Permutability of Backlund Transformation for N=1 Supersymmetric Sinh-Gordon
aa r X i v : . [ m a t h - ph ] F e b Permutability of Backlund Transformation for N = 1 Supersymmetric Sinh-Gordon J.F. Gomes , L.H. Ymai and A.H. ZimermanInstituto de F´ısica Te´orica-UNESPRua Pamplona 145fax (55)11 3177908001405-900 S˜ao Paulo, Brazil Abstract
The permutability of two Backlund transformations is employed to construct a nonlinear superposition formula to generate a class of solutions for the N = 1 super sinh-Gordon model. Backlund transformations (BT) relating two different soliton solutions are known to becharacteristic of certain class of nonlinear equations. A remarkable consequence is that froma particular soliton solution, a second solution can be generated by Backlund transformation.This second solution, in turn, generates a third one and such structure allows to constructconditions for the permutability of two sequences of BT.The study of superposition principle for soliton solutions of the sine-Gordon equation wasemployed to show that the order of two BT is, in fact, irrelevant [1]. Such condition becameknown as the permutability theorem as was applied to the KdV and for the N = 1 super KdVequations in [2] and in [3] respectively.Soliton solutions for the N = 1 super sine-Gordon were obtained in [4] from the superHirota’s formalism and in [5] using dressing transformation and vertex operators. Backlundsolutions for super mKdV were considered in [6]. In this paper we use the superfield approach key words:Backlund Transformation, Supersymmetric sinh-Gordon, non linear superposition formulaPacs 02.30.Ik corresponding author [email protected] N = 1 super sinh-Gordon model assuming that two sucessive BT commute.The model is described by the following equation of motion written within the superfieldformalism [7] D x D t Φ = 2 i sinh Φ , (1)where the bosonic superfield Φ is given in components byΦ = φ + θ ¯ ψ + iθ ψ − θ θ i sinh φ, (2)where θ and θ are Grassmann variables (i.e. θ = θ = 0 and θ θ + θ θ = 0 ) Thesuperderivatives D x = ∂ θ + θ ∂ x , D t = ∂ θ + θ ∂ t , (3)satisfy D x = ∂ x , D t = ∂ t , D x D t = − D t D x . (4)The Backlund transformation for eqn. (1) is given by [7] D x (Φ − Φ ) = − iβ f , cosh (cid:18) Φ + Φ (cid:19) , (5) D t (Φ + Φ ) = 2 β f , cosh (cid:18) Φ − Φ (cid:19) , (6)where β is an arbitrary parameter (spectral parameter) and the auxiliary fermionic superfield f , (i.e. f , = 0) f , = f (0 , + θ b (0 , + θ b (0 , + θ θ f (0 , , (7)2atisfy D x f , = 2 iβ sinh (cid:18) Φ + Φ (cid:19) , D t f , = β sinh (cid:18) Φ − Φ (cid:19) . (8)Consider now two successive Backlund transformations. The first one involving superfields Φ and Φ and the parameter β whilst the second involves Φ and Φ with β . The Permutabilitytheorem states that the order in which such Backlund transformations are employed is irrelevant,i.e. we might as well consider the first involving Φ and Φ with β followed by a second,involving Φ and Φ with β . Similar to (5) we have, D x (Φ − Φ ) = − iβ f , cosh (cid:18) Φ + Φ (cid:19) , (9) D x (Φ − Φ ) = − iβ f , cosh (cid:18) Φ + Φ (cid:19) , (10) D x (Φ − Φ ) = − iβ f , cosh (cid:18) Φ + Φ (cid:19) , (11) D x (Φ − Φ ) = − iβ f , cosh (cid:18) Φ + Φ (cid:19) . (12)and from (8), D x f , = 2 iβ sinh (cid:18) Φ + Φ (cid:19) , (13) D x f , = 2 iβ sinh (cid:18) Φ + Φ (cid:19) , (14) D x f , = 2 iβ sinh (cid:18) Φ + Φ (cid:19) , (15) D x f , = 2 iβ sinh (cid:18) Φ + Φ (cid:19) . (16)The equality of the sum of equations (9) and (10) with the sum of (11) and (12) yields thefollowing relation 1 β f , cosh (cid:18) Φ + Φ (cid:19) + 1 β f , cosh (cid:18) Φ + Φ (cid:19) = 1 β f , cosh (cid:18) Φ + Φ (cid:19) + 1 β f , cosh (cid:18) Φ + Φ (cid:19) . (17)3nalogously from (6) we find D t (Φ + Φ ) = 2 β f , cosh (cid:18) Φ − Φ (cid:19) , (18) D t (Φ + Φ ) = 2 β f , cosh (cid:18) Φ − Φ (cid:19) , (19) D t (Φ + Φ ) = 2 β f , cosh (cid:18) Φ − Φ (cid:19) , (20) D t (Φ + Φ ) = 2 β f , cosh (cid:18) Φ − Φ (cid:19) . (21)Equating now the difference of the first two, (18) and (19) and the last two equations, (20) and(21) we get β f , cosh (cid:18) Φ − Φ (cid:19) − β f , cosh (cid:18) Φ − Φ (cid:19) = β f , cosh (cid:18) Φ − Φ (cid:19) − β f , cosh (cid:18) Φ − Φ (cid:19) . (22)Solving (17) and (22), for f , and f , , we get f , = Λ (1)1 , f , + Λ (2)1 , f , , (23) f , = Λ (1)2 , f , + Λ (2)2 , f , , (24)where the coefficients Λ are given asΛ (1)1 , = − β β h cosh (cid:16) Φ +Φ (cid:17) cosh (cid:16) Φ − Φ (cid:17) + cosh (cid:16) Φ − Φ (cid:17) cosh (cid:16) Φ +Φ (cid:17)ih cosh (cid:16) Φ − Φ (cid:17) cosh (cid:16) Φ +Φ (cid:17) β − cosh (cid:16) Φ − Φ (cid:17) cosh (cid:16) Φ +Φ (cid:17) β i , Λ (2)1 , = h cosh (cid:16) Φ +Φ (cid:17) cosh (cid:16) Φ − Φ (cid:17) β + cosh (cid:16) Φ − Φ (cid:17) cosh (cid:16) Φ +Φ (cid:17) β ih cosh (cid:16) Φ − Φ (cid:17) cosh (cid:16) Φ +Φ (cid:17) β − cosh (cid:16) Φ − Φ (cid:17) cosh (cid:16) Φ +Φ (cid:17) β i , Λ (1)2 , = − h cosh (cid:16) Φ − Φ (cid:17) cosh (cid:16) Φ +Φ (cid:17) β + cosh (cid:16) Φ +Φ (cid:17) cosh (cid:16) Φ − Φ (cid:17) β ih cosh (cid:16) Φ − Φ (cid:17) cosh (cid:16) Φ +Φ (cid:17) β − cosh (cid:16) Φ − Φ (cid:17) cosh (cid:16) Φ +Φ (cid:17) β i , Λ (2)2 , = β β h cosh (cid:16) Φ +Φ (cid:17) cosh (cid:16) Φ − Φ (cid:17) + cosh (cid:16) Φ − Φ (cid:17) cosh (cid:16) Φ +Φ (cid:17)ih cosh (cid:16) Φ − Φ (cid:17) cosh (cid:16) Φ +Φ (cid:17) β − cosh (cid:16) Φ − Φ (cid:17) cosh (cid:16) Φ +Φ (cid:17) β i , (25)4cting with D x in eqn. (9)-(12) and using (13)- (16), we find ∂ x (Φ − Φ ) = 4 β sinh(Φ + Φ ) + 4 iβ f , D x (cid:20) cosh (cid:18) Φ + Φ (cid:19)(cid:21) . (26) ∂ x (Φ − Φ ) = 4 β sinh(Φ + Φ ) + 4 iβ f , D x (cid:20) cosh (cid:18) Φ + Φ (cid:19)(cid:21) . (27) ∂ x (Φ − Φ ) = 4 β sinh(Φ + Φ ) + 4 iβ f , D x (cid:20) cosh (cid:18) Φ + Φ (cid:19)(cid:21) . (28) ∂ x (Φ − Φ ) = 4 β sinh(Φ + Φ ) + 4 iβ f , D x (cid:20) cosh (cid:18) Φ + Φ (cid:19)(cid:21) . (29)Notice that equations (26)-(29) correspond formally to the pure bosonic case when the termsproportional to the fermionic superfields f i,j are neglected. Equating the R.H.S. of the sum ofeqns. (26) with (27) and (28) with (29) we find1 β (sinh(Φ + Φ ) − sinh(Φ + Φ )) + 1 β (sinh(Φ + Φ ) − sinh(Φ + Φ ))= − iβ f , D x (cid:20) cosh (cid:18) Φ + Φ (cid:19)(cid:21) − iβ f , D x (cid:20) cosh (cid:18) Φ + Φ (cid:19)(cid:21) + iβ f , D x (cid:20) cosh (cid:18) Φ + Φ (cid:19)(cid:21) + iβ f , D x (cid:20) cosh (cid:18) Φ + Φ (cid:19)(cid:21) . (30)Factorizing the L.H.S. of (30)1 β (sinh(Φ + Φ ) − sinh(Φ + Φ )) + 1 β (sinh(Φ + Φ ) − sinh(Φ + Φ ))= 2 cosh( Φ + Φ + Φ + Φ β sinh( Φ + Φ − Φ − Φ β sinh( − Φ + Φ − Φ + Φ ! = 2 cosh( Φ + Φ + Φ + Φ sinh( Φ − Φ − Φ β − β ) β β + sinh( Φ − Φ − Φ β + β β β ) ! . (31)The vanishing of this expression leads toΦ − Φ = 2 arctanh (cid:18) δ tanh( Φ − Φ (cid:19) ≡ Γ(Φ − Φ ) , δ = − β + β β − β ! . (32)5or the more general case taking into account the fermionic superfields f i,j we propose thefollowing ansatz, Φ = Φ + Γ(Φ − Φ ) + ∆ , (33)where ∆ is a bosonic superfield proportional to the product f , f , , i.e.,∆ = λf , f , , λ = λ (Φ − Φ ) . (34)Due to the fact that ∆ = 0, eqns. (25) take the general formΛ (1)1 , = − a + c f , f , , Λ (2)1 , = − b + c f , f , , Λ (1)2 , = b + c f , f , , Λ (2)2 , = a + c f , f , , (35)where c i = c i (Φ , Φ , Φ ) , i = 1 , · · · a = δ r − δ tanh (cid:16) Φ − Φ (cid:17) , b = δ sech (cid:16) Φ − Φ (cid:17)r − δ tanh (cid:16) Φ − Φ (cid:17) (36)where δ = β β ( β − β ) . Inserting (35) in (23) and (24) we find, since f = f = 0, f , = − af , − bf , ,f , = bf , + af , . (37)From the fact that δ − δ = 1, it follows that f , f , = f , f , . D x (Φ − Φ ) = 4 iβ f , cosh (cid:18) Φ + Φ (cid:19) + 4 iβ f , cosh (cid:18) Φ + Φ (cid:19) . (38)Substituting (33) and (37) in (38) we find f , Σ + f , Σ + ( D x λ ) f , f , = 0 , (39)where Σ = ∂ ξ Γ | ξ =(Φ1 − Φ2) iβ cosh (cid:18) Φ + Φ (cid:19) − λ iβ sinh (cid:18) Φ + Φ (cid:19) − iβ cosh (cid:18) Φ + Φ (cid:19) − Λ (1)1 , iβ cosh (cid:18) Φ + Φ + Γ2 (cid:19) , Σ = − ∂ ξ Γ | ξ =(Φ1 − Φ2) iβ cosh (cid:18) Φ + Φ (cid:19) + λ iβ sinh (cid:18) Φ + Φ (cid:19) − Λ (2)1 , iβ cosh (cid:18) Φ + Φ + Γ2 (cid:19) . The last term in (39) vanishes since D x λ = ∂ ξ λ | ξ =(Φ1 − Φ2) D x (Φ − Φ ) , and D x (Φ − Φ ), from (9) and (11), is proportional to f , and f , . Since f , and f , areindependent, (39) yields a pair of algebraic equations for λ , i.e. Σ = Σ = 0 which are satisfiedby λ = − (cid:16) Φ − Φ (cid:17) β β ( β + β ) β + β − − Φ ) β β . (40)and therefore Φ = Φ + 2 Arctanh " β + β β − β ! tanh (cid:18) Φ − Φ (cid:19) e Ωf , f , , (41)where Ω = δ sech (cid:18) Φ − Φ (cid:19) .
7n order to write eqn. (41) in components, we need to specify the superfields f , , f , in termsof the components of Φ , Φ , Φ . These are given by eqns. (5.91), (5.94), (5.96) and (5.98) inthe the appendix of ref. [8]. Introducing σ k = − β k ( k = 1 , φ = φ + 2 Arctanh " δ tanh φ − φ ! − ∆ √ σ σ ¯ ψ ( ¯ ψ − ¯ ψ ) + ¯ ψ ¯ ψ cosh (cid:16) φ + φ (cid:17) cosh (cid:16) φ + φ (cid:17) , (42)¯ ψ = ¯ ψ + ∆ ( ¯ ψ − ¯ ψ ) − ∆ s σ σ sinh (cid:16) φ + φ (cid:17) cosh (cid:16) φ + φ (cid:17) ( ¯ ψ − ¯ ψ ) − s σ σ sinh (cid:16) φ + φ (cid:17) cosh (cid:16) φ + φ (cid:17) ( ¯ ψ − ¯ ψ ) , (43) ψ = ψ + ∆ ( ψ − ψ ) − ∆ s σ σ sinh (cid:16) φ − φ (cid:17) cosh (cid:16) φ − φ (cid:17) ( ψ + ψ ) − s σ σ sinh (cid:16) φ − φ (cid:17) cosh (cid:16) φ − φ (cid:17) ( ψ + ψ ) , (44)where ∆ = 2sinh ( φ − φ ) δ tanh (cid:16) φ − φ (cid:17) − δ tanh (cid:16) φ − φ (cid:17) , ∆ = A sinh (cid:16) φ − φ (cid:17) B − sinh (cid:16) φ − φ (cid:17) ,δ = σ + σ σ − σ , A = σ + σ √ σ σ , B = ( σ − σ ) σ σ . (45) One soliton Solution
The Backlund equations in components for Φ = 0 take the form )from (7): ∂ x φ = 2 σ sinh φ , ∂ t φ = 2 σ sinh φ , (46)¯ ψ = 2 √ σ cosh ( φ / f (0 , , ψ = 2 s σ cosh ( φ / f (0 , (47) ∂ x f (0 , = r σ φ /
2) ¯ ψ , ∂ t f (0 , = 1 √ σ cosh ( φ / ψ (48)8y direct integration we obtain φ = ln (cid:18) E − E (cid:19) , E = b exp (cid:16) σ x + 2 σ − t (cid:17) , (49)¯ ψ = ǫ a b E (cid:18)
11 + E + 11 − E (cid:19) , ψ = ¯ ψ σ , (50)where a and b are arbitrary constants and ǫ is a fermionic parameter. Two soliton Solution
Choosing Φ = 0 and Φ , Φ as one soliton solutions with components φ k = ln (cid:18) E k − E k (cid:19) , E k = b k exp (cid:16) σ k x + 2 σ − k t (cid:17) , ¯ ψ k = ǫ k a k b k E k (cid:18)
11 + E k + 11 − E k (cid:19) ,ψ k = ¯ ψ k σ k , k = 1 , a k , b k are arbitrary constants and ǫ k fermiˆonic parameters we find from (41), φ = 2 Arctanh " δ tanh φ − φ ! − ∆ √ σ σ ¯ ψ ¯ ψ cosh (cid:16) φ (cid:17) cosh (cid:16) φ (cid:17) , (51)¯ ψ = ∆ + ∆ s σ σ sinh (cid:16) φ (cid:17) cosh (cid:16) φ (cid:17) ¯ ψ − ∆ + ∆ s σ σ sinh (cid:16) φ (cid:17) cosh (cid:16) φ (cid:17) ¯ ψ , (52) ψ = ∆ − ∆ s σ σ sinh (cid:16) φ (cid:17) cosh (cid:16) φ (cid:17) ψ − ∆ − ∆ s σ σ sinh (cid:16) φ (cid:17) cosh (cid:16) φ (cid:17) ψ , (53)By rescaling of parameters σ k → γ k , ǫ k → c k k = 1 , b → b γ − γ γ + γ ! , b → − b γ − γ γ + γ ! ,a → − γ γ − γ γ + γ ! , a → γ γ − γ γ + γ ! . we verify that solution (51) and (52) coincide precisely with solution (3.28)-(3.29) of ref. [5](after choosing γ = − γ and γ = − γ ). 9 cknowledgements LHY acknowledges support from Fapesp, JFG and AHZ thank CNPq for partial support.
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