Classical Integrable N=1 and $N= 2$ Super Sinh-Gordon Models with Jump Defects

Classical In tegrable N = 1 and N = 2 Sup er Sinh-Gordon Mo dels with Jump Defects J.F. Gomes, L.H. Ymai and A.H. Z imerman Instituto de F ´ ısic a T e´ oric a, Universidade Estadual Paulista Ru a Pamplo n a, 145 0140 5 -900 - S˜ ao Paulo, S P, Br azil (Dated: August 12, 2021) Abstract The structure of integrable field theories in the presence of jump defects is d iscussed in terms of b oundary functions und er the Lagrangian formalism. Explicit examples of b osonic and fermionic theories are considered. In p articular, the b ound ary f unctions f or the N = 1 and N = 2 sup er sinh-Gordon mo d els are constructed and sho wn to generate th e Bac klund transformations for its soliton solutions. As a new and in teresting example, a solution with an incoming b oson and an outgoing fermion f or the N = 1 case is present ed. The r esulting inte grable mo dels are sh o wn to b e in v arian t und er su p ersymmetric transformation. P A CS num b ers: 1 I. INTR ODUCTI ON The classical L a grangian formulation of a class of relativistic integrable field theories admiting certain discon tinuities (defects) has b een studied recen tly [1] - [3]. In particular, in ref. [1] the authors hav e considered a field theory in whic h differen t soliton solutions of the sine-Gordon mo del are link ed in such a wa y that the integrabilit y is preserv ed. The in tegrability of the total system imp oses sev ere constrain ts sp ecifying the p ossible ty p es of defects. These are c haracterized by Bac klund transformatio ns whic h are kno wn to connect t w o different solito n solutions. The presence of the defect indicates the breakdo wn of space isotropy and henceforth of momen tum conserv ation. The k ey ingredien t to classify in tegrable defects is to imp ose certain first order differen tial relations b et w een the differen t solutions (Bac klund tr a nsforma- tion). This in tro duces certain b oundary functions (BF) which are sp ecific of eac h integrable mo del and leads to the conserv ation of the total momen tum. Here, w e analize the structure of the p o ssible b oundary terms for v arious cases. W e first consider, f or p edagogical purposes, the pure b osonic case studied in refs. [1] - [3] and derive , according to them, the b order functions b y imp osing conserv ation of the tota l momen tum. Next, we consider a pure fermionic theory a nd prop ose Bac klund transformat io n in terms of an auxiliary f ermionic non lo cal field. Suc h structure is then generalized to include b oth b osonic and fermionic fields. In particular w e construct b oundary functions for the N = 1 sup ersymme tric sinh-Gordo n mo del and sho w that it leads to the Ba c klund transformation prop osed in [4]. W e extend the formalism to t he N = 2 supersymmetric sinh-Gordon mo del [5]. This system w a s originally prop osed in [6] and [7] (see also [8]). Later in [9] and [1 0] a system atic algebraic aproach w as dev elop ed. I I. GENERAL F ORMALISM - BOSO NIC CASE In this section we intro duce the La g rangian approac h prop osed in [1]. Consider a system described b y L = θ ( − x ) L p =1 + θ ( x ) L p =2 + δ ( x ) L D , (1) 2 where L p ( φ p , ∂ µ φ p ) = 1 2 ( ∂ x φ p ) 2 − 1 2 ( ∂ t φ p ) 2 − V ( φ p ) describes a set of fields denoted by φ 1 for x < 0 and φ 2 for x > 0. A defect placed at x = 0, is describ ed b y L D = 1 2 ( φ 2 ∂ t φ 1 − φ 1 ∂ t φ 2 ) + B 0 (2) where B 0 is the b order function. The equations o f motion are therefore given by ∂ 2 x φ 1 − ∂ 2 t φ 1 = ∂ φ 1 V ( φ 1 ) , x < 0 ∂ 2 x φ 2 − ∂ 2 t φ 2 = ∂ φ 2 V ( φ 2 ) , x > 0 (3) and x = 0, ∂ x φ 1 − ∂ t φ 2 = − ∂ φ 1 B 0 , ∂ x φ 2 − ∂ t φ 1 = ∂ φ 2 B 0 , (4) The momen tum is P = Z 0 −∞ ∂ x φ 1 ∂ t φ 1 + Z ∞ 0 ∂ x φ 2 ∂ t φ 2 . (5) Acting with time deriv ative and inserting eqns. of motion (3) w e find dP dt = Z 0 −∞ ( 1 2 ∂ x ( ∂ t φ 1 ) 2 + 1 2 ∂ x ( ∂ x φ 1 ) 2 − ∂ x φ 1 δ V 1 δ φ 1 ) dx + Z ∞ 0 ( 1 2 ∂ x ( ∂ t φ 2 ) 2 + 1 2 ∂ x ( ∂ x φ 2 ) 2 − ∂ x φ 2 δ V 2 δ φ 2 ) dx (6) Using eqns. (4) after integration, we find dP dt = [ − ∂ B 0 ∂ φ + ˙ φ + + ∂ B 0 ∂ φ − ˙ φ − + 1 2 ( ∂ B 0 ∂ φ 1 ) 2 − 1 2 ( ∂ B 0 ∂ φ 2 ) 2 − V 1 + V 2 ] | x =0 (7) where φ ± = φ 1 ± φ 2 and ˙ φ ± = ∂ t φ ± . If the b order function fa cto r izes in to B 0 = B + 0 ( φ + ) + B − 0 ( φ − ), the mo dified momen tum P = P +  B + 0 − B − 0  | x =0 is conserv ed provided B 0 satisfies its defining condition, i.e., [ 1 2 ( ∂ B 0 ∂ φ 1 ) 2 − 1 2 ( ∂ B 0 ∂ φ 2 ) 2 − V 1 + V 2 ] | x =0 = 0 (8) Let us illustrate the ab ov e structure b y first considering the free massiv e b osonic theory for whic h V p = 1 2 m 2 φ 2 p . 1 2 ( ∂ B 0 ∂ φ 1 ) 2 − 1 2 ( ∂ B 0 ∂ φ 2 ) 2 = m 2 2 ( φ 2 1 − φ 2 2 ) = m 2 2 φ + φ − (9) 3 The solution is easely found to b e B 0 = − mβ 2 4 φ 2 − − m 4 β 2 φ 2 + (10) and β 2 denotes a free (sp ectral) par a meter. As second example, consider t he sinh-Gordon mo del for whic h V p = 4 m 2 cosh(2 φ p ). The defining eqn. (8) indicates the nat ura l decomp osition 1 2 ( ∂ B 0 ∂ φ 1 ) 2 − 1 2 ( ∂ B 0 ∂ φ 2 ) 2 = 4 m 2 (cosh(2 φ 1 ) − cosh( 2 φ 2 )) = 8 m 2 sinh( φ + ) sinh( φ − ) (11) yielding B 0 = − mβ 2 cosh( φ − ) − 4 m β 2 cosh( φ + ) (12) and hence w e rederiv e from (4) the Back lund transformation for the sinh-G ordon mo del ∂ x φ 1 − ∂ t φ 2 = mβ 2 sinh( φ − ) + 4 m β 2 sinh( φ + ) , ∂ x φ 2 − ∂ t φ 1 = mβ 2 sinh( φ − ) − 4 m β 2 sinh( φ + ) , (13) I I I. FERMIONS AND THE N = 1 SUPER SINH- GORDON MODEL Before discuss ing t he Sup er sinh-G ordon Mo del let us consider the pure fermionic proto- t yp e describ ed by the Lagrang ian densit y L p = ¯ ψ p ∂ t ¯ ψ p − ¯ ψ p ∂ x ¯ ψ p + ψ p ∂ t ψ p + ψ p ∂ x ψ p + W ( ψ p , ¯ ψ p ) (14) where, for the free fermionic theory , W ( ψ p , ¯ ψ p ) = 2 m ¯ ψ p ψ p . F or the half line, x < 0 or x > 0 the equations of motion are given by ∂ x ψ p + ∂ t ψ p = − 1 2 ∂ ψ p W p , ∂ x ¯ ψ p − ∂ t ¯ ψ p = 1 2 ∂ ¯ ψ p W p (15) according to p = 1 or 2 resp ectiv ely . Let us prop o se the f o llo wing Bac klund t ransformation ψ 1 + ψ 2 = − iβ √ mf 1 = ∂ ψ 1 B 1 , ¯ ψ 1 − ¯ ψ 2 = − 2 i √ m β f 1 = − ∂ ¯ ψ 1 B 1 , (16) 4 where f 1 satisfies ˙ f 1 = − iβ √ m 2 ( ψ 1 − ψ 2 ) + i 2 β √ m ( ¯ ψ 1 + ¯ ψ 2 ) = − 1 4 ∂ f 1 B 1 , ∂ x f 1 = iβ √ m 2 ( ψ 1 − ψ 2 ) + i 2 β √ m ( ¯ ψ 1 + ¯ ψ 2 ) (17) written in terms of a b order function B 1 = B 1 ( ¯ ψ 1 , ¯ ψ 2 , ψ 1 , ψ 2 , f 1 ), whic h no w, due to the Grassmanian c hara cter of the fermions, dep ends up on the non lo cal fermionic field f 1 . By considering the Lagra ng ian system (1) with L p giv en b y (14) and L D = − ψ 1 ψ 2 − ¯ ψ 1 ¯ ψ 2 + 2 f 1 ∂ t f 1 + B 1 ( ¯ ψ 1 , ¯ ψ 2 , ψ 1 , ψ 2 , f 1 ) (18) construct the momen tum to b e P = Z 0 −∞ ( − ¯ ψ 1 ∂ x ¯ ψ 1 − ψ 1 ∂ x ψ 1 ) dx + Z ∞ 0 ( − ¯ ψ 2 ∂ x ¯ ψ 2 − ψ 2 ∂ x ψ 2 ) dx (19) In analising its conserv atio n, w e find dP dt = [ − W 1 + W 2 − ¯ ψ 1 ∂ t ¯ ψ 1 + ¯ ψ 2 ∂ t ¯ ψ 2 − ψ 1 ∂ t ψ 1 + ψ 2 ∂ t ψ 2 ] | x =0 (20) after using equations of motion (15) to eliminate time deriv a tiv es and integrating ov er x . Using the Bac klund tra nsformation (16) and (17), eqn. (20) b ecomes dP dt = − [ W 1 − W 2 − ∂ ¯ ψ 1 B 1 ˙ ¯ 1 ψ + ∂ ψ 1 B 1 ˙ ψ 1 − ∂ ¯ ψ 2 B 1 ˙ ¯ 2 ψ + ∂ ψ 2 B 1 ˙ ψ 2 + ∂ t ( ¯ ψ 2 ¯ ψ 1 ) − ∂ t ( ψ 2 ψ 1 )] | x =0 (21) If w e assume the b order function to decomp ose as B 1 = B + 1 ( ¯ ψ + , f 1 ) + B − 1 ( ψ − , f 1 ), where ¯ ψ ± = ¯ ψ 1 ± ¯ ψ 2 , ψ ± = ψ 1 ± ψ 2 , the mo dified momentum P = P + [ ¯ ψ 2 ¯ ψ 1 − ψ 2 ψ 1 + B + 1 − B − 1 ] x =0 (22) is conserv ed provided [ W 2 − W 1 − ∂ f 1 B − 1 ˙ f 1 + ∂ f 1 B + 1 ˙ f 1 ] | x =0 = 0 (23) F or the free fermi fields system (14 )-(15) eqn. (23) b ecomes 1 2 ( ∂ f 1 B + 1 )( ∂ f 1 B − 1 ) = 2 m ( ¯ ψ 1 ψ 1 − ¯ ψ 2 ψ 2 ) (24) 5 The solution is B 1 = − 2 i β √ mf 1 ¯ ψ + + iβ √ mf 1 ψ − (25) Let us now consider the sup er sinh-Gordon mo del describ ed by L p = 1 2 ( ∂ x φ p ) 2 − 1 2 ( ∂ t φ p ) 2 + ¯ ψ p ∂ t ¯ ψ p − ¯ ψ p ∂ x ¯ ψ p + ψ p ∂ t ψ p + ψ p ∂ x ψ p + V ( φ p ) + W ( φ p , ψ p , ¯ ψ p ) (26) and L D = 1 2 ( φ 2 ∂ t φ 1 − φ 1 ∂ t φ 2 ) − ψ 1 ψ 2 − ¯ ψ 1 ¯ ψ 2 + 2 f 1 ∂ t f 1 + B 0 ( φ 1 , φ 2 ) + B 1 ( φ 1 , φ 2 , ¯ ψ 1 , ¯ ψ 2 , ψ 1 , ψ 2 , f 1 ) (27) where V p = 4 m 2 cosh(2 φ p ) and W p = 8 m ¯ ψ p ψ p cosh( φ p ). Prop ose the follow ing Ba c klund transformation [11], ∂ x φ 1 − ∂ t φ 2 = − ∂ φ 1 ( B 0 + B 1 ) , ∂ x φ 2 − ∂ t φ 1 = ∂ φ 2 ( B 0 + B 1 ) ψ 1 + ψ 2 = ∂ ψ 1 B 1 , ¯ ψ 1 − ¯ ψ 2 = − ∂ ¯ ψ 2 B 1 , ˙ f 1 = − 1 4 ∂ f 1 B 1 (28) Assuming the decomp osition B 0 = B + 0 ( φ + ) + B − 0 ( φ − ) , B 1 = B + 1 ( φ + , ¯ ψ + , f 1 ) + B − 1 ( φ − , ψ − , f 1 ) (29) w e find that the mo dified momentum P = P + [ B + 0 ( φ + ) − B − 0 ( φ − ) + B + 1 ( φ + , ¯ ψ + , f 1 ) − B − 1 ( φ − , ψ − , f 1 ) − ¯ ψ 1 ¯ ψ 2 + ψ 1 ψ 2 ] x =0 (30) is conserv ed provided the b o r der functions B 0 and B 1 satisfy W 1 − W 2 = 1 2 ( ∂ f 1 B + 1 )( ∂ f 1 B − 1 ) + 2( ∂ φ + B + 0 )( ∂ φ − B − 1 ) + 2( ∂ φ − B − 0 )( ∂ φ + B + 1 ) V 1 − V 2 = 1 2 ( ∂ B 0 ∂ φ 1 ) 2 − 1 2 ( ∂ B 0 ∂ φ 2 ) 2 (31) The solution for (31) is fo und to b e B 0 = − mβ 2 cosh( φ − ) − 4 m β 2 cosh( φ + ) , B 1 = − 4 i β √ m cosh( φ + 2 ) f 1 ¯ ψ + + 2 iβ √ m cosh( φ − 2 ) f 1 ψ − (32) 6 The b oundary functions B 0 and B 1 in (32) generate the Backlund transforma t io n whic h agrees with the one prop osed in [4] for the N = 1 sup er sinh-Gordon mo del. The equations of motion obtained from (26) and ( 2 7) are v erified to b e in v ariant under the sup ersymmetry transformation δ ¯ ψ p = ǫ∂ z φ p , δ φ p = ǫ ¯ ψ p , δ ψ p = 2 ǫm sinh φ p (33) where ∂ z = 1 / 2( ∂ x + ∂ t ). In ref. [12] t he general soliton solutions for the N = 1 sup er sinh-Gordon mo del were constructed using v ertex functions tec hniques and in [11] differen t solutions w ere analysed in the conte xt of the Ba cklund tra nsforma t ion. Besides those examples discussed in [11], a new and in teresting case containing an inc oming b oson an d an outc omi n g fermion is giv en when φ 1 6 = 0 , ψ 1 = ¯ ψ 1 = 0 and φ 2 = 0 , ψ 2 , ¯ ψ 2 6 = 0. Under suc h conditions the Bac klund transformation (28) reads, ∂ x φ 1 = − 2( σ + 1 σ ) sinh φ 1 , ∂ t φ 1 = − 2( σ − 1 σ ) sinh φ 1 , ψ 2 = 2 r 2 σ cosh  φ 1 2  f 1 , ¯ ψ 2 = 2 √ 2 σ cosh  φ 1 2  f 1 , ∂ t f 1 = 2( σ − 1 σ ) cosh 2  φ 1 2  f 1 , ∂ x f 1 = 2( σ + 1 σ ) cosh 2  φ 1 2  f 1 (34) where w e to ok m = 1 and σ = − 2 β 2 . As solution of eqns. (34 ) w e find φ 1 = l n 1 + b 1 2 ρ 1 − b 1 2 ρ ! , f 1 = ǫρ − 1 r 1 − b 2 1 4 ρ 2 (35) where ρ = exp  − 2 σ ( x + t ) − 2 σ ( x − t )  , ǫ is a grassmaniann constant and ψ 2 and ¯ ψ 2 are obtained from the second eqn. ( 3 4) aft er inserting the ab o ve φ 1 and f 1 . The integrabilit y of the mo del is ve rified b y construction of the Lax pair represen tation of the equations of motion. This is achie ve d b y splitting the space in to tw o ov erlapping r egio ns, namely , x ≤ b and x ≥ a with a < b and defining a corresp onding Lax pair within eac h of t hem. The in tegrabilit y is ensured b y the exis tence o f a gauge transfor ma t ion relating the tw o sets of Lax pairs within the o v erlapping region. In ref. [11] we ha v e explicitly constructed suc h gauge transformation fo r the N = 1 sup er sinh-G ordon mo del in terms o f the S L (2 , 1) affine Lie algebra. 7 IV. N = 2 SUPER SINH- GORDON MODEL The Lag rangian densit y for the N = 2 Sup er sinh-Gordon Mo del is given by ( see for instance [8]) L p = 1 2 ( ∂ x φ p ) 2 − 1 2 ( ∂ t φ p ) 2 + 2 ¯ ψ p ∂ t ¯ ψ p + 2 ¯ ψ p ∂ x ¯ ψ p + 2 ψ p ∂ t ψ p − 2 ψ p ∂ x ψ p − 1 2 ( ∂ x ϕ p ) 2 + 1 2 ( ∂ t ϕ p ) 2 − 2 ¯ χ p ∂ t ¯ χ p − 2 ¯ χ p ∂ x ¯ χ p − 2 χ p ∂ t χ p + 2 χ p ∂ x χ p − 16( ψ p ¯ ψ p + χ p ¯ χ p ) cosh ϕ p cosh φ p − 4 cosh (2 ϕ p ) + 16( ψ p ¯ χ p + χ p ¯ ψ p ) sinh ϕ p sinh φ p + 4 cosh(2 φ p ) (36) whose equations of motion are inv arian t under the sup ersymmetry transformations δ ( φ p ± ϕ p ) = 2( ψ p ∓ χ p ) ǫ ± , δ ( ψ p ± χ p ) = − ∂ z ( φ p ∓ ϕ p ) ǫ ± (37) and δ ( ¯ ψ p ± ¯ χ p ) = 2 sinh( φ p ± ϕ p ) ǫ ∓ (38) Inspired from the N = 1 case (see eqn. (27)) w e prop ose the following La grangian description for the defect L D = 1 2 ( φ 2 ∂ t φ 1 − φ 1 ∂ t φ 2 ) − 2 ψ 1 ψ 2 − 2 ¯ ψ 1 ¯ ψ 2 + f 1 ∂ t f 2 − 1 2 ( ϕ 2 ∂ t ϕ 1 − ϕ 1 ∂ t ϕ 2 ) + 2 χ 1 χ 2 + 2 ¯ χ 1 ¯ χ 2 + f 2 ∂ t f 1 + B 0 ( φ p , ϕ p ) + B 1 ( φ p , ϕ p , ψ p , χ p , ¯ ψ p , ¯ χ p , f 1 , f 2 ) (39) 8 leading to the Bac klund transformation ∂ x φ 1 − ∂ t φ 2 = − ∂ φ 1 B , ∂ x φ 2 − ∂ t φ 1 = ∂ φ 2 B , ∂ x ϕ 1 − ∂ t ϕ 2 = ∂ ϕ 1 B , ∂ x ϕ 2 − ∂ t ϕ 1 = − ∂ ϕ 2 B , ψ 1 − ψ 2 = − 1 2 ∂ ψ 1 B = − 1 2 ∂ ψ 2 B χ 1 − χ 2 = 1 2 ∂ χ 1 B = 1 2 ∂ χ 2 B ¯ ψ 1 + ¯ ψ 2 = 1 2 ∂ ¯ ψ 1 B = − 1 2 ∂ ¯ ψ 2 B , ¯ χ 1 + ¯ χ 2 = − 1 2 ∂ ¯ χ 1 B = 1 2 ∂ ¯ χ 2 B , ˙ f 1 = − 1 2 ∂ f 2 B , ˙ f 2 = − 1 2 ∂ f 1 B (40) where B = B 0 + B 1 . The canonical momentum P = Z 0 −∞  ∂ x φ 1 ∂ t φ 1 − 2 ¯ ψ 1 ∂ x ¯ ψ 1 − 2 ψ 1 ∂ x ψ 1 − ∂ x ϕ 1 ∂ t ϕ 1 + 2 ¯ χ 1 ∂ x ¯ χ 1 + 2 χ 1 ∂ x χ 1 ) dx + Z ∞ 0  ∂ x φ 2 ∂ t φ 2 − 2 ¯ ψ 2 ∂ x ¯ ψ 2 − 2 ψ 2 ∂ x ψ 2 − ∂ x ϕ 2 ∂ t ϕ 2 + 2 ¯ χ 2 ∂ x ¯ χ 2 + 2 χ 2 ∂ x χ 2 ) dx (41) is no longer conserv ed. Prop osing the mo dified momen tum to b e P = P + [ B (+) 0 − B ( − ) 0 + B (+) 1 − B ( − ) 1 + 2 ¯ ψ 1 ¯ ψ 2 − 2 ψ 1 ψ 2 − 2 ¯ χ 1 ¯ χ 2 + 2 χ 1 χ 2 ] x =0 whic h is conserv ed pro vided the b order function satisfies ∂ φ + B (+) 0 ∂ φ − B ( − ) 0 − ∂ ϕ + B (+) 0 ∂ ϕ − B ( − ) 0 = 4 sinh φ + sinh φ − − 4 sinh ϕ + sinh ϕ − , (42) 9 and ∂ φ + B (+) 0 ∂ φ − B ( − ) 1 + ∂ φ − B ( − ) 0 ∂ φ + B (+) 1 − ∂ ϕ + B (+) 0 ∂ ϕ − B ( − ) 1 − ∂ ϕ − B ( − ) 0 ∂ ϕ + B (+) 1 + ∂ φ + B (+) 1 ∂ φ − B ( − ) 1 − ∂ ϕ + B (+) 1 ∂ ϕ − B ( − ) 1 − 1 2 ( ∂ f 1 B ( − ) 1 ∂ f 2 B (+) 1 + ∂ f 2 B ( − ) 1 ∂ f 1 B (+) 1 ) = − 2( ψ + ¯ ψ + + ψ − ¯ ψ − + χ + ¯ χ + + χ − ¯ χ − )Λ − − 2( ψ + ¯ ψ − + ψ − ¯ ψ + + χ + ¯ χ − + χ − ¯ χ + )Λ + + 2( ψ + ¯ χ + + ψ − ¯ χ − + χ + ¯ ψ + + χ − ¯ ψ − )∆ − + 2( ψ + ¯ χ − + ψ − ¯ χ + + χ + ¯ ψ − + χ − ¯ ψ + )∆ + (43) where χ ± = χ 1 ± χ 2 , ¯ χ ± = ¯ χ 1 ± ¯ χ 2 , ϕ ± = ϕ 1 ± ϕ 2 , · · · etc and Λ ± = cosh( φ + + φ − 2 ) cosh( ϕ + + ϕ − 2 ) ± cosh( φ + − φ − 2 ) cosh( ϕ + − ϕ − 2 ) , ∆ ± = sinh( φ + + φ − 2 ) sinh( ϕ + + ϕ − 2 ) ± sinh( φ + − φ − 2 ) sinh( ϕ + − ϕ − 2 ) , The solution of (42) and (43) is B (+) 0 = B (+) 0 ( φ + , ϕ + ) = 2 α 3 α 2 (cosh φ + − cosh ϕ + ) , (44) B ( − ) 0 = B ( − ) 0 ( φ − , ϕ − ) = 2 α 2 α 3 (cosh φ − − cosh ϕ − ) , (45) and B (+) 1 = B (+) 1 ( φ + , ϕ + , ψ + , χ + , f 1 , f 2 ) = i √ 2 f 1  − α 3 ( ψ + − χ + ) cosh 1 2 ( φ + + ϕ + )  + i √ 2 f 2  8 α 2 ( ψ + + χ + ) cosh 1 2 ( φ + − ϕ + )  , (46) B ( − ) 1 = B ( − ) 1 ( φ − , ϕ − , ¯ ψ − , ¯ χ − , f 1 , f 2 ) = i √ 2 f 1  α 2 ( ¯ ψ − + ¯ χ − ) cosh 1 2 ( φ − − ϕ − )  + i √ 2 f 2  − 8 α 3 ( ¯ ψ − − ¯ χ − ) cosh 1 2 ( φ − + ϕ − )  (47) where α 2 and α 3 are arbitrary constants. 10 These results are consisten t with the Ba cklund transformation for the N = 2 sup er sinh- Gordon obtained fr o m a superfield formalism [5]. W e should men tion that the Back lund transformation (40) with B 0 and B 1 giv en b y (44) - (47) are in v arian t under the sup ersym- metry transformation (37)-(3 8). In [5 ] some examples of Back lund solutions are discussed also. Ac knowle dgemen ts W e are gr a teful to CNPq a nd F APESP for financial supp ort. [1] P . Bo w co c k, E. Corr igan and C . Zamb on, Int. J . Mo d. P h ys. A19 (2004) 82, hep-th/030 5022 ; [2] P . Bo wcoc k, E. Corrigan and C. Z am b on, J. Hig h Ener gy Phys. JHEP 0401 (2004 )056, hep-th/040102 0 ; [3] E. Corr igan and C. Zam b on, J. Physics A37 (2004) L471, h ep-th/0407 199 [4] M. Chaichian and P . Kulish, Ph ys. Lett. 78B (1978) 413 [5] J.F. Gomes, L. H. Ym ai an d A.H. Zimerman , in pr eparation [6] T. Inami and Kanno, Nucl. Phys. B359 (1990 ) 201 [7] Kobay ashi and Uematsu, Phys. Lett. B264 (1991) 107 [8] R. Nep omec hie, Phys. Lett. 509B (2001) 183 [9] H. Arat yn , J.F. Gomes and A.H. Zimerman, Nucl. Phys. B67 6 (2004) 537 [10] J.F. Gomes, L . H. Ym ai and A.H. Zimerman,“ N = 2 and N = 4 Sup ers y m metric mKd V and sinh-Gordon Hierarc hies”, h ep-th/0409 171 [11] J.F. Gomes, L. H. Ymai and A.H. Zimerman, J. Ph ysics A39 (2006) 7471, hep-th/0601014 [12] J.F. Gomes, L. H. Ymai and A.H. Zimerman, Phys. Lett. 359A (200 6) 630, hep-th/0607107 11