1 ξ +(0 , 1) 2 ! ¯ ψ − 1 , 2 , C + 1 = sin ϕ + 1 , 2 2 ! A + ζ +(0 , 1) 1 ζ +(0 , 2) 1 ξ +(0 , 1) 2 ξ +(0 , 2) 2 , C + 2 = sin ϕ + 1 , 2 2 ! A + ζ +(0 , 1) 1 ζ +(0 , 2) 1  ξ +(0 , 2) 2 p +(0 , 1) 2 − ξ +(0 , 1) 2 p +(0 , 2) 2  , C + 3 = sin ϕ + 1 , 2 2 ! A + ξ +(0 , 1) 2 ξ +(0 , 2) 2  ζ +(0 , 1) 1 q +(0 , 2) 2 − ζ +(0 , 2) 1 q +(0 , 1) 2  , 8 A − 1 = cos ϕ − 1 , 2 2 ! sin ϕ + 1 , 2 2 !  ζ +(0 , 1) 1 ξ +(0 , 1) 2 + ζ +(0 , 2) 1 ξ +(0 , 2) 2  , A − 2 = − cos ϕ − 1 , 2 2 ! sin ϕ + 1 , 2 2 !  ∂ z q +(0 , 1) 1 ξ +(0 , 1) 2 + ∂ z q +(0 , 2) 1 ξ +(0 , 2) 2  − Σ −  ζ +(0 , 1) 1 ξ +(0 , 1) 2 + ζ +(0 , 2) 1 ξ +(0 , 2) 2  ψ + 1 , 2 , A − 3 = cos ϕ − 1 , 2 2 ! sin ϕ + 1 , 2 2 !  ζ +(0 , 1) 1 ∂ ¯ z p +(0 , 1) 1 + ζ +(0 , 2) 1 ∂ ¯ z p +(0 , 2) 1  − Σ −  ζ +(0 , 1) 1 ξ +(0 , 1) 2 + ζ +(0 , 2) 1 ξ +(0 , 2) 2  ¯ ψ + 1 , 2 , B − 1 = sin ϕ + 1 , 2 2 !  κ 2 κ 1 ζ +(0 , 1) 1 ξ +(0 , 2) 2 + κ 1 κ 2 ζ +(0 , 2) 1 ξ +(0 , 1) 2  , B − 2 = − sin ϕ + 1 , 2 2 !  κ 2 κ 1 ∂ z q +(0 , 1) 1 ξ +(0 , 2) 2 + κ 1 κ 2 ∂ z q +(0 , 2) 1 ξ +(0 , 1) 2  − Ω −  κ 2 κ 1 ζ +(0 , 1) 1 ξ +(0 , 2) 2 + κ 1 κ 2 ζ +(0 , 2) 1 ξ +(0 , 1) 2  ψ + 1 , 2 , B − 3 = sin ϕ + 1 , 2 2 !  κ 2 κ 1 ζ +(0 , 1) 1 ∂ ¯ z p +(0 , 2) 1 + κ 1 κ 2 ζ +(0 , 2) 1 ∂ ¯ z p +(0 , 1) 1  − Ω −  κ 2 κ 1 ζ +(0 , 1) 1 ξ +(0 , 2) 2 + κ 1 κ 2 ζ +(0 , 2) 1 ξ +(0 , 1) 2  ¯ ψ + 1 , 2 , C − 1 = sin ϕ − 1 , 2 2 ! A − ζ +(0 , 1) 1 ζ +(0 , 2) 1 ξ +(0 , 1) 2 ξ +(0 , 2) 2 , C − 2 = − sin ϕ − 1 , 2 2 ! A − ξ +(0 , 1) 2 ξ +(0 , 2) 2  ζ +(0 , 2) 1 ∂ z q +(0 , 1) 1 − ζ +(0 , 1) 1 ∂ z q +(0 , 2) 1  , C − 3 = − sin ϕ − 1 , 2 2 ! A − ζ +(0 , 1) 1 ζ +(0 , 2) 1  ξ +(0 , 2) 2 ∂ ¯ z p +(0 , 1) 1 − ξ +(0 , 1) 2 ∂ ¯ z p +(0 , 2) 1  , 9 Σ + = cos ϕ + 1 , 2 2 ! Ω + − 1 2 sin ϕ − 1 , 2 2 ! sin ϕ + 1 , 2 2 ! , Ω + = − sin ϕ − 1 , 2 2 ! " 1 ˜ η + sin ϕ + 1 , 2 + ϕ − 1 , 2 2 ! + 1 ˜ η − sin ϕ + 1 , 2 − ϕ − 1 , 2 2 !# , A + = cos ϕ − 1 , 2 2 !  a + cos ϕ + 1 , 2 − cos ϕ − 1 , 2  − 2 µ + cos ϕ + 1 , 2 2 ! , Σ − = cos ϕ − 1 , 2 2 ! Ω − − 1 2 sin ϕ − 1 , 2 2 ! sin ϕ + 1 , 2 2 ! , Ω − = − sin ϕ + 1 , 2 2 ! " 1 ˜ η + sin ϕ + 1 , 2 + ϕ − 1 , 2 2 ! − 1 ˜ η − sin ϕ + 1 , 2 − ϕ − 1 , 2 2 !# , A − = cos ϕ + 1 , 2 2 !  a − cos ϕ + 1 , 2 + cos ϕ − 1 , 2  − 2 µ + cos ϕ − 1 , 2 2 ! , and denoted ϕ ± 1 , 2 = ϕ ± 1 − ϕ ± 2 , ψ ± 1 , 2 = ψ ± 1 − ψ ± 2 , ¯ ψ ± 1 , 2 = ¯ ψ ± 1 − ¯ ψ ± 2 . F rom the Bac klund eqns. w e get (see app. A) ζ +(0 ,k ) 1 = − κ k 8 ( ψ − 0 − ψ − k ) cos  ϕ − 0 + ϕ − k 2  , ξ +(0 ,k ) 2 = 1 κ k ( ¯ ψ − 0 + ¯ ψ − k ) cos  ϕ − 0 − ϕ − k 2  , ∂ z q +(0 ,k ) 1 = λ k g 4 sin ϕ + 0 + ϕ + k 2 ! , p +(0 ,k ) 2 = 2 g κ k sin ϕ − 0 + ϕ − k 2 ! , q +(0 ,k ) 2 = − κ k g 4 sin ϕ − 0 − ϕ − k 2 ! , ∂ ¯ z p +(0 ,k ) 1 = 2 g λ k sin ϕ + 0 − ϕ + k 2 ! . 4 1 -Solito n So lution Setting φ ± 0 = 0 in t he Bac klund eqns. (14)-(21) w e find in comp onen ts, ∂ ¯ z ζ +(0 , 1) 1 = − g 2 γ 1 cos ϕ + 1 2 ! cos ϕ − 1 2 ! ζ +(0 , 1) 1 , ∂ z ζ +(0 , 1) 1 = γ 1 cos ϕ + 1 2 ! cos ϕ − 1 2 ! ζ +(0 , 1) 1 , ∂ ¯ z ξ +(0 , 1) 2 = − g 2 γ 1 cos ϕ + 1 2 ! cos ϕ − 1 2 ! ξ +(0 , 1) 2 , ∂ z ξ +(0 , 1) 2 = γ 1 cos ϕ + 1 2 ! cos ϕ − 1 2 ! ξ +(0 , 1) 2 , 10 ∂ ¯ z ϕ ± 1 = − 2 g 2 γ 1 sin ϕ ± 1 2 ! cos ϕ ∓ 1 2 ! , ∂ z ϕ ± 1 = 2 γ 1 sin ϕ ± 1 2 ! cos ϕ ∓ 1 2 ! . In tegrating the ab ov e eqns. w e g et the 1-solito n solution, ψ − 1 = 8 κ 1 cos ϕ − 1 2 ! ζ +(0 , 1) 1 , ψ + 1 = − λ 1 cos ϕ + 1 2 ! ξ +(0 , 1) 2 , ¯ ψ − 1 = κ 1 cos ϕ − 1 2 ! ξ +(0 , 1) 2 , ¯ ψ + 1 = − 8 λ 1 cos ϕ + 1 2 ! ζ +(0 , 1) 1 , ϕ ± 1 = 2 ar ctan( a 1 ρ 1 ) ± 2 arctan( b 1 ρ 1 ) , ζ +(0 , 1) 1 = ξ +(0 , 1) 2 = ǫ 1 χ 1 , χ 1 = ρ 1 q (1 + a 2 1 ρ 2 1 )(1 + b 2 1 ρ 2 1 ) , where a 1 and b 1 are arbitrary constants , ǫ 1 is a grassmann parameter and ρ 1 = exp γ 1 z − g 2 γ 1 ¯ z ! . The 1-soliton solution constructed in this section can b e obtained from those of [7] b y relating parameters since they b oth in v o lve a single grassmann parameter. 5 2 -Solito n So lution F or the 2-solito n case we obtain from the sup erp osition form ula e (32) ϕ + 3 = ϕ +(0) 3 + ϕ +(1) 3 ǫ 1 ǫ 2 , ϕ +(0) 3 = 2 ar ctan " δ tan ϕ + 1 , 2 + ϕ − 1 , 2 4 !# + 2 a rctan " δ tan ϕ + 1 , 2 − ϕ − 1 , 2 4 !# , ϕ +(1) 3 = 8 µ − g ˜ η + ˜ η − sin ϕ − 1 , 2 2 ! λ 2 λ 1 − λ 1 λ 2 ! χ 1 χ 2 , ψ − 3 = ǫ 1 ψ − (1) 3 + ǫ 2 ψ − (2) 3 , ψ − (1) 3 = 8 κ 1 F 1 , 2 cos ϕ − 1 2 ! χ 1 + 16 κ 1 γ 1 µ − ˜ η + ˜ η − sin ϕ − 1 , 2 2 ! χ 1 " γ 2 sin ϕ − 2 2 ! − γ 1 cos ϕ + 1 , 2 2 ! sin ϕ − 1 2 !# , ψ − (2) 3 = − 8 κ 2 F 1 , 2 cos ϕ − 2 2 ! χ 2 + 16 κ 2 γ 2 µ − ˜ η + ˜ η − sin ϕ − 1 , 2 2 ! χ 2 " γ 1 sin ϕ − 1 2 ! − γ 2 cos ϕ + 1 , 2 2 ! sin ϕ − 2 2 !# , 11 ¯ ψ − 3 = ǫ 1 ¯ ψ − (1) 3 + ǫ 2 ¯ ψ − (2) 3 , ¯ ψ − (1) 3 = κ 1 F 1 , 2 cos ϕ − 1 2 ! χ 1 + 2 κ 1 γ 2 µ − ˜ η + ˜ η − sin ϕ − 1 , 2 2 ! χ 1 " γ 1 sin ϕ − 2 2 ! − γ 2 cos ϕ + 1 , 2 2 ! sin ϕ − 1 2 !# , ¯ ψ − (2) 3 = − κ 2 F 1 , 2 cos ϕ − 2 2 ! χ 2 + 2 κ 2 γ 1 µ − ˜ η + ˜ η − sin ϕ − 1 , 2 2 ! χ 2 " γ 2 sin ϕ − 1 2 ! − γ 1 cos ϕ + 1 , 2 2 ! sin ϕ − 2 2 !# , ϕ − 3 = ϕ − (0) 3 + ϕ − (1) 3 ǫ 1 ǫ 2 , ϕ − (0) 3 = 2 ar ctan " δ tan ϕ + 1 , 2 + ϕ − 1 , 2 4 !# − 2 arctan " δ tan ϕ + 1 , 2 − ϕ − 1 , 2 4 !# , ϕ − (1) 3 = − 8 µ − g ˜ η + ˜ η − sin ϕ + 1 , 2 2 !  κ 2 κ 1 − κ 1 κ 2  χ 1 χ 2 , ψ + 3 = ǫ 1 ψ +(1) 3 + ǫ 2 ψ +(2) 3 , ψ +(1) 3 = − λ 1 F 1 , 2 cos ϕ + 1 2 ! χ 1 − 2 λ 1 γ 1 µ − ˜ η + ˜ η − sin ϕ + 1 , 2 2 ! χ 1 " γ 2 sin ϕ + 2 2 ! − γ 1 cos ϕ − 1 , 2 2 ! sin ϕ + 1 2 !# , ψ +(2) 3 = λ 2 F 1 , 2 cos ϕ + 2 2 ! χ 2 − 2 λ 2 γ 2 µ − ˜ η + ˜ η − sin ϕ + 1 , 2 2 ! χ 2 " γ 1 sin ϕ + 1 2 ! − γ 2 cos ϕ − 1 , 2 2 ! sin ϕ + 2 2 !# , ¯ ψ + 3 = ǫ 1 ¯ ψ +(1) 3 + ǫ 2 ¯ ψ +(2) 3 , ¯ ψ +(1) 3 = − 8 λ 1 F 1 , 2 cos ϕ + 1 2 ! χ 1 − 16 λ 1 γ 2 µ − ˜ η + ˜ η − sin ϕ + 1 , 2 2 ! χ 1 " γ 1 sin ϕ + 2 2 ! − γ 2 cos ϕ − 1 , 2 2 ! sin ϕ + 1 2 !# , ¯ ψ +(2) 3 = 8 λ 2 F 1 , 2 cos ϕ + 2 2 ! χ 2 − 16 λ 2 γ 1 µ − ˜ η + ˜ η − sin ϕ + 1 , 2 2 ! χ 2 " γ 2 sin ϕ + 1 2 ! − γ 1 cos ϕ − 1 , 2 2 ! sin ϕ + 2 2 !# , 12 where ϕ ± k = 2 arctan( a k ρ k ) ± 2 arctan( b k ρ k ) , χ k = ρ k q (1 + a 2 k ρ 2 k )(1 + b 2 k ρ 2 k ) , k = 1 , 2, a k and b k are arbitrary constants , ǫ k is a grassmann constan t and ρ k = exp γ k z − g 2 γ k ¯ z ! . Notice that the 2-soliton solution constructed in this section generalizes those constructed in ref. [7]) in volving a single grassmann para meter. Both 1- and 2- soliton solutions presen ted ab ov e w ere v erified to satisfy the equations of motion. Ac knowledgm ents LHY ac kno wledges supp ort from F ap esp, JFG and AHZ thank CNPq for partia l supp o rt. App endix A In order to simplify no t a tion let us in tro duce ϕ ( − ) ± = ϕ − 1 ± ϕ − 2 , ϕ (+) ± = ϕ + 1 ± ϕ + 2 and similar notation for the other fields. In comp onen ts eqn. (5) b ecomes • ¯ D + F = − κg 4 sin  φ − 1 − φ − 2 2  , ⇓ q + 2 = − κg 4 sin  ϕ ( − ) − 2  , ∂ ¯ z ζ + 1 = − κg 8 cos  ϕ ( − ) − 2  ¯ ψ (+) − , ∂ z ζ + 2 = κg 8 cos  ϕ ( − ) − 2  ψ (+) − , ∂ ¯ z ∂ z q + 1 = κg 8 cos  ϕ ( − ) − 2  F ( − ) − + κg 16 sin  ϕ ( − ) − 2  ψ (+) − ¯ ψ (+) − . • D + G = − 2 g κ sin  φ − 1 + φ − 2 2  , ⇓ p + 2 = 2 g κ sin  ϕ ( − ) + 2  , ∂ ¯ z ξ + 1 = g κ cos  ϕ ( − ) + 2  ¯ ψ (+) + , ∂ z ξ + 2 = − g κ cos  ϕ ( − ) + 2  ψ (+) + , ∂ ¯ z ∂ z p + 1 = − g κ cos  ϕ ( − ) + 2  F ( − ) + − g 2 κ sin  ϕ ( − ) + 2  ψ (+) + ¯ ψ (+) + . 13 Similarly we find for (8), • ¯ D − G = 2 g λ sin  φ + 1 − φ + 2 2  , ⇓ q − 2 = 2 g λ sin  ϕ (+) − 2  , ∂ ¯ z ζ − 1 = g λ cos  ϕ (+) − 2  ¯ ψ ( − ) − , ∂ z ζ − 2 = − g λ cos  ϕ (+) − 2  ψ ( − ) − , ∂ ¯ z ∂ z q − 1 = − g λ cos  ϕ (+) − 2  F (+) − − g 2 λ sin  ϕ (+) − 2  ψ ( − ) − ¯ ψ ( − ) − . • D − F = λg 4 sin  φ + 1 + φ + 2 2  , ⇓ p − 2 = − λg 4 sin  ϕ (+) + 2  , ∂ ¯ z ξ − 1 = − λg 8 cos  ϕ (+) + 2  ¯ ψ ( − ) + , ∂ z ξ − 2 = λg 8 cos  ϕ (+) + 2  ψ ( − ) + , ∂ ¯ z ∂ z p − 1 = λg 8 cos  ϕ (+) + 2  F (+) + + λg 16 sin  ϕ (+) + 2  ψ ( − ) + ¯ ψ ( − ) + . F rom (3) and (4) , • D + φ + 1 = D + φ + 2 − 8 κ F cos  φ − 1 + φ − 2 2  , ⇓ ψ ( − ) − = − 8 κ ζ + 1 cos  ϕ ( − ) + 2  , F (+) − = − 8 κ q + 2 cos  ϕ ( − ) + 2  , ∂ z ϕ (+) − = − 4 κ sin  ϕ ( − ) + 2  ζ + 1 ψ (+) + − 8 κ ∂ z q + 1 cos  ϕ ( − ) + 2  . • ¯ D + φ + 1 = − ¯ D + φ + 2 + κ G cos  φ − 1 − φ − 2 2  , ⇓ ¯ ψ ( − ) + = κξ + 2 cos  ϕ ( − ) − 2  , F (+) + = κp + 2 cos  ϕ ( − ) − 2  , ∂ ¯ z ϕ (+) + = κ 2 sin  ϕ ( − ) − 2  ξ + 2 ¯ ψ (+) − + κ∂ ¯ z p + 1 cos  ϕ ( − ) − 2  . F rom (6) and (7) , • D − φ − 1 = D − φ − 2 + λ G cos  φ + 1 + φ + 2 2  , ⇓ ψ (+) − = λζ − 1 cos  ϕ (+) + 2  , F ( − ) − = λq − 2 cos  ϕ (+) + 2  , ∂ z ϕ ( − ) − = λ 2 sin  ϕ (+) + 2  ζ − 1 ψ ( − ) + + λ∂ z q − 1 cos  ϕ (+) + 2  . • ¯ D − φ − 1 = − ¯ D − φ − 2 − 8 λ F cos  φ + 1 − φ + 2 2  , ⇓ ¯ ψ (+) + = − 8 λ ξ − 2 cos  ϕ (+) − 2  , F ( − ) + = − 8 λ p − 2 cos  ϕ (+) − 2  , ∂ ¯ z ϕ ( − ) + = − 4 λ sin  ϕ (+) − 2  ξ − 2 ¯ ψ ( − ) − − 8 λ ∂ ¯ z p − 1 cos  ϕ (+) − 2  . 14 App endix B Applying the permutabilit y theorem to eqn s. (22) and (2 4 ) after neglecting the con tribution prop ortional to f ermionic sup erfields, we obtain the following relations γ 1 s + 0 , 1 c − 0 , 1 + γ 2 s + 1 , 3 c − 1 , 3 = γ 2 s + 0 , 2 c − 0 , 2 + γ 1 s + 2 , 3 c − 2 , 3 , γ 1 s − 0 , 1 c + 0 , 1 + γ 2 s − 1 , 3 c + 1 , 3 = γ 2 s − 0 , 2 c + 0 , 2 + γ 1 s − 2 , 3 c + 2 , 3 . Summing and subtracting the ab ov e eqns., w e find γ 1 h ( s + 0 , 1 c − 0 , 1 ± s − 0 , 1 c + 0 , 1 ) − ( s + 2 , 3 c − 2 , 3 ± s − 2 , 3 c + 2 , 3 ) i + γ 2 h ( s + 1 , 3 c − 1 , 3 ± s − 1 , 3 c + 1 , 3 ) − ( s + 0 , 2 c − 0 , 2 ± s − 0 , 2 c + 0 , 2 ) i = 0 . (35) Using the iden tity sin a cos b ± sin b cos a = sin ( a ± b ) , (36) and eqns. (26) and (27) we can rewrite (35) as γ 1 ( sin " φ + 0 + φ + 1 2 ! ± φ − 0 + φ − 1 2 !# − sin " φ + 2 + φ + 3 2 ! ± φ − 2 + φ − 3 2 !#) γ 2 ( sin " φ + 1 + φ + 3 2 ! ± φ − 1 + φ − 3 2 !# − sin " φ + 0 + φ + 2 2 ! ± φ − 0 + φ − 2 2 !#) = 0 . Using the fact tha t sin a − sin b = 2 cos a + b 2 ! sin a − b 2 ! , yields 2 cos  Y + ± Y −  n γ 1 sin h ( X + 1 , 2 ± X − 1 , 2 ) − ( X + 3 , 0 ± X − 3 , 0 ) i + γ 2 sin h ( X + 1 , 2 ± X − 1 , 2 ) + ( X + 3 , 0 ± X − 3 , 0 ) io = 0 , where w e hav e denoted Y ± = φ ± 0 + φ ± 1 + φ ± 2 + φ ± 3 4 , X ± j,k = φ ± j − φ ± k 4 . from where it fo llo ws that ( γ 1 + γ 2 ) sin  X + 1 , 2 ± X − 1 , 2  cos  X + 3 , 0 ± X − 3 , 0  = ( γ 1 − γ 2 ) sin  X + 3 , 0 ± X − 3 , 0  cos  X + 1 , 2 ± X − 1 , 2  , 15 or, tan  X + 3 , 0 ± X − 3 , 0  = γ 1 + γ 2 γ 1 − γ 2 ! tan  X + 1 , 2 ± X − 1 , 2  , and therefore φ + 3 − φ + 0 4 ! ± φ − 3 − φ − 0 4 ! = arctan h δ tan  X + 1 , 2 ± X − 1 , 2 i , where δ =  γ 1 + γ 2 γ 1 − γ 2  . Adding and subtracting the a b ov e expressions w e obtain φ ± 3 = φ ± 0 + Γ ± , with Γ ± = 2 arcta n h δ tan  X + 1 , 2 + X − 1 , 2 i ± 2 arctan h δ tan  X + 1 , 2 − X − 1 , 2 i . App endix C Relations (29)and (31 ) can b e written in matrix form, F (1 , 3) F (2 , 3) ! = 1 Z A − B C − D ! F (0 , 1) F (0 , 2) ! (37) where A = κ 2 λ 2 ( ¯ c + 0 , 1 c − 2 , 3 + ¯ c + 2 , 3 c − 0 , 1 ) , B = κ 2 λ 1 ¯ c + 0 , 2 c − 2 , 3 + κ 1 λ 2 ¯ c + 2 , 3 c − 0 , 2 , C = κ 2 λ 1 ¯ c + 1 , 3 c − 0 , 1 + κ 1 λ 2 ¯ c + 0 , 1 c − 1 , 3 , D = κ 1 λ 1 ( ¯ c + 0 , 2 c − 1 , 3 + ¯ c + 1 , 3 c − 0 , 2 ) , Z = κ 2 λ 1 ¯ c + 1 , 3 c − 2 , 3 − κ 1 λ 2 ¯ c + 2 , 3 c − 1 , 3 . (38) In tro duce eqn. (32) in to expressions (38). Consider now the follo wing expansions c − k , 3 = c k , Γ − 1 − ∆ 2 − 8 ! − ∆ − 2 s k , Γ − , ¯ c + k , 3 = ¯ c k , Γ − 1 − ∆ 2 + 8 ! + ∆ + 2 ¯ s k , Γ + , where w e hav e denoted c k , Γ − = cos φ − k + φ − 0 + Γ − 2 ! = c − k , 0 σ + − s − k , 0 ρ − , s k , Γ − = sin φ − k + φ − 0 + Γ − 2 ! = s − k , 0 σ + + c − k , 0 ρ − , ¯ c k , Γ + = cos φ + k − φ + 0 − Γ + 2 ! = ¯ c + k , 0 σ − + ¯ s + k , 0 ρ + ¯ s k , Γ + = sin φ + k − φ + 0 − Γ + 2 ! = ¯ s + k , 0 σ − − ¯ c + k , 0 ρ + , 16 and σ ± = 1 ± δ 2 tan  x + y 4  tan  x − y 4  r 1 + δ 2 tan 2  x + y 4  r 1 + δ 2 tan 2  x − y 4  , ρ ± = δ h tan  x + y 4  ± tan  x − y 4 i r 1 + δ 2 tan 2  x + y 4  r 1 + δ 2 tan 2  x − y 4  . Next, w e expand the expressions fo r A , B , C , D and Z in p o w er series of f obta ining A = A 0 + 2 X j,k =1 A j,k f j,k + O ( f 0 ) , A 0 = κ 2 λ 2 ( ¯ c + 0 , 1 c 2 , Γ − + c − 0 , 1 ¯ c 2 , Γ + ) , A j,k = 1 2 κ 2 λ 2 ( c − 0 , 1 ¯ s 2 , Γ + Λ + j,k − ¯ c + 0 , 1 s 2 , Γ − Λ − j,k ) , B = B 0 + 2 X j,k =1 B j,k f j,k + O ( f 0 ) , B 0 = κ 2 λ 1 ¯ c + 0 , 2 c 2 , Γ − + κ 1 λ 2 ¯ c 2 , Γ + c − 0 , 2 , B j,k = 1 2 ( κ 1 λ 2 c − 0 , 2 ¯ s 2 , Γ + Λ + j,k − κ 2 λ 1 ¯ c + 0 , 2 s 2 , Γ − Λ − j,k ) , C = C 0 + 2 X j,k =1 C j,k f j,k + O ( f 0 ) , C 0 = κ 1 λ 2 ¯ c + 0 , 1 c 1 , Γ − + κ 2 λ 1 ¯ c 1 , Γ + c − 0 , 1 , C j,k = 1 2 ( κ 2 λ 1 c − 0 , 1 ¯ s 1 , Γ + Λ + j,k − κ 1 λ 2 ¯ c + 0 , 1 s 1 , Γ − Λ − j,k ) , D = D 0 + 2 X j,k =1 D j,k f j,k + O ( f 0 ) , D 0 = κ 1 λ 1 ( ¯ c + 0 , 2 c 1 , Γ − + c − 0 , 2 ¯ c 1 , Γ + ) , D j,k = 1 2 κ 1 λ 1 ( c − 0 , 2 ¯ s 1 , Γ + Λ + j,k − ¯ c + 0 , 2 s 1 , Γ − Λ − j,k ) , Z = Z 0 + 2 X j,k =1 Z j,k f j,k + O ( f 0 ) , Z 0 = κ 2 λ 1 ¯ c 1 , Γ + c 2 , Γ − − κ 1 λ 2 c 1 , Γ − ¯ c 2 , Γ + , Z j,k = 1 2 ( κ 2 λ 1 c 2 , Γ − ¯ s 1 , Γ + − κ 1 λ 2 c 1 , Γ − ¯ s 2 , Γ + )Λ + j,k − 1 2 ( κ 2 λ 1 s 2 , Γ − ¯ c 1 , Γ + − κ 1 λ 2 s 1 , Γ − ¯ c 2 , Γ + )Λ − j,k , 17 where O ( f 0 ) denotes terms pro p ortional to f 0 . It then follow s X Z = X 0 Z 0   1 + 2 X j,k =1  X j,k X 0 − Z j,k Z 0  f j,k   + O ( f 0 ) , where X = { A, B , C , D } . Substituting (37 ), w e obtain F (1 , 3) = A 0 Z 0 F (0 , 1) − B 0 Z 0 F (0 , 2) + ω (1) 1 F (0 , 1) f 2 , 1 + ω (1) 2 F (0 , 1) f 2 , 2 , (39) F (2 , 3) = C 0 Z 0 F (0 , 1) − D 0 Z 0 F (0 , 2) + ω (2) 1 F (0 , 1) f 2 , 1 + ω (2) 2 F (0 , 1) f 2 , 2 , (40) where ω (1) 1 = A 0 Z 0  A 2 , 1 A 0 − Z 2 , 1 Z 0  + B 0 Z 0  B 1 , 1 B 0 − Z 1 , 1 Z 0  , ω (1) 2 = A 0 Z 0  A 2 , 2 A 0 − Z 2 , 2 Z 0  + B 0 Z 0  B 1 , 2 B 0 − Z 1 , 2 Z 0  , ω (2) 1 = C 0 Z 0  C 2 , 1 C 0 − Z 2 , 1 Z 0  + D 0 Z 0  D 1 , 1 D 0 − Z 1 , 1 Z 0  , ω (2) 2 = C 0 Z 0  C 2 , 2 C 0 − Z 2 , 2 Z 0  + D 0 Z 0  D 1 , 2 D 0 − Z 1 , 2 Z 0  . F rom eqns. (28), w e get D + ( φ + 3 − φ + 0 ) = 8 κ 1 F (0 , 1) c − 0 , 1 + 8 κ 2 F (1 , 3) c − 1 , 3 , D + ( φ + 1 − φ + 2 ) = 8 κ 1 F (0 , 1) c − 0 , 1 − 8 κ 2 F (0 , 2) c − 0 , 2 . In tro ducing solution (32) in t he first eqn. ab ov e, w e find D + ( φ + 3 − φ + 0 ) = D + (Γ + + ∆ + ) = = ∂ x Γ + D + ( φ + 1 − φ + 2 ) + D + ∆ + = = 8 κ 1 F (0 , 1) c − 0 , 1 + 8 κ 2 F (1 , 3) c − 1 , 3 . Using eqn . (39) in the ab o ve expression, and taking in to a ccoun t that F (0 , 1) , F (0 , 2) , F (0 , 1) f 2 , 1 and F (0 , 1) f 2 , 2 are indep enden t, we ar riv e at the follow ing conditions, c − 0 , 1 κ 1 ( ∂ x Γ + − 1) − c 1 , Γ − κ 2 A 0 Z 0 + g s − 0 , 2 4 κ 2 Λ + 1 , 2 + g s − 0 , 1 4 κ 1 Λ + 1 , 1 = 0 , c − 0 , 2 κ 2 ∂ x Γ + − c 1 , Γ − κ 2 B 0 Z 0 − g s − 0 , 1 4 κ 1 Λ + 2 , 1 − g s − 0 , 2 4 κ 2 Λ + 2 , 2 = 0 , (41) c − 0 , 2 κ 2 ∂ x Λ + 1 , 1 + c − 0 , 1 κ 1 ∂ x Λ + 2 , 1 + g s − 0 , 2 4 κ 2 Λ + 0 − c 1 , Γ − κ 2 ω (1) 1 + s 1 , Γ − 2 κ 2  A 0 Z 0 Λ − 2 , 1 + B 0 Z 0 Λ − 1 , 1  = 0 , c − 0 , 2 κ 2 ∂ x Λ + 1 , 2 + c − 0 , 1 κ 1 ∂ x Λ + 2 , 2 − g s − 0 , 1 4 κ 1 Λ + 0 − c 1 , Γ − κ 2 ω (1) 2 + s 1 , Γ − 2 κ 2  A 0 Z 0 Λ − 2 , 2 + B 0 Z 0 Λ − 1 , 2  = 0 . 18 Moreo v er, the chiralit y condition o n (32) give s ¯ D − ( φ + 3 − φ + 0 ) = ¯ D − (Γ + + ∆ + ) = 0 , from where w e obtain the following eqns. ¯ c + 0 , 1 λ 1 ∂ y Γ + + g ¯ s + 0 , 1 4 λ 1 Λ + 1 , 1 + g ¯ s + 0 , 2 4 λ 2 Λ + 1 , 2 = 0 , ¯ c + 0 , 2 λ 2 ∂ y Γ + − g ¯ s + 0 , 1 4 λ 1 Λ + 2 , 1 − g ¯ s + 0 , 2 4 λ 2 Λ + 2 , 2 = 0 , ¯ c + 0 , 2 λ 2 ∂ y Λ + 1 , 1 + ¯ c + 0 , 1 λ 1 ∂ y Λ + 2 , 1 + g ¯ s + 0 , 2 4 λ 2 Λ + 0 = 0 , ¯ c + 0 , 2 λ 2 ∂ y Λ + 1 , 2 + ¯ c + 0 , 1 λ 1 ∂ y Λ + 2 , 2 − g ¯ s + 0 , 1 4 λ 1 Λ + 0 = 0 . (42) The tw o sets of eqns. namely , (41) and (42) giv e the follo wing solutions, Λ + 1 , 1 = Λ + 2 , 2 = − 8 µ − g η + η − cos  x 2  sin  y 2  , Λ + 1 , 2 = 8 µ − g η + η − λ 2 λ 1 ! sin  y 2  , Λ + 2 , 1 = 8 µ − g η + η − λ 1 λ 2 ! sin  y 2  , Λ + 0 = − 32 µ − ( g η + η − ) 2 sin  x 2   cos  y 2  ( a + cos x − cos y ) − 2 µ + cos  x 2  , where µ ± = γ 1 γ 2 ± γ 2 γ 1 , a = 1 2 γ 2 1 γ 2 2 + γ 2 2 γ 2 1 ! + 3 , η ± = µ + − 2 cos  x ± y 2  . (43) In order to determine the co efficien ts Λ − w e mak e use of ¯ D − ( φ − 3 − φ − 0 ) = 8 λ 1 F (0 , 1) ¯ c + 0 , 1 − 8 λ 2 F (1 , 3) ¯ c + 1 , 3 , ¯ D − ( φ − 1 − φ − 2 ) = − 8 λ 1 F (0 , 1) ¯ c + 0 , 1 + 8 λ 2 F (0 , 2) ¯ c + 0 , 2 , whic h a re obtained fro m (30 ). In tro ducing (32) in the first o f these eqns. w e find ¯ D − ( φ − 3 − φ − 0 ) = ¯ D − (Γ − + ∆ − ) = = ∂ y Γ − ¯ D − ( φ − 1 − φ − 2 ) + ¯ D − ∆ − = = 8 λ 1 F (0 , 1) ¯ c + 0 , 1 − 8 λ 2 F (1 , 3) ¯ c + 1 , 3 . 19 Using eqn. (3 9) in the ab o v e expression a nd taking into accoun t tha t F (0 , 1) , F (0 , 2) , F (0 , 1) f 2 , 1 and F (0 , 1) f 2 , 2 are indep enden t, we ar riv e at the follow ing expressions, ¯ c + 0 , 1 λ 1 ( ∂ y Γ − + 1) − ¯ c 1 , Γ + λ 2 A 0 Z 0 + g ¯ s + 0 , 1 4 λ 1 Λ − 1 , 1 + g ¯ s + 0 , 2 4 λ 2 Λ − 1 , 2 = 0 , ¯ c + 0 , 2 λ 2 ∂ y Γ − − ¯ c 1 , Γ + λ 2 B 0 Z 0 − g ¯ s + 0 , 1 4 λ 1 Λ − 2 , 1 − g ¯ s + 0 , 2 4 λ 2 Λ − 2 , 2 = 0 , (44) ¯ c + 0 , 2 λ 2 ∂ y Λ − 1 , 1 + ¯ c + 0 , 1 λ 1 ∂ y Λ − 2 , 1 + g ¯ s + 0 , 2 4 λ 2 Λ − 0 − ¯ c 1 , Γ + λ 2 ω (1) 1 − ¯ s 1 , Γ + 2 λ 2  A 0 Z 0 Λ + 2 , 1 + B 0 Z 0 Λ + 1 , 1  = 0 , ¯ c + 0 , 2 λ 2 ∂ y Λ − 1 , 2 + ¯ c + 0 , 1 λ 1 ∂ y Λ − 2 , 2 − g ¯ s + 0 , 1 4 λ 1 Λ − 0 − ¯ c 1 , Γ + λ 2 ω (1) 2 − ¯ s 1 , Γ + 2 λ 2  A 0 Z 0 Λ + 2 , 2 + B 0 Z 0 Λ + 1 , 2  = 0 . The c hiral condition D + ( φ − 3 − φ − 0 ) = D + (Γ − + ∆ − ) = 0 , leads us to c − 0 , 1 κ 1 ∂ x Γ − + g s − 0 , 1 4 κ 1 Λ − 1 , 1 + g s − 0 , 2 4 κ 2 Λ − 1 , 2 = 0 , c − 0 , 2 κ 2 ∂ x Γ − − g s − 0 , 1 4 κ 1 Λ − 2 , 1 − g s − 0 , 2 4 κ 2 Λ − 2 , 2 = 0 , c − 0 , 2 κ 2 ∂ x Λ − 1 , 1 + c − 0 , 1 κ 1 ∂ x Λ − 2 , 1 + g s − 0 , 2 4 κ 2 Λ − 0 = 0 , c − 0 , 2 κ 2 ∂ x Λ − 1 , 2 + c − 0 , 1 κ 1 ∂ x Λ − 2 , 2 − g s − 0 , 1 4 κ 1 Λ − 0 = 0 . (45) Solving (44) and (45) for Λ − , w e find Λ − 1 , 1 = Λ − 2 , 2 = 8 µ − g η + η − cos  y 2  sin  x 2  , Λ − 1 , 2 = − 8 µ − g η + η −  κ 2 κ 1  sin  x 2  , Λ − 2 , 1 = − 8 µ − g η + η −  κ 1 κ 2  sin  x 2  , Λ − 0 = − 32 µ − ( g η + η − ) 2 sin  y 2   cos  x 2  ( a − cos x + cos y ) − 2 µ + cos  y 2  , where µ ± , a and η ± are giv en in (43). 20 Bibliograph y [1] C. Rogers, in “Soliton Theory: a su r v ey of results ” , Ed. A. F o rdy , Manc hester Univ. Press . (1990) [2] H. W ahlquist and F. Estabro ok, Ph ys. Rev. 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