Recursion Operator and Bäcklund Transformation for Super mKdV Hierarchy
A.R. Aguirre, J.F. Gomes, A.L. Retore, N.I. Spano, A.H. Zimerman
aa r X i v : . [ n li n . S I] A p r Recursion Operator and B¨acklundTransformation for Super mKdVHierarchy
A.R. Aguirre, J.F. Gomes, A.L. Retore, N.I. Spano, and A.H. Zimerman
Abstract
In this paper we consider the N = 1 supersymmetric mKdV hi-erarchy composed of positive odd flows embedded within an affine ˆ sl (2 , The algebraic formulation for integrable hierarchies presents itself as a pow-erful framework in order to discuss its integrable properties, symmetries andsoliton solutions. In particular the supersymmetric mKdV hierarchy consistsof a set of time evolution (flows) equations obtained from a zero curvaturerepresentation involving a two dimensional gauge potential lying within anaffine ˆ sl (2 ,
1) Kac-Moody algebra and a common infinite set of conservationlaws [1, 2, 3].Moreover, B¨acklund transformation can be employed to construct an infi-nite sequence of solitons solutions by purely superposition principle and also
A.R. AguirreInstituto de F´ısica e Qu´ımica, Universidade Federal de Itajub´a - IFQ/UNIFEI, Av.BPS 1303, 37500-903, Itajub´a, MG, Brasil. e-mail: [email protected]. RetorePhysics Department of the University of Miami, Coral Gables, FL 33124 USA. e-mail:[email protected]. Spano, J.F. Gomes, and A.H. ZimermanInstituto de F´ısica Te´orica - IFT/UNESP, Rua Dr. Bento Teobaldo Ferraz 271, BlocoII, 01140-070, S˜ao Paulo, Brasil. e-mail: [email protected], [email protected],[email protected] 1 A.R. Aguirre, J.F. Gomes, A.L. Retore, N.I. Spano, and A.H. Zimerman to link nonlinear equations to canonical forms as discussed for many examplesin [4]. For the supersymmetric mKdV hierarchy, the B¨acklund transformationwas derived for the entire hierarchy by an universal gauge transformation thatpreserves the zero curvature representation and henceforth the equations ofmotion [1]. The results obtained in [2, 5, 6] for the super sinh-Gordon wereextended to the entire smKdV hierarchy by the construction of a B¨acklund-gauge transformation which connects two field configurations of the sameequations of motion [1]. Such structure was first introduced in [7, 8] for thebosonic sine-Gordon theory in order to describe integrable defects in the sensethat two solitons solutions are interpolated by a defect, as a set of internalboundary conditions derived from a Lagrangian density located at certainspatial position connecting two types of solutions. The integrability of themodel is guaranteed by the gauge invariance of the zero curvature represen-tation.The N = 1 supersymmetric modified Korteweg de-Vries (smKdV) hierar-chy in the presence of defects was investigated in [1] through the constructionof gauge transformation in the form of a B¨acklund-defect matrix approach.Firstly, we employ the defect matrix associated to the hierarchy which turnsout to be the same as for the super sinh-Gordon (sshG) model. The methodis general for all flows and as an example we have derived explicitly theB¨acklund equations in components for the first few flows of the hierarchy,namely t , t and t . Finally, this super B¨acklund transformation is employedto introduce type I defects for the supersymmetric mKdV hierarchy.In this note we propose an alternative derivation for the B¨acklund trans-formation obtained in [1] by employing a recursion operator. For the bosoniccase of the mKdV hierarchy the recursion operator was constructed in [12]and it relates equations of motion for two consecutive time evolutions. Weshow that the same philosophy can be applied to the supersymmetric mKdVhierarchy to relate B¨acklund transformations for two consecutive flows.In what follows, we first derive the recursion operator for the supersym-metric mKdV hierarchy directly from the zero curvature representation.For technical reasons we change variables u ( x, t N ) of mKdV equation as u ( x, t N ) = ∂ x φ ( x, t N ) which seems more suitable to deal with B¨acklundtransformations. We next conjecture that the B¨acklund transformations forconsecutive flows are also related by the same recursion operator. In fact weverify our conjecture for the first few flows generated by t , t and t . An integrable hierarchy can be obtained from the zero curvature condition[ ∂ x + A x , ∂ t N + A t N ] = 0 (1) ecursion Operator and B¨acklund Transformation for smKdV Hierarchy 3 where, A x and A t N are the Lax pair lying into an affine Kac-Moody super-algebra ( b G ) and t N represents the time flow of an integrable equation.Another important key ingredient to construct an integrable hierarchy is agrading operator Q and a constant grade one element E (1) that decomposesthe affine superalgebra into the following subspaces b G = ⊕ b G m = K ( E ) ⊕ M ( E ) (2)where m is the degree of the subspace b G m according to Q , i.e., [ Q, G m ] = m G m , K ( E ) = (cid:8) x ∈ b G / [ x, E (1) ] = 0 (cid:9) is the kernel of E (1) and M ( E ) is itscomplement ( image).Now we can define the Lax pair as A x = E (1) + A + A / , (3) A t N = D ( N ) N + D ( N − / N + ... + D (1 / N + D (0) N , (4)where A ∈ b G ∩M ( E ) , A / ∈ b G / ∩M ( E ) with their respective componentsdefining the bosonic and fermionic fields of the theory and D ( m ) N ∈ b G m .The decomposition of the equation (1) into graded subspaces yields thefollowing system( N + 1) : h E (1) , D ( N ) N i = 0 , ( N + 1 /
2) : h E (1) , D ( N − / N i + h A / , D ( N ) N i = 0 , ( N ) : ∂ x D NN + h A , D ( N ) N i + h E (1) , D ( N − N i + h A / , D ( N − / N i = 0 , ...(1) : ∂ x D (1) N + h A , D (1) N i + h E (1) , D (0) N i + h A / , D (1 / N i = 0 , (1 /
2) : ∂ x D (1 / N + h A , D (1 / N i + h A / , D (0) N i − ∂ t N A / = 0 , (0) : ∂ x D (0) N + h A , D (0) N i − ∂ t N A = 0 . (5)The set of equations (5) can be recursively solved yielding the time evo-lution equations for the fields in A and A / as the zero and 1 / b G = b sl (2 , Q = 2 d + h (0)1 and the constant ele-ment E (1) = K (1) + K (2) . Its generators may be regrouped as A.R. Aguirre, J.F. Gomes, A.L. Retore, N.I. Spano, and A.H. Zimerman F ( n + ) = E ( n + ) α + α − E ( n +1) α + E ( n +1) − α − α − E ( n + ) − α ,F ( n + ) = − E ( n ) α + α + E ( n + ) α + E ( n + ) − α − α − E ( n ) − α ,G ( n + ) = E ( n ) α + α + E ( n + ) α + E ( n + ) − α − α + E ( n ) − α G ( n + ) = − E ( n + ) α + α − E ( n +1) α + E ( n +1) − α − α + E ( n + ) − α ,K (2 n +1)1 = − E ( n +1) − α − E ( n ) α ,K (2 n +1)2 = h ( n + ) − h ( n + ) ,M (2 n +1)1 = E ( n +1) − α − E ( n ) α ,M (2 n )2 = 2 h (2 n )1 (6)and decomposed as follows (see [9] for details), M bos = n M (2 n )2 , M (2 n +1)1 o , M fer = n G (2 n + )1 , G (2 n + )2 o , K bos = n K (2 n +1)1 , K (2 n +1)2 o , K fer = n F (2 n + )1 , F (2 n + )2 o (7)Notice that the fermionic generators F i and G i , i = 1 , Z structure in their affine indices,in the sense that the semi integers indices N + 1 / n + 1 / n + 3 / Z structure arises, nowdecomposing the integers N into odd (2 n + 1) and even (2 n ) subsets. Assignto the bosonic generators { K , K , M } and { M } the grades 2 n + 1 and 2 n respectively. The affine algebra displayed in the appendix is shown to closeconsistently with the Z structures described above.The x part of the Lax pair is then constructed from A = u ( x, t ) M (0)2 and A / = ¯ ψ ( x, t ) G (1 / .The first equation in the system (5) implies that D ( N ) N ∈ K ( E ) and hence N = 2 n +1. In order to solve equations in (5) we expand D ( m ) N according to itsbosonic or fermionic character using latin or greek coefficients, respectivelyfollowing the grading given in equation (7) , i.e., Moreover we use α m , β m for grades m = 2 n + 1 / γ m , δ m for m = 2 n + 3 / a m , b m , c m for m = 2 n + 1 and d m for m = 2 n .ecursion Operator and B¨acklund Transformation for smKdV Hierarchy 5 D (2 n + ) N = γ n + F ( n + ) + δ n + G ( n + ) ,D (2 n +1) N = a n +1 K (2 n +1)1 + b n +1 K (2 n +1)2 + c n +1 M (2 n +1)1 ,D (2 n + ) N = α n + F ( n + ) + β n + G ( n + ) ,D (2 n ) N = d n M (2 n )2 ,D (2 n − ) N = γ n − F ( n − ) + δ n − G ( n − ) ,D (2 n − N = a n − K (2 n − + b n − K (2 n − + c n − M (2 n − ,D (2 n − ) N = α n − F ( n − ) + β n − G ( n − ) ,D (2 n − N = d n − M (2 n − , ... D ( ) N = γ F ( ) + δ G ( ) ,D (1) N = a K (1)1 + b K (1)2 + c M (1)1 ,D ( ) N = α F ( ) + β G ( ) D (0) N = d M (0)2 . (8)where the a m , b m , c m , d m and α m , β m , γ m , δ m are functionals of the fields u and ¯ ψ .Substituting this parameterization in the equation (5), one solve recur-sively for all D ( m ) , m = 0 , · · · N . Starting with the highest grade equation in(5) in which N = 2 n + 1,[ K (1)1 + K (1)2 , a n +1 K (2 n +1)1 + b n +1 K (2 n +1)2 + c n +1 M (2 n +1)1 ] = 0 (9)We obtain after using the comutation relations given in the appendix that c n +1 = 0. Now substituting this result in the next equation in (5), i.e, theequation for degree N + 1 / β n + = 12 ¯ ψ ( a n +1 + b n +1 ) (10)From the equation for degree N we find that a n +1 , b n +1 are constantsand d n = ua n +1 + ¯ ψα n + . Proceeding in this way until we reach theequation for degree N −
2, we get
A.R. Aguirre, J.F. Gomes, A.L. Retore, N.I. Spano, and A.H. Zimerman ( N − /
2) : ∂ x α n + − uβ n + + ¯ ψd n = 0 ∂ x β n + − uα n + + 2 δ n − = 0 (11)( N −
1) : ∂ x d n − c n − + 2 ¯ ψγ n − = 0 (12)( N − /
2) : ∂ x γ n − − uδ n − + ¯ ψc n − = 0 ∂ x δ n − − uγ n − + 2 β n − − ¯ ψ ( a n − + b n − ) = 0(13)( N −
2) : ∂ x a n − + 2 uc n − − ψβ n − = 0 ∂ x b n − + 2 ¯ ψβ n − = 0 ∂ x c n − − d n − + 2 ua n − + 2 ¯ ψα n − = 0 (14)The subsequent equations are all similar to the set above, in the sensethat the equations for even grade will correspond to (12), the odd ones willbe similar to the set in (14).For the semi-integer degree equations the following combinations are al-lowed: if ( N − − m ) then it corresponds to the set (11) and if the gradecan be written as ( N − − m −
1) it seems like (13) where m ∈ Z + .Then for a specific n ∈ Z + these results can be written in the followingway, c n +1 = 0 , β n + = √ i ψ ( a n +1 + b n +1 ) ,a n +1 = constant b n +1 = constantd n = ua n +1 + √ i ¯ ψα n + ∂ x α n + − j − uβ n + − j + √ i ¯ ψd n +1 − j = 0 (odd j) ∂ x β n + − j − uα n + − j + 2 δ n + − j = 0 (odd j) ∂ x d n +1 − j − c n − j + 2 √ i ¯ ψγ n + − j = 0 (odd j) ∂ x γ n + − j − uδ n + − j + √ i ¯ ψc n +1 − j = 0 (even j) ∂ x δ n + − j − uγ n + − j + 2 β n + − j − √ i ¯ ψ ( a n +1 − j + b n +1 − j ) = 0 (even j) ∂ x a n +1 − j + 2 uc n +1 − j − √ i ¯ ψβ n + − j = 0 (even j) ∂ x b n +1 − j + 2 √ i ¯ ψβ n + − j = 0 (even j) ∂ x c n +1 − j − d n − j + 2 ua n +1 − j + 2 √ i ¯ ψα n + − j = 0 (even j) (15)where j = 1 , ..., n .We proceed in this way until the grade (1 /
2) equation in (5) to get ∂ x α = uβ − ¯ ψd (16) ∂ t n +1 ¯ ψ = ∂ x β − uα (17) ecursion Operator and B¨acklund Transformation for smKdV Hierarchy 7 and the zero grade equation to obtain ∂ t n +1 u = ∂ x d (18)Therefore the problem is to recursively solve this set of equations, findingthe respective coefficients for a given value of n and then substitute them in(18) and (17) to obtain the time evolution of the fields u , ¯ ψ . For example, if n = 0 the equations of motion are ∂ t ¯ ψ = ∂ x ¯ ψ, ∂ t u = ∂ x u (19)For n = 1 we have the supersymmetric mKdV equation4 ∂ t u = u x − u u x + 3 i ¯ ψ∂ x (cid:0) u ¯ ψ x (cid:1) , (20)4 ∂ t ¯ ψ = ¯ ψ x − u∂ x (cid:0) u ¯ ψ (cid:1) . (21)Then, for n = 2 we have16 ∂ t u = u x − u x ) − u ( u x )( u x ) − u ( u x ) + 30 u ( u x )+ 5 i ¯ ψ∂ x ( u ¯ ψ x − u ¯ ψ x + u x ¯ ψ x + u x ¯ ψ x ) + 5 i ¯ ψ x ∂ x ( u ¯ ψ x ) , (22)16 ∂ t ¯ ψ = ¯ ψ x − u∂ x ( u ¯ ψ x + 2 u x ¯ ψ x + u x ¯ ψ ) + 10 u ∂ x ( u ¯ ψ ) − u x ) ∂ x ( u x ¯ ψ ) . (23) We shall now consider the construction of a set of supersymmetric integrableequations by solving the system in (15). Since the solution of (15) is similarfor all values of n it is expected that there exists a connection among thetime flows. The recursion operator is the mathematical object responsible forsuch connection and will be constructed in this section.In order to see this we consider the equations for N = 2 n +1 and N = 2 n +3 A.R. Aguirre, J.F. Gomes, A.L. Retore, N.I. Spano, and A.H. Zimerman + + c n +1 = 0 , c n +3 = 0 (24) a n +1 = b n +1 = 1 , a n +3 = b n +3 = 1 , (25) β n + = ¯ ψ β n + = ¯ ψ (26) d n = u + ¯ ψα n + , d n +2 = u + ¯ ψα n + (27) ∂ x α n + − uβ n + + ¯ ψd n = 0 , ∂ x α n + − uβ n + + ¯ ψd n +2 = 0 (28) ∂ x β n + − uα n + + 2 δ n − = 0 , ∂ x β n + − uα n + + 2 δ n + = 0 (29) ∂ x d n − c n − + 2 ¯ ψγ n − = 0 , ∂ x d n +2 − c n +1 + 2 ¯ ψγ n + = 0 (30)... ... ∂ x α / − uβ / + ¯ ψd = 0 , ∂ x α / − uβ / + ¯ ψd = 0 (31) ∂ x β / − uα / + 2 δ / = 0 (32) ∂ x d − c + 2 ¯ ψγ / = 0 (33) ∂ x γ / − uδ / + ¯ ψc = 0 (34) ∂ x δ / − uγ / − ¯ ψ ( a + b ) + 2 β / = 0(35) ∂ x a − ψβ / + 2 uc = 0 (36) ∂ x b + 2 ¯ ψβ / = 0 (37) ∂ x c − d + 2 ua + 2 ¯ ψα / = 0 (38) ∂ x α / − uβ / + ¯ ψd = 0 (39) ∂ t n +1 ¯ ψ = ∂ x β / − uα / , ∂ t n +3 ¯ ψ = ∂ x β / − uα / (40) ∂ t n +1 u = ∂ x d , ∂ t n +3 u = ∂ x d (41)Notice that until the equation (32) the alligned equations have the samesolution, in such a way that we can make the following useful identifications, d (cid:12)(cid:12)(cid:12) n +3 = d (cid:12)(cid:12)(cid:12) n +1 , β / (cid:12)(cid:12)(cid:12) n +3 = β / (cid:12)(cid:12)(cid:12) n +1 , α / (cid:12)(cid:12)(cid:12) n +3 = α / (cid:12)(cid:12)(cid:12) n +1 . (42)The case for N = 2 n + 3 has eight additional equations (32)-(39), whichcan be solved in terms of the coefficients for N = 2 n + 1 by the relations (42).Then we will be able to relate the time evolution equations for t n +3 to thetime evolution equations for t n +1 .Starting with the equation (32) by using (42) we get δ / (cid:12)(cid:12)(cid:12) n +3 = − ∂ t n +1 ¯ ψ. (43)Substituting in (33) and (34) ecursion Operator and B¨acklund Transformation for smKdV Hierarchy 9 c (cid:12)(cid:12)(cid:12) n +3 = 12 ∂ t n +1 u + ¯ ψγ / (cid:12)(cid:12)(cid:12) n +3 (44) γ / (cid:12)(cid:12)(cid:12) n +3 = − Z dx ∂ t n +1 ( u ¯ ψ ) . (45)Recursively solving the equations (35)-(39) we get the following coefficients β / (cid:12)(cid:12)(cid:12) n +3 = 14 ∂ x ∂ t n +1 ¯ ψ − u Z dx ∂ t n +1 ( u ¯ ψ ) + 12 ¯ ψ ( a + b ) (cid:12)(cid:12)(cid:12) n +3 , (46) a (cid:12)(cid:12)(cid:12) n +3 = − Z dx u∂ t n +1 u + 12 Z dx ′ u ¯ ψ Z dx ∂ t n +1 ( u ¯ ψ )+ 12 Z dx ¯ ψ∂ x ∂ t n +1 ¯ ψ, (47) b (cid:12)(cid:12)(cid:12) n +3 = − Z dx ¯ ψ∂ x ∂ t n +1 ¯ ψ + 12 Z dx ′ u ¯ ψ Z dx ∂ t n +1 ( u ¯ ψ ) , (48) d (cid:12)(cid:12)(cid:12) n +3 = 14 ∂ x ∂ t n +1 u −
14 ¯ ψ∂ t n +1 ( u ¯ ψ ) − u Z dx u∂ t n +1 u + ¯ ψα / (cid:12)(cid:12)(cid:12) n +3 + u Z dx ′ u ¯ ψ Z dx ∂ t n +1 ( u ¯ ψ ) − ∂ x ¯ ψ Z dx ∂ t n +1 ( u ¯ ψ )+ u Z dx ¯ ψ∂ x ∂ t n +1 ¯ ψ, (49) α / (cid:12)(cid:12)(cid:12) n +3 = 14 Z dx ( u∂ x ∂ t n +1 ¯ ψ − ¯ ψ∂ x ∂ t n +1 u )+ 12 Z dx ′ u ¯ ψ Z dx ( u∂ t n +1 u − ¯ ψ∂ x ∂ t n +1 ¯ ψ ) − Z dx ′ (cid:0) u − ¯ ψ∂ x ¯ ψ (cid:1) Z dx ∂ t n +1 ( u ¯ ψ ) . (50)Finally putting these coefficients in the equations of motion (40) and (41)we obtain that the t n +3 equation of the smKdV hierarchy is given by ∂u∂t n +3 = R ∂u∂t n +1 + R ∂ ¯ ψ∂t n +1 , ∂ ¯ ψ∂t n +3 = R ∂u∂t n +1 + R ∂ ¯ ψ∂t n +1 (51)where { R , R } , { R , R } are the bosonic and fermionic components of therecursion operator, respectively, which are given by R = 14 D − u − u x D − u + i ψ ¯ ψ x + i u ¯ ψ D − ¯ ψ + i u x D − u ¯ ψ D − ¯ ψ − − i ψ x D − ¯ ψ − i ψ x D − ¯ ψ D − i ψ x D − u D − ¯ ψ + i ψ x D − u ¯ ψ D − u −
14 ¯ ψ x D − ¯ ψ ¯ ψ x D − ¯ ψ, (52) R = i u ¯ ψ D − i u ¯ ψ x − i u x ¯ ψ + i u ¯ ψ D − u + i u x D − ¯ ψ D + i u x D − u ¯ ψ D − u − i ψ x D − u + i ψ x D − u D − i ψ x D − u D − u + 12 ¯ ψ x D − u ¯ ψ D − ¯ ψ D −
14 ¯ ψ x D − ¯ ψ ¯ ψ x D − u, (53) R = − u ¯ ψ − u x D − ¯ ψ −
12 ¯ ψ x D − u + i ψ x D − u ¯ ψ D − ¯ ψ + 14 u D − ¯ ψ D − u D − u ¯ ψ D − u − i u D − ¯ ψ ¯ ψ x D − ¯ ψ + 14 u D − u D − ¯ ψ, (54) R = 14 D − u − u x D − u − u D − u D + 14 u D − u D − u − i u D − ¯ ψ ¯ ψ x D − u + i ψ x D − u ¯ ψ D − u + i u D − u ¯ ψ D − ¯ ψ D . (55)where D = ∂ x and D − is its inverse.In terms of u = φ x we get ∂φ∂t n +3 = R ∂φ∂t n +1 + R ∂ ¯ ψ∂t n +1 , ∂ ¯ ψ∂t n +3 = R ∂φ∂t n +1 + R ∂ ¯ ψ∂t n +1 (56)where R = D − R D , R = D − R , R = R D , R = R , with R = 14 D − φ x − φ x D − φ x + i ψ ¯ ψ x + i φ x ¯ ψ D − ¯ ψ − i ψ x D − ¯ ψ − i ψ x D − ¯ ψ D − i ψ x D − φ x D − ¯ ψ + i ψ x D − φ x ¯ ψ D − φ x −
14 ¯ ψ x D − ¯ ψ ¯ ψ x D − ¯ ψ + i φ x D − φ x ¯ ψ D − ¯ ψ, (57) R = i φ x ¯ ψ D − i φ x ¯ ψ x − i φ x ¯ ψ + i φ x ¯ ψ D − φ x + i φ x D − ¯ ψ D − i ψ x D − φ x + i ψ x D − φ x D − i ψ x D − φ x D − φ x + 12 ¯ ψ x D − φ x ¯ ψ D − ¯ ψ D −
14 ¯ ψ x D − ¯ ψ ¯ ψ x D − φ x + i φ x D − φ x ¯ ψ D − φ x , (58) R = − φ x ¯ ψ − φ x D − ¯ ψ −
12 ¯ ψ x D − φ x + i ψ x D − φ x ¯ ψ D − ¯ ψ + 14 φ x D − ¯ ψ D − φ x D − φ x ¯ ψ D − φ x − i φ x D − ¯ ψ ¯ ψ x D − ¯ ψ + 14 φ x D − φ x D − ¯ ψ, (59) ecursion Operator and B¨acklund Transformation for smKdV Hierarchy 11 R = 14 D − φ x − φ x D − φ x − φ x D − φ x D + 14 φ x D − φ x D − φ x − i φ x D − ¯ ψ ¯ ψ x D − φ x + i ψ x D − φ x ¯ ψ D − φ x + i φ x D − φ x ¯ ψ D − ¯ ψ D . (60)We have explicitly checked that by employing equation (51) for n = 0we recover the smKdV equation (20), (21). Also it was verified that (51) for n = 1, yields the t flow of the hierarchy (22), (23) as predicted. In this section we will start by reviewing the systematic construction of theB¨acklund transformation for the smKdV hierarchy, based on the invarianceof the zero curvature equation (1) under the gauge transformation, ∂ µ K = KA µ ( φ , ¯ ψ ) − A µ ( φ , ¯ ψ ) K. (61)where A = A x , A = A t n +1 , ∂ = ∂ x , ∂ = ∂ t n +1 and assuming theexistence of a defect matrix K ( φ , ¯ ψ , φ , ¯ ψ ) which maps a field configuration { φ , ¯ ψ } into another { φ , ¯ ψ } .It is important to point out that the spatial Lax operator A x is commonto all members of the smKdV hierarchy and is given by A x = λ / − φ x − √ i ¯ ψ − λ λ / + φ x √ i λ / ¯ ψ √ i λ / ¯ ψ √ i ¯ ψ λ / , (62)Moreover, based on this fact it has been shown that the spatial compo-nent of the B¨acklund transformation, and consequently the associated defectmatrix, are also common and henceforth universal within the entire bosonichierarchy [10, 11]. This agrees More recently, in [1], this result has beenextended to the supersymmetric mKdV hierarchy with the following defectmatrix K = λ / − ω e φ + λ − / − √ iω e φ +2 f − ω e − φ + λ / λ / − √ iω e − φ +2 f λ / √ iω e − φ +2 f λ / √ iω e φ +2 f ω + λ / (63) where φ ± = φ ± φ , ω represents the B¨acklund parameter, and f is anauxiliary fermionic field.We can now substitute (62) and (63) in the x -part of the gauge transfor-mation (61), to get ∂ x φ − = 4 ω sinh( φ + ) − iω sinh (cid:18) φ + (cid:19) f ¯ ψ + , (64)¯ ψ − = 4 ω cosh (cid:18) φ + (cid:19) f , (65) ∂ x f = 1 ω cosh (cid:18) φ + (cid:19) ¯ ψ + . (66)which is the spatial part of the B¨acklund transformations, where we havedenoted ¯ ψ ± = ¯ ψ ± ¯ ψ .The corresponding temporal part of the B¨acklund is obtained from thegauge transformation in (61) for µ = 0, i.e., ∂ t n +1 K = KA t n +1 ( φ , ¯ ψ ) − A t n +1 ( φ , ¯ ψ ) K. (67)where we consider the corresponding temporal part of the Lax pair A t n +1 .Now, we will consider some examples.For n = 0 we have that A x = A t so the temporal part of the B¨acklundis, ∂ t φ + = ∂ x φ + , (68) ∂ t f = ∂ x f . (69)This implies that ∂ t φ − = ∂ x φ − . (70)Then, for n = 0 the x and t components of the B¨acklund are the same.The next non-trivial example is the smKdV equation ( n = 1), A t = p + λ / p / − λφ x + λ / p + − λ µ + + λ / ν + + λ √ i ¯ ψ − λp − − λ − p + λ / p / + λφ x + λ / λ / µ − + λν − + λ / √ i ¯ ψλ / µ − − λν − + λ / √ i ¯ ψ µ + − λ / ν + + λ √ i ¯ ψ λ / p / + 2 λ / , (71) where p = 14 (cid:0) φ x ) − φ x − iφ x ¯ ψ∂ x ¯ ψ (cid:1) , p / = − i ψ∂ x ¯ ψ, ν ± = √ i (cid:0) ∂ x ¯ ψ ± ¯ ψ∂ x φ (cid:1) p ± = 12 (cid:0) φ x ± ( φ x ) ∓ i ¯ ψ∂ x ¯ ψ (cid:1) , µ ± = √ i (cid:0) ∂ x ¯ ψ ± φ x ∂ x ¯ ψ ∓ ¯ ψφ x − ψ ( φ x ) (cid:1) . (72) ecursion Operator and B¨acklund Transformation for smKdV Hierarchy 13 By substituting (71) and (63) in (67), we obtain4 ∂ t φ − = iω (cid:20) φ (+)2 x cosh (cid:16) φ + (cid:17) − (cid:16) φ (+) x (cid:17) sinh (cid:16) φ + (cid:17)(cid:21) ¯ ψ + f − ω sinh φ + − iω (cid:20) φ (+) x cosh (cid:16) φ + (cid:17) ¯ ψ (+) x − (cid:16) φ + (cid:17) ¯ ψ (+)2 x (cid:21) f + 2 ω h φ (+)2 x cosh φ + − (cid:16) φ (+) x (cid:17) sinh φ + + i ¯ ψ + ¯ ψ (+) x sinh φ + i − iω (cid:20) sinh (cid:16) φ + (cid:17) + 4 sinh (cid:16) φ + (cid:17) + 3 sinh (cid:16) φ + (cid:17)(cid:21) ¯ ψ + f , (73)4 ∂ t f = 12 ω cosh (cid:16) φ + (cid:17) h ψ (+)2 x − ¯ ψ + ( φ (+) x ) i + 12 ω sinh φ + cosh (cid:16) φ + (cid:17) ¯ ψ + + 12 ω sinh (cid:16) φ + (cid:17) h ¯ ψ + φ (+)2 x − φ (+) x ¯ ψ (+) x i − ω sinh φ + cosh (cid:16) φ + (cid:17) φ (+) x f . (74)the corresponding temporal part of the B¨acklund transformation for thesmKdV equation, where φ ( ± ) ix = ∂ ix φ ± and ¯ ψ ( ± ) ix = ∂ ix ¯ ψ ± . In [1] this pro-cedure have been aplied to obtain these transformations for the t memberof the smKdV hierarchy. In this section we will extend the idea of recursion operator to generate theB¨acklund transformation for smKdV hierarchy as an alternative method.In order to construct the recursion operator for the B¨acklund transforma-tions we consider two different solutions of the equation (56) as ∂φ ∂t n +3 = R (1)1 ∂φ ∂t n +1 + R (1)2 ∂ ¯ ψ ∂t n +1 , ∂ ¯ ψ ∂t n +3 = R (1)3 ∂φ ∂t n +1 + R (1)4 ∂ ¯ ψ ∂t n +1 (75) ∂φ ∂t n +3 = R (2)1 ∂φ ∂t n +1 + R (2)2 ∂ ¯ ψ ∂t n +1 , ∂ ¯ ψ ∂t n +3 = R (2)3 ∂φ ∂t n +1 + R (2)4 ∂ ¯ ψ ∂t n +1 (76)where R ( p ) i = R i (cid:16) φ ( p ) x , φ ( p )2 x , ¯ ψ p , ¯ ψ ( p ) x , ¯ ψ ( p )2 x (cid:17) , i = 1 , ..., , p = 1 , ∂ t n +3 φ − = (cid:16) R (1)1 + R (2)1 (cid:17) ∂ t n +1 φ − + (cid:16) R (1)2 + R (2)2 (cid:17) ∂ t n +1 ¯ ψ − + (cid:16) R (1)1 − R (2)1 (cid:17) ∂ t n +1 φ + + (cid:16) R (1)2 − R (2)2 (cid:17) ∂ t n +1 ¯ ψ + , (77) ∂ t n +3 ¯ ψ − = (cid:16) R (1)3 + R (2)3 (cid:17) ∂ t n +1 φ − + (cid:16) R (1)4 + R (2)4 (cid:17) ∂ t n +1 ¯ ψ − + (cid:16) R (1)3 − R (2)3 (cid:17) ∂ t n +1 φ + + (cid:16) R (1)4 − R (2)4 (cid:17) ∂ t n +1 ¯ ψ + . (78)At this point, we conjecture that the equations (77) and (78) correspondto the temporal part of the super B¨acklund transformation for an superintegrable equation especified by n . We note that as well as the consecutiveequations of motion within the hierarchy are connected by the same recursionoperator, here the same occurs to the B¨acklund transformations. In order toclarify this hypothesis we next consider some examples.For n = 0 we have2 ∂ t φ − = (cid:16) R (1)1 + R (2)1 (cid:17) ∂ t φ − + (cid:16) R (1)2 + R (2)2 (cid:17) ∂ t ¯ ψ − + (cid:16) R (1)1 − R (2)1 (cid:17) ∂ t φ + + (cid:16) R (1)2 − R (2)2 (cid:17) ∂ t ¯ ψ + , (79)2 ∂ t ¯ ψ − = (cid:16) R (1)3 + R (2)3 (cid:17) ∂ t φ − + (cid:16) R (1)4 + R (2)4 (cid:17) ∂ t ¯ ψ − + (cid:16) R (1)3 − R (2)3 (cid:17) ∂ t φ + + (cid:16) R (1)4 − R (2)4 (cid:17) ∂ t ¯ ψ + . (80)By using (68)-(70) we recover equations (73) and (74), ie the time compo-nent of the B¨acklund transformantions for n = 2 (smKdV).Next we consider the case for n = 2 and using again (73)-(74) we obtainfrom (77),16 ∂ t φ − = φ ( − )5 x + 38 (cid:16) φ ( − ) x (cid:17) − φ ( − ) x (cid:18)(cid:16) φ ( − )2 x (cid:17) + (cid:16) φ (+)2 x (cid:17) (cid:19) − φ ( − )2 x φ (+)2 x φ (+) x + 5 φ ( − ) x φ (+) x (cid:18) (cid:16) φ (+) x (cid:17) − φ (+)3 x (cid:19) − φ ( − )3 x (cid:18)(cid:16) φ ( − ) x (cid:17) + (cid:16) φ (+) x (cid:17) (cid:19) + 5 i (cid:16) ¯ ψ − ¯ ψ ( − ) x + ¯ ψ + ¯ ψ (+) x (cid:17) (cid:20) φ ( − )3 x − φ ( − ) x (cid:18)(cid:16) φ ( − ) x (cid:17) + 3 (cid:16) φ (+) x (cid:17) (cid:19)(cid:21) + 5 i (cid:16) ¯ ψ − ¯ ψ (+) x + ¯ ψ + ¯ ψ ( − ) x (cid:17) (cid:20) φ (+)3 x − φ (+) x (cid:18)(cid:16) φ (+) x (cid:17) + 3 (cid:16) φ ( − ) x (cid:17) (cid:19)(cid:21) + 5 i ψ + (cid:16) ¯ ψ ( − )3 x φ (+) x + ¯ ψ (+)3 x φ ( − ) x (cid:17) + 5 i ψ + (cid:16) ¯ ψ ( − )2 x φ (+)2 x + ¯ ψ (+)2 x φ ( − )2 x (cid:17) + 154 (cid:16) φ ( − ) x (cid:17) (cid:16) φ (+) x (cid:17) + 5 i ψ − (cid:16) φ ( − )2 x ¯ ψ ( − )2 x + φ (+)2 x ¯ ψ (+)2 x (cid:17) + 5 i ψ − (cid:16) φ ( − ) x ¯ ψ ( − )3 x + φ (+) x ¯ ψ (+)3 x (cid:17) (81)And for the equation (78) we get ecursion Operator and B¨acklund Transformation for smKdV Hierarchy 15 ∂ t ¯ ψ − = ¯ ψ ( − )5 x −
54 ¯ ψ − (cid:16) φ ( − ) x φ ( − )4 x + φ (+) x φ (+)4 x (cid:17) −
54 ¯ ψ + (cid:16) φ ( − ) x φ (+)4 x + φ (+) x φ ( − )4 x (cid:17) − (cid:16) ¯ ψ − φ (+)2 x + ¯ ψ + φ ( − )2 x (cid:17) (cid:20) φ (+)3 x − φ (+) x (cid:18)(cid:16) φ (+) x (cid:17) + 3 (cid:16) φ ( − ) x (cid:17) (cid:19)(cid:21) − (cid:16) ¯ ψ − φ ( − )2 x + ¯ ψ + φ (+)2 x (cid:17) (cid:20) φ ( − )3 x − φ ( − ) x (cid:18)(cid:16) φ ( − ) x (cid:17) + 3 (cid:16) φ (+) x (cid:17) (cid:19)(cid:21) −
58 ¯ ψ ( − ) x φ ( − ) x (cid:20) φ ( − )3 x − φ ( − ) x (cid:18)(cid:16) φ ( − ) x (cid:17) + 6 (cid:16) φ (+) x (cid:17) (cid:19)(cid:21) −
58 ¯ ψ ( − ) x φ (+) x (cid:18) φ (+)3 x − (cid:16) φ (+) x (cid:17) (cid:19) −
54 ¯ ψ (+) x (cid:16) φ ( − )2 x φ (+)2 x + 3 φ (+)3 x φ x (cid:17) −
54 ¯ ψ (+) x φ (+) x (cid:20) φ ( − )3 x − (cid:18)(cid:16) φ ( − ) x (cid:17) + (cid:16) φ (+) x (cid:17) (cid:19)(cid:21) −
154 ¯ ψ (+)2 x (cid:16) φ (+)2 x φ ( − ) x + φ ( − )2 x φ (+) x (cid:17) −
54 ¯ ψ ( − )3 x (cid:18)(cid:16) φ ( − ) x (cid:17) + (cid:16) φ (+) x (cid:17) (cid:19) −
52 ¯ ψ (+)3 x φ ( − ) x φ (+) x −
52 ¯ ψ ( − ) x (cid:18)(cid:16) φ ( − )2 x (cid:17) + (cid:16) φ (+)2 x (cid:17) (cid:19) −
154 ¯ ψ ( − )2 x (cid:16) φ ( − )2 x φ ( − ) x + φ (+)2 x φ (+) x (cid:17) (82)Now, using the x -part of the B¨acklund transformation (64)-(66) in thesetwo equations we end up with the corresponding B¨acklund transformationfor n = 3, that was obtained in [1]. Conclusions
In this note we have considered a hierarchy of supersymmetric equations ofmotion underlined by an affine construction of a Kac-Moody algebra ˆ sl (2 , Z structure assigned to bosonicand fermionic generators defined in (6). For higher rank algebras we expect tosystematize such construction decomposing both integers and semi-integersin disjoint subsets compatible with the closure of the algebra, e.g. Z k forˆ sl ( k, Acknowledgements
ALR thanks the Sao Paulo Research Foundation FAPESP forfinancial support under the process 2015/00025-9. JFG, NIS and AHZ thank CNPqfor financial support.
Appendix
Here we resume the commutation and Anti-commutation relations for the b sl (2,1) affine Lie superalgebra[ K (2 n +1)1 , K (2 m +1)2 ] = 0 , [ M (2 n +1)1 , K (2 m +1)1 ] = 2 M n + m +1)2 + ( n + m ) δ n + m +1 , ˆ c, [ M (2 n +1)1 , K (2 m +1)2 ] = 0 , [ K (2 n +1)2 , K (2 m +1)2 ] = − ( n − m ) δ n + m +1 , ˆ c, [ M (2 n )2 , K (2 m +1)1 ] = 2 M n + m )+11 , [ M (2 n )2 , K (2 m +1)2 ] = 0 , [ M (2 n +1)1 , M (2 m )2 ] = − K n + m )+11 , [ M (2 n +1)1 , M (2 m +1)1 ] = − ( n − m ) δ n + m +1 , ˆ c, [ M (2 n )2 , M (2 m )2 ] = ( n − m ) δ n + m, ˆ c, [ K (2 n +1)1 , K (2 m +1)1 ] = ( n − m ) δ n + m +1 , ˆ c, [ F (2 n +3 / , K (2 m +1)1 ] = − [ F (2 n +3 / , K (2 m +1)2 ] = F n + m +1)+1 / , [ F (2 n +1 / , K (2 m +1)1 ] = − [ F (2 n +1 / , K (2 m +1)2 ] = F n + m )+3 / , [ M (2 n +1)1 , F (2 m +3 / ] = G n + m +1)+1 / , [ M (2 n +1)1 , F (2 m +1 / ] = − [ M (2 n )2 , F (2 m +3 / ] = G n + m )+3 / , [ M (2 n )2 , F (2 m +1 / ] = − G n + m )+1 / , [ G (2 n +1 / , K (2 m +1)1 ] = − G n + m )+3 / , [ G (2 n +1 / , K (2 m +1)2 ] = − G n + m )+3 / , [ G (2 n +3 / , K (2 m +1)1 ] = − G n + m +1)+1 / , [ G (2 n +3 / , K (2 m +1)2 ] = − G n + m +1)+1 / , [ M (2 n +1)1 , G (2 m +1 / ] = − F n + m )+3 / , [ M (2 n +1)1 , G (2 m +3 / ] = − F n + m +1)+1 / , [ M (2 n )2 , G (2 m +1 / ] = − F n + m )+1 / , [ M (2 n )2 , G (2 m +3 / ] = − F n + m )+3 / , { F (2 n +3 / , F (2 m +1 / } = [(2 n + 1) − m ] δ n + m +1 , ˆ c, ecursion Operator and B¨acklund Transformation for smKdV Hierarchy 17 { F (2 n +3 / , F (2 m +3 / } = 2( K n + m +1)+12 + K n + m +1)+11 ) , { F (2 n +1 / , F (2 m +1 / } = − K n + m )+12 + K n + m )+11 ) , { F (2 n +1 / , G (2 m +1 / } = 2 M n + m )+11 , { F (2 n +3 / , G (2 m +3 / } = − M n + m +1)+11 , { F (2 n +3 / , G (2 m +1 / } = 2 M n + m +1)2 + [(2 n + 1) + 2 m ] δ n + m +1 , ˆ c, { F (2 n +1 / , G (2 m +3 / } = − M n + m +1)2 − [2 n + (2 m + 1)] δ n + m +1 , ˆ c, { G (2 n +1 / , G (2 m +3 / } = [2 n − (2 m + 1)] δ n + m +1 , ˆ c, { G (2 n +1 / , G (2 m +1 / } = 2( K n + m )+12 − K n + m )+11 ) , { G (2 n +3 / , G (2 m +3 / } = − K n + m +1)+12 − K n + m +1)+11 ) . (83) References
1. A.R. Aguirre, A.L. Retore, J.F. Gomes, N.I. Spano, A.H. Zimerman, “De-fects in the supersymmetric mKdV hierarchy via Backlund transformations”, J.High Energ. Phys. (2018) 2018: 18., https://doi.org/10.1007/JHEP01(2018)018,arXiv:1709.055682. A.R. Aguirre, J.F. Gomes, N.I. Spano, A.H. Zimerman
N=1 super sinh-Gordonmodel with defects revisited
JHEP 02 (2015) 175 [ arXiv:1412.2579 ].3. H. Aratyn, J. F. Gomes, and A.H. Zimerman,
Supersymmetry and the KdV equa-tions for Integrable Hierarchies with a Half-integer Gradation , Nucl. Phys.
B 676 (2004) 537 [hep-th/0309099].4. C. Rogers and W.F. Shadwick,
B¨acklund transformations and their applications ,New York, Academic Press, 1982.5. A.R. Aguirre, T.R. Araujo, J.F. Gomes, and A.H. Zimerman,
Type-IIB¨acklund transformations via gauge transformations , JHEP (2011) 056[ nlin/1110.1589 ].6. A. R. Aguirre, J. F. Gomes, N. I. Spano and A. H. Zimerman, Type-II Super-B¨acklund Transformation and Integrable Defects for the N = , JHEP , 125 (2015) [ arXiv:1504.07978 [math-ph] ].7. P. Bowcock, E. Corrigan and C. Zambon, Classically integrable field theories withdefects , Int. J. Mod. Phys.
A19 (2004) 82 [ hep-th/0305022 ].8. P. Bowcock, E. Corrigan and C. Zambon,
Affine Toda field theories with defects , JHEP (2004) 056 [ hep-th/0401020 ].9. J.F. Gomes, L.H. Ymai, and A.H. Zimerman, Soliton Solutions for the SupermKdV and sinh-Gordon Hierarchy , Phys. Lett.
A 359 (2006) 630-637 [hep-th/0607107].10. J.F. Gomes, A.L. Retore, N.I. Spano, and A.H. Zimerman,
B¨acklund Transfor-mation for Integrable Hierarchies: example - mKdV Hierarchy , J. Phys.: Conf.Ser. (2015) 012039 [arXiv:1501.00865].11. J.F. Gomes, A.L. Retore, and A.H. Zimerman,
Construction of type-II B¨acklundtransformation for the mKdV hierarchy , J. Phys.: Math. Theor. (2015) 405203[arXiv:1505.01024].12. P.J. Olver, Evolution equations possessing infinitely many symmetries , J. Math.Phys.18