A Darboux transformation of the sl(2|1) super KdV hierarchy and a super lattice potential KdV equation

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A Darboux transformation of the sl(2|1) super KdV hierarchy and a super lattice potential KdV equation Ruguang Zhou School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu 221116, PR China

a r t i c l e

i n f o

Article history: Received 26 August 2013 Received in revised form 1 April 2014 Accepted 25 April 2014 Available online xxxx Communicated by A.P. Fordy

a b s t r a c t A hierarchy of super KdV equations is derived from sl(2|1) supermatrix-valued spectral problem. Each equation in the hierarchy is shown to be bi-super-Hamiltonian. Moreover, a Darboux transformation of the hierarchy is constructed. As the compatibility condition of a pair of the Darboux transformations, we obtain a super lattice potential KdV (lpKdV) equation with two discrete variables. © 2014 Elsevier B.V. All rights reserved.

Keywords: The super KdV hierarchy Bi-super-Hamiltonian structure Darboux transformation Lattice integrable system The lpKdV equation

1. Introduction There have existed several versions of the super or fermionic integrable extensions for the celebrated Korteweg–de Vries (KdV) equation

ut = −u xxx + 6uu x .

(1)

The first one is the Kuper–KdV equation



ut = −u xxx + 6uu x − 3ξ ξxx ,

ξt = −4ξxxx + 3u x ξ + 6u ξx ,

(2)

ut = −u xxx + 6uu x − 3ξ ξxx ,

ξt = −ξxxx + 3u x ξ + 3u ξx ,

(3)

which was presented by Manin and Radul in 1985 as a reduction of the general supersymmetric Kadomtsev–Petviashvili hierarchy [6]. The third one with two fermionic superfunctions α , β is

E-mail address: [email protected]. http://dx.doi.org/10.1016/j.physleta.2014.04.052 0375-9601/© 2014 Elsevier B.V. All rights reserved.

(4)

which was obtained in 1989 by Holod and Pakuliak as a reduction of the general fermionic KP equation [7]. Recently, starting from a 3 × 3 matrix-valued spectral problem, Geng and Wu constructed another new super KdV equation [8]



which was first proposed by Kupershmidt in 1984 [1], where u is a bosonic (or commuting or even) superfunction and ξ a fermionic (or anticommuting, or odd) superfunction. It has been shown that the Kuper–KdV equation has an osp(2|1) matrix-valued spectral problem [2–4], a bi-super-Hamiltonian structure [5] and infinitely many conversed laws. Hence it is an integrable super-Hamiltonian system. The second one is the (Manin–Radul) supersymmetric KdV equation



⎧ ⎨ ut2 = −u xxx + 6uu x + 6(α βxx − αxx β), αt = −4αxxx + 6u αx + 3u x α , ⎩ 2 βt2 = −4βxxx + 6u βx + 3u x β,

ut = −u xxx + 6uu x + 12u ξxx ξ + 6u x ξx ξ − 3ξxxxx ξ − 6ξxxx ξx ,

ξt = −4ξxxx + 3u x ξ + 6u ξx . (5) For simplicity, all these equations are called the super KdV equations. However, only the Manin–Radul supersymmetric KdV equation preserves an invariant of supersymmetric transformation [9] and, precisely speaking, only the Manin–Radul KdV equation is a supersymmetric system. So far, no one knows the relation among these super KdV equations except that α = β = − 12 ξ reduces (4) to (2). It is well known that Darboux transformation (DT) is a powerful tool to construct exact solutions of integrable systems [10,11] and the compatibility of two consistent DTs usually may lead to lattice equations with two discrete independent variables [12–14]. The well-known lattice potential KdV (lpKdV) equation or H1 model in the Adler–Bobenko–Suris (ABS) list of Z2 -lattice equations defined on the square lattice may be obtained from the DTs of the KdV equation in this way [13–15]. As early as in 1997, Liu and Manas have constructed the DTs of a number of supersymmetric systems

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such as the Manin–Radul supersymmetric KdV and the supersymmetric sine-Gordon equations [16–19]. However, the study of the DT of the non-supersymmetric super systems and the super lattice full-discrete integrable system only recently has been considered. In [20] Grahovski and Mikhailov constructed two elementary Darboux transformations of a class of nonlinear Schrödinger equations on Grassmann algebras and derived a super lattice system. In this letter, a hierarchy of super KdV equations from a Lie superalgebra sl(2|1) supermatrix-valued spectral problem is derived. Like the Kaup–KdV equation, every equation in the hierarchy is not supersymmetric but bi-super-Hamiltonian. Moreover, the Holod– Pakuliak super KdV equation (4) is just a member of the hierarchy. In Section 3, a DT of the hierarchy of equations and a super lattice potential KdV (lpKdV) equation as the compatibility condition of DTs are constructed. Some concluding remarks are given in Section 4. 2. The super KdV hierarchy of equations We begin with considering the following sl(2|1) supermatrixvalued spectral problem

⎛

φx = U ( w , λ)φ,

0

1 U ( w , λ) = ⎝ u − λ 0 β 0

0

⎞

0

(6)

where w = (u , α , β) T , λ is a bosonic spectral parameter, φ1 (x, t ), φ2 (x, t ) are bosonic superfunction and φ3 (x, t ) an fermionic superfunction, u (x, t ) is a bosonic superfield and α , β fermionic superfields. The background in superalgebra and supergeometry may refer to [21–23]. Throughout the paper, we always assume 3 × 3 matrices to be of the even form, i.e. any 3 × 3 matrix X is of the block form



A C

B D



2

⎧ 1 1 ⎪ am = − bm,x + ∂ −1 (β ρm + αηm ), ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ ⎨ em = ∂ −1 (β ρm + αηm ), cm = −bm+1 + am,x + ubm − β ρm , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ δm = ρm,x + α bm , ⎪ ⎩ ξm = −ηm,x + β bm ,

δ1 = 4 α ,

ξ2 = −4βxx + 2u β, 1

(7)

with sl(2|1) supermatrix

ξ

η

c 1 = 2u ,

ρ1 = η1 = 0,

3

η2 = 4βx ,

2

b3 = − u xx + u + 3(α βx − αx β). 2 2 Furthermore we take an auxiliary spectral problem as follows

V (n) =

n j =0

ρ

⎞

δ ⎠=

e

∞

⎛

aj

⎝ cj ξj j =0

bj

−a j + e j

ηj

ρj

aj ⎝ cj

⎛

2∂ J =⎝ 0 0

ej

ρj

ηj

ej

⎞

⎛

0

δ j ⎠ λn− j + ⎝ bn+1 0

0 0 0

⎞

0 0 ⎠. 0

  + U , V (n) = 0,

(12)

which is equivalent to the following super KdV hierarchy of equations

where

δ j ⎠ λ− j ,

bj

−a j + e j

ξj

w t n = J G n +1 ,

⎞

where a, b, c , e , ρ , δ, ξ, η are undetermined superfunctions. From (7) we arrive at

a0 = b0 = e 0 = ρ0 = δ0 = ξ0 = η0 = 0,

⎛

(11)

The compatibility condition of (6) and (11) gives rise to the zero curvature equation

U tn − V x

V x = [U , V ] ≡ U V − V U ,

b

b 1 = 4,

a2 = −u x + 2α β, b2 = 2u , 1 1 2 c 2 = − u xx + u + αx β − α βx , 2 2 e 2 = 4α β, ρ2 = −4αx , δ2 = −4αxx + 2u α ,

where

−a + e

(10)

ξ1 = 4β,

(n)

a V =⎝c

m ≥ 0.

φtn = V (n) ( w , λ)φ,

where element D and the entries of 2 × 2 matrix A are bosonic, and the elements of column vector B and row vector C are fermionic. With this notation, X ∈ sl(2|1) means str X ≡ tr A − D = 0. To derive the hierarchy of isospectral evolution equations, we first look for solutions of the equation

⎛

m ≥ 0, (9)

and

a 1 = e 1 = 0,

φ1 φ = ⎝ φ2 ⎠ , φ3

X=

2

1 1 ⎪ ⎪ ⎪ + α ∂ −1 β ρm + α ∂ −1 αηm , ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ ⎪ 3 1 ⎪ ⎪ ηm+1 = βx bm + β bm,x + β∂ −1 β ρm − ηm,xx ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ ⎪ 1 −1 ⎪ ⎩ + u ηm + β∂ αηm ,

If choosing an allowable initial condition: c 0 = −4, then we can work out a j , b j , c j , ρ j , δ j , ξ j , η j recursively. The first few are as follows:

α ⎠,

⎞

⎛

⎧ 1 1 1 ⎪ ⎪ bm+1,x = − bm,xxx + ubm,x + u x bm − βx ρm ⎪ ⎪ ⎪ 4 2 4 ⎪ ⎪ ⎪ 3 1 3 ⎪ ⎪ − β ρm,x + αx ηm + αηm,x , ⎪ ⎪ ⎪ 4 4 4 ⎪ ⎪ ⎪ 3 ⎪ ⎪ ⎨ ρm+1 = −αx bm − α bm,x − ρm,xx + u ρm

n  2,

⎞

0 0 0 1 ⎠, 1 0

(13)

⎛ Gn = ⎝

bn

⎞

ηn ⎠ .

(14)

−ρn

In particular, we find V (1) = 4U and



(8)

V

(2 )

=

−u x + 2α β 4λ + 2u −4αx −4λ2 + 2u λ − u xx + 2(u 2 + α βx − αx β) u x + 2α β 4α λ − 4αxx + 2α u −4βxx + 2u β + 4λβ 4βx 4α β

 .

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Substituting (21) into (20) we find that

From the zero curvature equation (2 )

U t2 − V x



+ U, V

 (2 )

=0

τ = α,

we arrive at nothing but the sl(2|1) super KdV equation (4). It is easy to see that if α = β = 0 then Eq. (4) becomes the well-known KdV equation; while if making the reduction α = β = − 12 ξ , Eq. (4) reduces to the Kuper–KdV equation. For convenience, we call (13) the sl(2|1) super KdV hierarchy. In the following we construct bi-super-Hamiltonian structure of the sl(2|1) KdV hierarchy. To this end, we apply the super trace identity [24,25]

δ δw

 

 ∂U ∂ ∂U λγ Str , dx = λ−γ V ∂λ ∂λ ∂w





Str V

2

δ 2n − 1 δ w

bn+1 dx.

Hn =

2 2n − 1



αx + βx +

3 α∂ 2 3 β∂ 2

1 a 0

0

⎞

α ⎠,

(23)

1

and the updated superfields are given by (22). In the following, we construct a super lpKdV equation. It is easy to see that

from (22). Therefore, like the KdV equation, we consider the potential case by setting v = 2u x and thus a = v˜ − v. Consider a pair of Darboux transformations

(16)

1 α + 32 α ∂ 2 x − 12 α ∂ −1 α 2 −∂ + u + 12 β∂ −1 α

1 β 2 x

+ 32 β∂ −∂ 2 + u + 12 α ∂ −1 β − 12 β∂ −1 β

⎞ ⎟ ⎠.

(17) With a direct check, we find that K , J are a pair of compatible super-Hamiltonian operators. Hence, the super sl(2|1) KdV hierarchy is of a bi-super-Hamiltonian form

w tn = J

a

2ax = u˜ − u

bn+1 dx.

where

+ 2u ∂ + u x

(22)

β˜

K G m = J G m +1 ,

⎜ K =⎝

⎛



δ Hn , δw

− 12 ∂ 3

⎧ ˜ u = a2 − ax + k + α β, ⎪ ⎪ ⎪ ⎨˜ 2 ˜ u = a + k + ax + α β, ⎪ αx = α˜ − aα , ⎪ ⎪ ⎩ β˜x = β˜ a − β.

˜ k) = ⎝ −λ + k + a2 + α β˜ T = T (a, α , β;

Moreover, from (9) we get

⎛

and

Finally, the Darboux matrix is

Therefore, the hierarchy (13) can be written as a super-Hamiltonian form:

w tn = J

˜ χ = β,

(15)

where w = (u , α , β) T . After some algebras, from (15) we get

Gn =

3

H n −1 δ Hn =K . δw δw

˜ k1 )φ φ˜ = T ( v˜ − v , α , β;

(24)

and

ˆ k2 )φ, φˆ = T ( vˆ − v , α , β;

(25)

where k1 , k2 are two arbitrary bosonic constants,

⎛

v˜ − v ˜ k1 ) = ⎝ −λ + k1 + ( v˜ − v )2 + α β˜ T ( v˜ − v , α , β;

β˜

1 v˜ − v 0

⎞

α ⎠, 1

and

⎛

3. A Darboux transformation and a super lpKdV equation We proceed now to construct Darboux transformation of the spectral problem (6). By a Darboux transformation of spectral problem (6) we mean a transformation

φ˜ = T φ,

(18)

such that

vˆ − v ⎝ ˆ T ( vˆ − v , α , β; k2 ) = −λ + k2 + ( vˆ − v )2 + α βˆ

βˆ

1 vˆ − v 0

(19)

where U˜ has the same form as U but with an updated potentials u , α , β . This immediately implies that Darboux matrix T must satisfy

T x = U˜ T − T U .

(20)

Let us take a one-parameter Darboux matrix T being of the form

a

T (a, τ , χ ; k) = ⎝ −λ + k + a2 + τ χ

χ

1 a 0

0

⎞

τ ⎠,

(21)

1

where k is an even parameter, a, τ , χ are undetermined superfunctions.

0

⎞

α ⎠. 1

The compatibility condition of (24) and (25) leads to

˜ k1 ) T ( vˆ − v , α , β; ˆ k2 ) Tˆ ( v˜ − v , α , β; ˆ k2 ) T ( v˜ − v , α , β; ˜ k1 ), = T˜ ( vˆ − v , α , β;

φ˜ x = U˜ φ,

⎛

0

(26)

which is equivalent to the following super lpKdV equation

⎧ ˜ = k1 − k2 , ⎨ ( v˜ˆ − vˆ )( vˆ − v˜ ) + α (βˆ − β) α ( vˆ − v˜ ) = αˆ − α˜ , ⎩˜ ˆ ˆ vˆ − v˜ ) = β˜ − β. β(

(27)

If we set v , α , β are superfunctions with two discrete variables n, m ∈ Z: v = v (n, m), α = α (n, m), β = β(n, m) and define the shift operators: ˜f (n, m) = f (n + 1, m), ˆf (n, m) = f (n, m + 1) for any superfunction f (n, m), then Eq. (27) can be regarded as a Z2 -lattice equations defined on the square lattice. It is clear that if α = β = 0 then (27) becomes the lpKdV equation. Therefore, it is appropriate to call (27) the super lpKdV equation or the super H1 model. Eq. (26) is the Lax representation of (27).

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4. Concluding remarks We have presented a hierarchy of super KdV of equations associated with Lie superalgebra sl(2|1) supermatrix-valued spectral problem. The first nontrivial equation in the hierarchy is just the Holod–Pakuliak super KdV equation. It is shown that the hierarchy possesses bi-super-Hamiltonian structure. By the theory of Magri, the bi-super-Hamiltonian structure implies that the hierarchy has a recursion operator and infinite hierarchy of commuting symmetries. A one parameter DT for the resulting super KdV hierarchy and a super lpKdV equation as the compatibility condition of two DTs has been proposed. In [24] it has shown that super integrable systems may be derived various Lie subsuperalgebras supermatrixvalued spectral problems. It should be interesting to discuss their Darboux transformations and the related subjects. Finally, I would like to mention that much progress has been made during past few months. Xue, Levi and Liu have obtained the Darboux transformation and discrete systems of the Manin–Radul supersymmetric KdV equation [26]. Very recently Xue and Liu have reached three Darboux transformations and discrete systems for the sl(2|1) super KdV equation which partially overlaps our results [27]. Acknowledgements This work is supported by the National Natural Science Foundation of China (Grant No. 11271168). The author would like to express his sincere thanks to the referees for their suggestions and comments. References [1] B.A. Kupershmidt, Phys. Lett. A 102 (1984) 213.

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