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A 2-component or N=2 supersymmetric Camassa–Holm equation
Physics Letters A 354 (2006) 110–114 www.elsevier.com/locate/pla
A 2-component or N = 2 supersymmetric Camassa–Holm equation Ziemowit Popowicz Institute of Theoretical Physics, University of Wrocław, pl. M. Borna 9, 50-205 Wrocław, Poland Received 29 September 2005; received in revised form 27 December 2005; accepted 11 January 2006 Available online 19 January 2006 Communicated by A.P. Fordy
Abstract The extended N = 2 supersymmetric Camassa–Holm equation is presented. It is accomplished by formulating the supersymmetric version of the Fuchssteiner method. In this framework we use two supersymmetric recursion operators of the N = 2, α = −2, 4 Korteweg–de Vries equation and construct two different versions of the supersymmetric Camassa–Holm equation. The bosonic sector of N = 2, α = 4 supersymmetric Camassa– Holm equation contains the two component generalization of this equation proposed by Chen, Liu and Zhang and as a special case the two component generalized Hunter–Saxton equation considered by Aratyn, Gomes and Zimerman. As a byproduct of our analysis we defined the N = 2 supersymmetric Hunter–Saxton equation. The bihamiltonian structure is constructed for the supersymmetric N = 2, α = 4 Camassa–Holm equation. © 2006 Elsevier B.V. All rights reserved.
1. Introduction Camassa and Holm introduced [1] in 1993 the integrable non-linear partial differential equation ut − uxxt = −3uux + 2ux uxx + uuxxx 1 2 3 2 = uuxx + ux − u 2 2 x
uxxt = −2ux uxx − uuxxx − ρρx , (1)
which describes a special approximation of shallow water theory and has been extensively studied recently [2–7]. It was shown that this equation possesses the bihamiltonian structure, could be solved using the inverse scattering method and has the so-called peakon solutions. The two component generalization of Camassa–Holm equation mt = −umx − 2mux + ρρx , ρt = −(ρu)x ,
(2)
where m = u − uxx , has been proposed by Chen et al. [3]. This generalization, similarly to the Camassa–Holm equation, is the first negative flow of the AKNS hierarchy and possesses the interesting peakon and multi-kink solutions [3,4,6]. Moreover the E-mail address: [email protected] (Z. Popowicz). 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.01.027
system (2) is connected with the energy dependent Schrödinger spectral problem [3,8]. Quite recently Aratyn et al. [4] showed that the modification of the Schrödinger spectral problem for Eq. (2) leads to two-component Hunter–Saxton equation [9]
ρt = −(ρu)x .
(3)
In this Letter we show that both mentioned generalizations are contained in the bosonic sector of N = 2 extended supersymmetric version of the Camassa–Holm equation. The idea of using extended supersymmetry for the generalization of the soliton equations appeared almost in parallel to the usage of this symmetry in the quantum field theory [10]. The main idea of the supersymmetry is to treat boson and fermion operators equally. In order to get supersymmetric theory we have to add to a system of k bosonic equations kN fermions and k(N − 1) boson fields (k = 1, 2, . . . , N = 1, 2, . . .) in such a way that the final theory becomes supersymmetric invariant. From the soliton point of view we can distinguish two important classes of the supersymmetric equations: the non-extended (N = 1) and extended (N > 1) cases. Consideration of the extended case may imply new bosonic equations whose properties need further investigation. This may be viewed as a bonus, but this extended case is in no way more fundamental than the non-extended one.
Z. Popowicz / Physics Letters A 354 (2006) 110–114
Interestingly enough, some typical supersymmetric effects may occur in the supersymmetrical generalization of the soliton theory, compared to the classical case. We mention two of them: the ambiguity of the roots for the supersymmetric Lax operator [11] and appearance odd Poisson brackets [12]. There are many different methods for supersymmetrization of classical equations. The most popular one is to use the gradation arguments in which to each dependent and independent variables, in a given equation, we associate some weights. We then replace the bosonic fields by superboson fields. For example, for the extended N = 2 supersymmetric case we consider the supersymmetric analog of dependent variable u(x, t), which can be thought of as Φ(x, t, θ1 , θ2 ) = v(x, t) + θ1 ξ1 (x, t) + θ2 ξ2 (x, t) + θ2 θ1 u(x, t)
(4)
where θ1 and θ2 are two anticommuting variables while ξ1 and ξ2 are Grassmann valued functions and v is an additional “new” bosonic field. In the next step using the usual and supersymmetric derivatives D1 = ∂θ1 + θ1 ∂x ,
D2 = ∂θ2 + θ2 ∂x ,
D1 D2 + D2 D1 = ∂
(5)
we consider the most general assumption on the supersymmetric version of the given classical system in such a way as to preserve the given gradation of the supersymmetric equation. In the last step, we assume, that our supersymmetric generalization should possesses some special properties such as the existence a bihamiltonian structure or a Lax operator, for example. However this procedure cannot be applied to the Camassa– Holm equation, because this equation does not preserve gradation with respect to weight. In order to overcome this problem, we supersymmetrize the Fuchssteiner method [7], in which the hereditary operator, responsible for Camassa–Holm hierarchy is constructed out of two different hereditary recursion operators. We introduce the supersymmetry to the theory, considering the supersymmetric analog of these two operators. The Letter is organized as follows. In Section 2 we summarize the Fuchssteiner method. In Section 3, using the supersymmetric recursion operators, which create two different supersymmetric N = 2 Korteweg–de Vries equations, we construct two different supersymmetric versions of Camassa–Holm equation. As a byproduct of our analysis we define the extended N = 2 version of the supersymmetric Hunter–Saxton equation. In the same section we investigate the bosonic sector of this equation. In the last section we describe the bihamiltonian structure of the supersymmetric Camassa–Holm equation. 2. Camassa–Holm equation In order to construct the Camassa–Holm equation, we briefly describe the method used by Fuchssteiner [7]. This method based on the following observation. If we have two different hereditary recursion operators R1 and R2 and if one of them,
111
for example R2 , is invertible then R = R1 R2−1 is also the hereditary recursion operator. Therefore this operator can be used to construct some new integrable hierarchy of equations. Let us present this method for the Camassa–Holm equation. First let us consider the hereditary recursion operator for the Korteweg–de Vries hierarchy R = c∂ 2 + λ ∂u∂ −1 + u , (6) where c and λ are an arbitrary constants. This operator generates the hierarchy of integrable equations in which the first member is ∂u = Rux = cuxxx + 3λuux . (7) ∂t For c = 1 and λ = 2 we have famous Korteweg–de Vries equation. The second recursion operator can be extracted from the first operator shifting the function u → u + γ in recursion operator (6) where γ is a constant. Indeed after shifting the function u in (6) the R operator transforms to R(u + γ ) = R1 + R2 = c1 ∂ 2 + λ ∂u∂ −1 + u + c2 ∂ 2 + λγ ,
(8)
where c = c1 + c2 . It turns out that the recursion operator R = R1 R2−1 generates a new hierarchy of integrable equations n utn = R1 R2−1 ux . (9) Assuming that u = R2 v = c2 vxx + λγ v the first member of the hierarchy is λγ vt + c2 vxxt λc2 2 3λ2 γ 2 = c1 vxx + λc2 vxx v + . vx + v 2 2 x
(10)
Now assuming that c1 = 0, c2 = γ = 1 and λ = −1 the last equation is exactly the Camassa–Holm equation. The second choice γ = 0, c1 = 0, λ = −1 leads us to the Hunter–Saxton equation [9] 1 vtxx = − vxx v + vx2 . (11) 2 x The bihamiltonian structure of Eq. (1) has been constructed in [1] δH1 δH1 3 = c1 ∂ + λ(∂m + m∂) δm δm δH2 δH2 3 = c2 ∂ + λγ ∂ , = J1 (12) δm δm where m = c2 vxx + λγ v and H1 = 12 dx (c2 vxx + λγ v)v, H2 = 12 dx (2c1 λvxx v 2 + 3c1 vxx v + c2 λvx2 v + 3γ λ2 v 3 ). As we see the second Hamiltonian operator is the same as for the Korteweg–de Vries equation and is connected with the Virasoro algebra. For the centerless Virasoro c1 = 0 we algebra √ have additional conserved Hamiltonian 2 dx m which is the Casimir for this algebra as well. Using the first Hamiltonian operator to this quantity we obtain the Harry Dym type equation mt = J2
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3.1. α = 4
of motion 1 mt = J1 √ m
(13)
which reduces to the Harry Dym equation when c2 = 1, γ = 0. 3. N = 2 supersymmetric Camassa–Holm and Hunter–Saxton equation We have four different integrable extended N = 2 supersymmetric extensions of the Korteweg–de Vries equation. Three of them are connected with the supersymmetric Virasoro algebra [13] while the fourth is connected with the odd version of the supersymmetric Virasoro algebra [12]. The first three mentioned equations are
R4 (Φ + γ ) = R1 + R2 ,
δH1 = (D1 D2 ∂ + 2∂Φ + 2Φ∂ − D1 ΦD1 − D2 ΦD2 ) δΦ (α − 1) = −Φxx + 3Φ(D1 D2 Φ) + D1 D2 Φ 2 + αΦ 3 2 x (14) where H1 = 12 dx dθ1 dθ2 (ΦΦ(D1 D2 Φ) + 13 Φ 3 ) and the parameter α can take three different values 1, 4, −2. This system has been extensively studied from different points of view in many papers. The properties of these generalizations are different for different values of α. The Lax operator for α = 4 has two different roots [11]. For the case α = 4 instead of the bihamiltonian formulation, we deal with the inverse first Hamiltonian structure [11], while for the case α = 1 we have the non-standard Lax operator [14] and higher order non-local recursion operator [15]. The hereditary recursion operator for α = 4 and α = −2 has been constructed in [11] −1 R4 = J2 J1,4
= cD1 D2 − λ 2∂Φ∂ −1 − (D1 Φ)D1 ∂ −1 − (D2 Φ)D2 ∂ −1 ,
R(−2) = J2 J1,−2 ,
(16)
where R1 = c1 D1 D2 − λ 2∂Φ∂ −1 − (D1 Φ)D1 ∂ −1 − (D2 Φ)D2 ∂ −1 , R2 = c2 D1 D2 − 2λγ
δH1 Φt = J2 δΦ
(17)
and c = c1 + c2 . Obviously R2 is a hereditary operator and we can consider the hierarchy of equations generated by n Φtn = R1 R2−1 Φx . (18) Assuming that Φ = R2 Υ = c2 (D1 D2 Υ ) − 2λγ Υ we obtain the first member in this hierarchy as c2 (D1 D2 Υt ) − 2λγ Υt = c1 (D1 D2 Υ )x − 2λ(ΦΥ )x + λ(D2 Φ)(D2 Υ ) + λ(D1 Φ)(D1 Υ )
(19)
or more explicitly as c2 (D1 D2 Υt ) − 2λγ Υt = c1 (D1 D2 Υ ) − 2c2 Υ (D1 D2 Υ ) + 4γ λ2 Υ 2 + c2 λ(D2 Υ )(D1 Υ ) x .
(20)
It is our supersymmetric N = 2, α = 4 Camassa–Holm equation. Let us compute the bosonic sector of the previous equation where all odd functions disappear. Assuming that Υ = v + θ2 θ1 u in this sector, we obtain (c2 u − 2γ λv)t = c1 u + 4γ λ2 v 2 − 2c2 λvu x , (−c2 vxx − 2γ λu)t = −c2 λu2 + 2c2 λvxx v − c1 vxx + c2 λvx2 + 8γ λ2 vu x .
J2 = cD1 D2 ∂ − λ(2∂Φ + 2Φ∂ − D1 ΦD1 − D2 ΦD2 ),
(21) Now introducing a new function ρ = c2 u − 2γ λv our system (21) can be rewritten as
J1,4 = ∂, J1,(−2) = cD1 D2 ∂ −1 − λ ∂ −1 D1 ΦD1 ∂ −1 + ∂ −1 D2 ΦD2 ∂ −1 .
Interestingly the supersymmetric Korteweg–de Vries equation for this value of α is not the first member of hierarchy Φtn = R n φ. The first member is Φt = ((D1 D2 Φ) + Φ 2 )x while the second is our supersymmetric Korteweg–de Vries equation. However let us use the supersymmetric recursion operator R4 for the construction of the supersymmetric analogue of the Camassa–Holm recursion operator. In order to find the analogue of the operator R2 let us shift the Φ superfunction to Φ → Φ + γ in R4 obtaining
(15)
Here J2 is the second Hamiltonian operator, which is connected with the extended N = 2 supersymmetric version of the Virasoro algebra, J1,4 is the first Hamiltonian operator for the α = 4 while J1,−2 is an inverse operator of the first Hamiltonian operator for the α = −2 case. In the next we will use these operators in order to adopt the Fuchssteiner method in the supersymmetric case. For these purposes we consider these operators independently.
1 ρt = (c1 ρ + 2c1 γ λv − 2c2 λρv)x , c2 4γ 2 λ2 −c2 vxx − v c2 t 2c1 γ λ 4c1 γ 2 λ2 = ρ+ v − c1 vxx + 2c2 λvvxx c2 c2 12γ 2 λ3 2 λ 2 2 + c2 λvx + v − ρ . c2 c2 x
(22)
Z. Popowicz / Physics Letters A 354 (2006) 110–114
113
We obtained an even more broader generalization of the Camassa–Holm equation than Eq. (2). Interestingly our Eq. (22) contains Eq. (2) as a special case in which c1 = 0, c22 = −1, λ = 12 , γ = 1. For γ = c1 = 0, λ = 12 , c22 = −1 our system of Eq. (22) reduces to the two component version of Hunter–Saxton (3) and for that reason Eq. (20) when λ = 12 , c1 = 0
(γ u − uxx )t 1 = 2uxx u − 2uxx + u2x − 3γ u2 − 2vxxx vx u − 2vxxx vx 2 2 − 3vxx + 2vxx vx ux − 2γ vxx vu + vx2 uxx + γ vx2 u + γ vx2 − 2vx vux − γ v 2 uxx + γ 2 v 2 u x + vxxxx vx
1 (23) (D2 Υ )(D1 Υ ) − 2Υ (D1 D2 Υ ) x 2 could be considered as the N = 2 supersymmetric Hunter– Saxton equation.
For v = 0 this system reduces to the classical Camassa– Holm equation (1) while for v = 0 it gives us new generalization of Camassa–Holm equation.
(D1 D2 Υ )t =
3.2. α = −2 In contrast to the previous case, now the supersymmetric Korteweg–de Vries equation is a first member of the hierarchy Φt = R(−2) Φx . Similarly to the α = 4 case we can shift the superfunction Φ → Φ + γ in R−2 in order to obtain R2 operator. However then we find that the operator R2 contains the Φ superfunction and therefore will be not considered here. The next choice is to use the R2 operator from the α = 4 case. In this situation we obtain an integro-differential supersymmetric equation which does not reduce in the bosonic sector to the Camassa–Holm equation and for that reason we will not consider such possibility. On the other hand, we can try to choose R2 = γ − ∂ 2 operator, the same as in the classical situation, in order to consider the hierarchy of equations generated by Φt = R(−2) R2−1 Φx
(24)
where now Φ = R2 Υ = γ Υ − Υxx . Assuming that λ = 12 and c = 1 the last equation reads (γ Υ − Υxx )t 1 = 4(D1 D2 Υ )Υxx − 2Υxx + Υxx Υx2 − 4γ (D1 D2 Υ )Υ 2 − γ (D2 Υ )(D1 Υ )(D1 D2 Υ ) − γ Υxx Υ 2 − γ Υx2 Υ − 2γ 3 Υ 3 + γ (D2 Υ )(D1 Υ ) x 1 + (D2 Υxx )(D2 Υ )Υxx − 2(D2 Υxxx )(D1 Υ ) 4 − (D2 Υxx )(D2 Υx )Υx − (D2 Υxx )(D1 Υx )(D1 D2 Υ )
+ 2vxxx vxx u − 3γ vxx vx u.
3.3. α = 1 As we mentioned earlier, the recursion operator for this case, has a higher order non-locality. For that reason, the R2 operator obtained by shifting the Φ superfunction, analogously to the previous case, leads us to a very complicated operator. If we proceed in the same way as in the case α = 4 and use the same operator R2 then we obtain a very complicated system also. For that reason we will not study this case further. 4. Bihamiltonian structure We have constructed this structure for the case α = 4 only. This structure could be obtained from the recursion operator R4 in a similar manner to the classical case [1]. In order to end this let us make the following observation. Notice that the first member in the hierarchy (18) could be rewritten as −1 −1 R 2 Mx Mt = J2 (M)J1,4
= J2 Υ = J2 R2−1
Mt = ∂ where
− (D2 Υ )(D1 Υxx )(D1 D2 Υx ) + 2(D1 Υx )(D1 Υ )Υxxx + γ (D2 Υ )(D1 Υ ) x (D1 D2 Υ ) − 3γ (D2 Υx )(D2 Υ )Υx + (D1 ⇒ D2 , D2 ⇒ −D1 ) . (25)
H2 =
(γ v − vxx )t =
1 4vxx u − 2vxx + vxx vx2 − γ vxx v 2 − γ vx2 v 2 + γ v 3 − 4γ vu x , (26)
δH1 δH1 = J2 , δΥ δM
(28)
where M = R2 Υ = c2 (D1 D2 Υ − λγ Υ ) and H1 = 12 × dx dθ1 dθ2 ((R2 Υ )Υ ). As we see this is the second Hamiltonian structure and is generated by the same supersymmetric operator which is responsible for the second Hamiltonian structure of the Korteweg–de Vries equation. The first Hamiltonian structure follows from the following observation. Notice that Eq. (20) could be rewritten also as
+ (D2 Υxx )(D1 Υ )(D1 D2 Υx )
It is our supersymmetric N = 2, α = −2 Camassa–Holm equation. The bosonic sector in which Υ = v + θ1 θ2 u reads
(27)
δH2 δH2 δH2 = ∂R2 R2−1 = ∂R2 , δΥ δΥ δM
(29)
dx dθ1 dθ2 (−4c2 λ(D1 D2 Υ ) Υ 2 + 3c1 (D1 D2 Υ )Υ + 8γ λ2 Υ 3 + 2c2 λ(D2 Υ )(D1 Υ ) . (30)
1 6
Now the Hamiltonian operators J2 and (c2 D1 D2 ∂ − λγ ∂) are compatible. We have checked the compatibility condition by brute force verifying the supersymmetric Jacobi identity using the computer algebra Reduce [16] and special computer package SUSY2 [17]. Therefore we can apply these operators to the construction of the recursion operator J2 ∂ −1 R2−1 . Correspondingly this operator generates an infinite sequence of conservation laws.
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Z. Popowicz / Physics Letters A 354 (2006) 110–114
However in this supersymmetric structure there is a fundamental difference compared with the classical situation. The classical second Hamiltonian operator of the √ Camassa–Holm equation has a Casimir in the form of dx m and it is a Casimir for the centerless Virasoro algebra also. The extended N = 2 centerless supersymmetric Virasoro algebra does not possesses such a Casimir [18]. Hence it is impossible to construct the supersymmetric analog of the classical hierarchy which contains the Harry Dym type equation. 5. Conclusion In this Letter the extended N = 2 supersymmetric generalization of Camassa–Holm equation was presented. It was accomplished by extending the Fuchssteiner method for the generation of the Camassa–Holm equation to the supersymmetric case. In this framework we used two different recursion operators of the N = 2 supersymmetric α = −2, 4 Korteweg– de Vries equation and constructed two different versions of the supersymmetric Camassa–Holm equation. The bosonic sector of the N = 2, α = 4 supersymmetric Camassa–Holm equation contains the two component generalization of this equation proposed by Chen, Liu and Zhang and the two component Hunter– Saxton equation considered by Aratyn, Gomes and Zimerman. As a byproduct of our analysis we defined the N = 2 supersymmetric Hunter–Saxton equation. We have constructed the
bihamiltonian structure for the supersymmetric N = 2, α = 4 Camassa–Holm equation only. For the α = −2 case, the supersymmetric Korteweg–de Vries has the inverse first Hamiltonian formulation. For that reason we expect that the same may occur in the supersymmetric N = 2, α = −2 version of the Camassa– Holm equation. However this point needs further investigation. References [1] R. Camassa, D. Holm, Phys. Rev. Lett. 71 (1993) 1661. [2] A. Constantin, Proc. R. Soc. London Ser. A: Math. Phys. Eng. Sci. 457 (2001) 953. [3] M. Chen, S.-Q. Liu, Y. Zhang, nlin.SI/0501028. [4] H. Aratyn, J.F. Gomes, A.H. Zimerman, nlin.SI/0507062. [5] A. Hone, J. Phys. A: Math. Gen. 32 (1999) L307. [6] G. Falqui, nlin.SI/0505059. [7] B. Fuchssteiner, Physica D 95 (1996) 229. [8] M. Antonowicz, A. Fordy, Commun. Math. Phys. 124 (1989) 465. [9] J. Hunter, Y. Zheng, Physica D 79 (1994) 361. [10] J. Wess, J. Bagger, Supersymmetry and Supergravity, Princeton Univ. Press, Princeton, NJ, 1990. [11] W. Oevel, Z. Popowicz, Commun. Math. Phys. 139 (1991) 441. [12] Z. Popowicz, Phys. Lett. B 459 (1999) 150. [13] C. Laberge, P. Mathieu, Phys. Lett. B 212 (1988) 718. [14] Z. Popowicz, Phys. Lett. A 174 (1993) 411. [15] A. Sorin, P. Kersten, Phys. Lett. A 300 (2002) 397. [16] A. Hearn, Reduce 3.8 Manual, Santa Monica, CA, USA, 2004. [17] Z. Popowicz, Comput. Phys. Commun. 100 (1997) 277. [18] A. Das, Z. Popowicz, J. Math. Phys. 40 (2005) 082702.
A 2-component or N = 2 supersymmetric Camassa–Holm equation Ziemowit Popowicz Institute of Theoretical Physics, University of Wrocław, pl. M. Borna 9, 50-205 Wrocław, Poland Received 29 September 2005; received in revised form 27 December 2005; accepted 11 January 2006 Available online 19 January 2006 Communicated by A.P. Fordy
Abstract The extended N = 2 supersymmetric Camassa–Holm equation is presented. It is accomplished by formulating the supersymmetric version of the Fuchssteiner method. In this framework we use two supersymmetric recursion operators of the N = 2, α = −2, 4 Korteweg–de Vries equation and construct two different versions of the supersymmetric Camassa–Holm equation. The bosonic sector of N = 2, α = 4 supersymmetric Camassa– Holm equation contains the two component generalization of this equation proposed by Chen, Liu and Zhang and as a special case the two component generalized Hunter–Saxton equation considered by Aratyn, Gomes and Zimerman. As a byproduct of our analysis we defined the N = 2 supersymmetric Hunter–Saxton equation. The bihamiltonian structure is constructed for the supersymmetric N = 2, α = 4 Camassa–Holm equation. © 2006 Elsevier B.V. All rights reserved.
1. Introduction Camassa and Holm introduced [1] in 1993 the integrable non-linear partial differential equation ut − uxxt = −3uux + 2ux uxx + uuxxx 1 2 3 2 = uuxx + ux − u 2 2 x
uxxt = −2ux uxx − uuxxx − ρρx , (1)
which describes a special approximation of shallow water theory and has been extensively studied recently [2–7]. It was shown that this equation possesses the bihamiltonian structure, could be solved using the inverse scattering method and has the so-called peakon solutions. The two component generalization of Camassa–Holm equation mt = −umx − 2mux + ρρx , ρt = −(ρu)x ,
(2)
where m = u − uxx , has been proposed by Chen et al. [3]. This generalization, similarly to the Camassa–Holm equation, is the first negative flow of the AKNS hierarchy and possesses the interesting peakon and multi-kink solutions [3,4,6]. Moreover the E-mail address: [email protected] (Z. Popowicz). 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.01.027
system (2) is connected with the energy dependent Schrödinger spectral problem [3,8]. Quite recently Aratyn et al. [4] showed that the modification of the Schrödinger spectral problem for Eq. (2) leads to two-component Hunter–Saxton equation [9]
ρt = −(ρu)x .
(3)
In this Letter we show that both mentioned generalizations are contained in the bosonic sector of N = 2 extended supersymmetric version of the Camassa–Holm equation. The idea of using extended supersymmetry for the generalization of the soliton equations appeared almost in parallel to the usage of this symmetry in the quantum field theory [10]. The main idea of the supersymmetry is to treat boson and fermion operators equally. In order to get supersymmetric theory we have to add to a system of k bosonic equations kN fermions and k(N − 1) boson fields (k = 1, 2, . . . , N = 1, 2, . . .) in such a way that the final theory becomes supersymmetric invariant. From the soliton point of view we can distinguish two important classes of the supersymmetric equations: the non-extended (N = 1) and extended (N > 1) cases. Consideration of the extended case may imply new bosonic equations whose properties need further investigation. This may be viewed as a bonus, but this extended case is in no way more fundamental than the non-extended one.
Z. Popowicz / Physics Letters A 354 (2006) 110–114
Interestingly enough, some typical supersymmetric effects may occur in the supersymmetrical generalization of the soliton theory, compared to the classical case. We mention two of them: the ambiguity of the roots for the supersymmetric Lax operator [11] and appearance odd Poisson brackets [12]. There are many different methods for supersymmetrization of classical equations. The most popular one is to use the gradation arguments in which to each dependent and independent variables, in a given equation, we associate some weights. We then replace the bosonic fields by superboson fields. For example, for the extended N = 2 supersymmetric case we consider the supersymmetric analog of dependent variable u(x, t), which can be thought of as Φ(x, t, θ1 , θ2 ) = v(x, t) + θ1 ξ1 (x, t) + θ2 ξ2 (x, t) + θ2 θ1 u(x, t)
(4)
where θ1 and θ2 are two anticommuting variables while ξ1 and ξ2 are Grassmann valued functions and v is an additional “new” bosonic field. In the next step using the usual and supersymmetric derivatives D1 = ∂θ1 + θ1 ∂x ,
D2 = ∂θ2 + θ2 ∂x ,
D1 D2 + D2 D1 = ∂
(5)
we consider the most general assumption on the supersymmetric version of the given classical system in such a way as to preserve the given gradation of the supersymmetric equation. In the last step, we assume, that our supersymmetric generalization should possesses some special properties such as the existence a bihamiltonian structure or a Lax operator, for example. However this procedure cannot be applied to the Camassa– Holm equation, because this equation does not preserve gradation with respect to weight. In order to overcome this problem, we supersymmetrize the Fuchssteiner method [7], in which the hereditary operator, responsible for Camassa–Holm hierarchy is constructed out of two different hereditary recursion operators. We introduce the supersymmetry to the theory, considering the supersymmetric analog of these two operators. The Letter is organized as follows. In Section 2 we summarize the Fuchssteiner method. In Section 3, using the supersymmetric recursion operators, which create two different supersymmetric N = 2 Korteweg–de Vries equations, we construct two different supersymmetric versions of Camassa–Holm equation. As a byproduct of our analysis we define the extended N = 2 version of the supersymmetric Hunter–Saxton equation. In the same section we investigate the bosonic sector of this equation. In the last section we describe the bihamiltonian structure of the supersymmetric Camassa–Holm equation. 2. Camassa–Holm equation In order to construct the Camassa–Holm equation, we briefly describe the method used by Fuchssteiner [7]. This method based on the following observation. If we have two different hereditary recursion operators R1 and R2 and if one of them,
111
for example R2 , is invertible then R = R1 R2−1 is also the hereditary recursion operator. Therefore this operator can be used to construct some new integrable hierarchy of equations. Let us present this method for the Camassa–Holm equation. First let us consider the hereditary recursion operator for the Korteweg–de Vries hierarchy R = c∂ 2 + λ ∂u∂ −1 + u , (6) where c and λ are an arbitrary constants. This operator generates the hierarchy of integrable equations in which the first member is ∂u = Rux = cuxxx + 3λuux . (7) ∂t For c = 1 and λ = 2 we have famous Korteweg–de Vries equation. The second recursion operator can be extracted from the first operator shifting the function u → u + γ in recursion operator (6) where γ is a constant. Indeed after shifting the function u in (6) the R operator transforms to R(u + γ ) = R1 + R2 = c1 ∂ 2 + λ ∂u∂ −1 + u + c2 ∂ 2 + λγ ,
(8)
where c = c1 + c2 . It turns out that the recursion operator R = R1 R2−1 generates a new hierarchy of integrable equations n utn = R1 R2−1 ux . (9) Assuming that u = R2 v = c2 vxx + λγ v the first member of the hierarchy is λγ vt + c2 vxxt λc2 2 3λ2 γ 2 = c1 vxx + λc2 vxx v + . vx + v 2 2 x
(10)
Now assuming that c1 = 0, c2 = γ = 1 and λ = −1 the last equation is exactly the Camassa–Holm equation. The second choice γ = 0, c1 = 0, λ = −1 leads us to the Hunter–Saxton equation [9] 1 vtxx = − vxx v + vx2 . (11) 2 x The bihamiltonian structure of Eq. (1) has been constructed in [1] δH1 δH1 3 = c1 ∂ + λ(∂m + m∂) δm δm δH2 δH2 3 = c2 ∂ + λγ ∂ , = J1 (12) δm δm where m = c2 vxx + λγ v and H1 = 12 dx (c2 vxx + λγ v)v, H2 = 12 dx (2c1 λvxx v 2 + 3c1 vxx v + c2 λvx2 v + 3γ λ2 v 3 ). As we see the second Hamiltonian operator is the same as for the Korteweg–de Vries equation and is connected with the Virasoro algebra. For the centerless Virasoro c1 = 0 we algebra √ have additional conserved Hamiltonian 2 dx m which is the Casimir for this algebra as well. Using the first Hamiltonian operator to this quantity we obtain the Harry Dym type equation mt = J2
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3.1. α = 4
of motion 1 mt = J1 √ m
(13)
which reduces to the Harry Dym equation when c2 = 1, γ = 0. 3. N = 2 supersymmetric Camassa–Holm and Hunter–Saxton equation We have four different integrable extended N = 2 supersymmetric extensions of the Korteweg–de Vries equation. Three of them are connected with the supersymmetric Virasoro algebra [13] while the fourth is connected with the odd version of the supersymmetric Virasoro algebra [12]. The first three mentioned equations are
R4 (Φ + γ ) = R1 + R2 ,
δH1 = (D1 D2 ∂ + 2∂Φ + 2Φ∂ − D1 ΦD1 − D2 ΦD2 ) δΦ (α − 1) = −Φxx + 3Φ(D1 D2 Φ) + D1 D2 Φ 2 + αΦ 3 2 x (14) where H1 = 12 dx dθ1 dθ2 (ΦΦ(D1 D2 Φ) + 13 Φ 3 ) and the parameter α can take three different values 1, 4, −2. This system has been extensively studied from different points of view in many papers. The properties of these generalizations are different for different values of α. The Lax operator for α = 4 has two different roots [11]. For the case α = 4 instead of the bihamiltonian formulation, we deal with the inverse first Hamiltonian structure [11], while for the case α = 1 we have the non-standard Lax operator [14] and higher order non-local recursion operator [15]. The hereditary recursion operator for α = 4 and α = −2 has been constructed in [11] −1 R4 = J2 J1,4
= cD1 D2 − λ 2∂Φ∂ −1 − (D1 Φ)D1 ∂ −1 − (D2 Φ)D2 ∂ −1 ,
R(−2) = J2 J1,−2 ,
(16)
where R1 = c1 D1 D2 − λ 2∂Φ∂ −1 − (D1 Φ)D1 ∂ −1 − (D2 Φ)D2 ∂ −1 , R2 = c2 D1 D2 − 2λγ
δH1 Φt = J2 δΦ
(17)
and c = c1 + c2 . Obviously R2 is a hereditary operator and we can consider the hierarchy of equations generated by n Φtn = R1 R2−1 Φx . (18) Assuming that Φ = R2 Υ = c2 (D1 D2 Υ ) − 2λγ Υ we obtain the first member in this hierarchy as c2 (D1 D2 Υt ) − 2λγ Υt = c1 (D1 D2 Υ )x − 2λ(ΦΥ )x + λ(D2 Φ)(D2 Υ ) + λ(D1 Φ)(D1 Υ )
(19)
or more explicitly as c2 (D1 D2 Υt ) − 2λγ Υt = c1 (D1 D2 Υ ) − 2c2 Υ (D1 D2 Υ ) + 4γ λ2 Υ 2 + c2 λ(D2 Υ )(D1 Υ ) x .
(20)
It is our supersymmetric N = 2, α = 4 Camassa–Holm equation. Let us compute the bosonic sector of the previous equation where all odd functions disappear. Assuming that Υ = v + θ2 θ1 u in this sector, we obtain (c2 u − 2γ λv)t = c1 u + 4γ λ2 v 2 − 2c2 λvu x , (−c2 vxx − 2γ λu)t = −c2 λu2 + 2c2 λvxx v − c1 vxx + c2 λvx2 + 8γ λ2 vu x .
J2 = cD1 D2 ∂ − λ(2∂Φ + 2Φ∂ − D1 ΦD1 − D2 ΦD2 ),
(21) Now introducing a new function ρ = c2 u − 2γ λv our system (21) can be rewritten as
J1,4 = ∂, J1,(−2) = cD1 D2 ∂ −1 − λ ∂ −1 D1 ΦD1 ∂ −1 + ∂ −1 D2 ΦD2 ∂ −1 .
Interestingly the supersymmetric Korteweg–de Vries equation for this value of α is not the first member of hierarchy Φtn = R n φ. The first member is Φt = ((D1 D2 Φ) + Φ 2 )x while the second is our supersymmetric Korteweg–de Vries equation. However let us use the supersymmetric recursion operator R4 for the construction of the supersymmetric analogue of the Camassa–Holm recursion operator. In order to find the analogue of the operator R2 let us shift the Φ superfunction to Φ → Φ + γ in R4 obtaining
(15)
Here J2 is the second Hamiltonian operator, which is connected with the extended N = 2 supersymmetric version of the Virasoro algebra, J1,4 is the first Hamiltonian operator for the α = 4 while J1,−2 is an inverse operator of the first Hamiltonian operator for the α = −2 case. In the next we will use these operators in order to adopt the Fuchssteiner method in the supersymmetric case. For these purposes we consider these operators independently.
1 ρt = (c1 ρ + 2c1 γ λv − 2c2 λρv)x , c2 4γ 2 λ2 −c2 vxx − v c2 t 2c1 γ λ 4c1 γ 2 λ2 = ρ+ v − c1 vxx + 2c2 λvvxx c2 c2 12γ 2 λ3 2 λ 2 2 + c2 λvx + v − ρ . c2 c2 x
(22)
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We obtained an even more broader generalization of the Camassa–Holm equation than Eq. (2). Interestingly our Eq. (22) contains Eq. (2) as a special case in which c1 = 0, c22 = −1, λ = 12 , γ = 1. For γ = c1 = 0, λ = 12 , c22 = −1 our system of Eq. (22) reduces to the two component version of Hunter–Saxton (3) and for that reason Eq. (20) when λ = 12 , c1 = 0
(γ u − uxx )t 1 = 2uxx u − 2uxx + u2x − 3γ u2 − 2vxxx vx u − 2vxxx vx 2 2 − 3vxx + 2vxx vx ux − 2γ vxx vu + vx2 uxx + γ vx2 u + γ vx2 − 2vx vux − γ v 2 uxx + γ 2 v 2 u x + vxxxx vx
1 (23) (D2 Υ )(D1 Υ ) − 2Υ (D1 D2 Υ ) x 2 could be considered as the N = 2 supersymmetric Hunter– Saxton equation.
For v = 0 this system reduces to the classical Camassa– Holm equation (1) while for v = 0 it gives us new generalization of Camassa–Holm equation.
(D1 D2 Υ )t =
3.2. α = −2 In contrast to the previous case, now the supersymmetric Korteweg–de Vries equation is a first member of the hierarchy Φt = R(−2) Φx . Similarly to the α = 4 case we can shift the superfunction Φ → Φ + γ in R−2 in order to obtain R2 operator. However then we find that the operator R2 contains the Φ superfunction and therefore will be not considered here. The next choice is to use the R2 operator from the α = 4 case. In this situation we obtain an integro-differential supersymmetric equation which does not reduce in the bosonic sector to the Camassa–Holm equation and for that reason we will not consider such possibility. On the other hand, we can try to choose R2 = γ − ∂ 2 operator, the same as in the classical situation, in order to consider the hierarchy of equations generated by Φt = R(−2) R2−1 Φx
(24)
where now Φ = R2 Υ = γ Υ − Υxx . Assuming that λ = 12 and c = 1 the last equation reads (γ Υ − Υxx )t 1 = 4(D1 D2 Υ )Υxx − 2Υxx + Υxx Υx2 − 4γ (D1 D2 Υ )Υ 2 − γ (D2 Υ )(D1 Υ )(D1 D2 Υ ) − γ Υxx Υ 2 − γ Υx2 Υ − 2γ 3 Υ 3 + γ (D2 Υ )(D1 Υ ) x 1 + (D2 Υxx )(D2 Υ )Υxx − 2(D2 Υxxx )(D1 Υ ) 4 − (D2 Υxx )(D2 Υx )Υx − (D2 Υxx )(D1 Υx )(D1 D2 Υ )
+ 2vxxx vxx u − 3γ vxx vx u.
3.3. α = 1 As we mentioned earlier, the recursion operator for this case, has a higher order non-locality. For that reason, the R2 operator obtained by shifting the Φ superfunction, analogously to the previous case, leads us to a very complicated operator. If we proceed in the same way as in the case α = 4 and use the same operator R2 then we obtain a very complicated system also. For that reason we will not study this case further. 4. Bihamiltonian structure We have constructed this structure for the case α = 4 only. This structure could be obtained from the recursion operator R4 in a similar manner to the classical case [1]. In order to end this let us make the following observation. Notice that the first member in the hierarchy (18) could be rewritten as −1 −1 R 2 Mx Mt = J2 (M)J1,4
= J2 Υ = J2 R2−1
Mt = ∂ where
− (D2 Υ )(D1 Υxx )(D1 D2 Υx ) + 2(D1 Υx )(D1 Υ )Υxxx + γ (D2 Υ )(D1 Υ ) x (D1 D2 Υ ) − 3γ (D2 Υx )(D2 Υ )Υx + (D1 ⇒ D2 , D2 ⇒ −D1 ) . (25)
H2 =
(γ v − vxx )t =
1 4vxx u − 2vxx + vxx vx2 − γ vxx v 2 − γ vx2 v 2 + γ v 3 − 4γ vu x , (26)
δH1 δH1 = J2 , δΥ δM
(28)
where M = R2 Υ = c2 (D1 D2 Υ − λγ Υ ) and H1 = 12 × dx dθ1 dθ2 ((R2 Υ )Υ ). As we see this is the second Hamiltonian structure and is generated by the same supersymmetric operator which is responsible for the second Hamiltonian structure of the Korteweg–de Vries equation. The first Hamiltonian structure follows from the following observation. Notice that Eq. (20) could be rewritten also as
+ (D2 Υxx )(D1 Υ )(D1 D2 Υx )
It is our supersymmetric N = 2, α = −2 Camassa–Holm equation. The bosonic sector in which Υ = v + θ1 θ2 u reads
(27)
δH2 δH2 δH2 = ∂R2 R2−1 = ∂R2 , δΥ δΥ δM
(29)
dx dθ1 dθ2 (−4c2 λ(D1 D2 Υ ) Υ 2 + 3c1 (D1 D2 Υ )Υ + 8γ λ2 Υ 3 + 2c2 λ(D2 Υ )(D1 Υ ) . (30)
1 6
Now the Hamiltonian operators J2 and (c2 D1 D2 ∂ − λγ ∂) are compatible. We have checked the compatibility condition by brute force verifying the supersymmetric Jacobi identity using the computer algebra Reduce [16] and special computer package SUSY2 [17]. Therefore we can apply these operators to the construction of the recursion operator J2 ∂ −1 R2−1 . Correspondingly this operator generates an infinite sequence of conservation laws.
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However in this supersymmetric structure there is a fundamental difference compared with the classical situation. The classical second Hamiltonian operator of the √ Camassa–Holm equation has a Casimir in the form of dx m and it is a Casimir for the centerless Virasoro algebra also. The extended N = 2 centerless supersymmetric Virasoro algebra does not possesses such a Casimir [18]. Hence it is impossible to construct the supersymmetric analog of the classical hierarchy which contains the Harry Dym type equation. 5. Conclusion In this Letter the extended N = 2 supersymmetric generalization of Camassa–Holm equation was presented. It was accomplished by extending the Fuchssteiner method for the generation of the Camassa–Holm equation to the supersymmetric case. In this framework we used two different recursion operators of the N = 2 supersymmetric α = −2, 4 Korteweg– de Vries equation and constructed two different versions of the supersymmetric Camassa–Holm equation. The bosonic sector of the N = 2, α = 4 supersymmetric Camassa–Holm equation contains the two component generalization of this equation proposed by Chen, Liu and Zhang and the two component Hunter– Saxton equation considered by Aratyn, Gomes and Zimerman. As a byproduct of our analysis we defined the N = 2 supersymmetric Hunter–Saxton equation. We have constructed the
bihamiltonian structure for the supersymmetric N = 2, α = 4 Camassa–Holm equation only. For the α = −2 case, the supersymmetric Korteweg–de Vries has the inverse first Hamiltonian formulation. For that reason we expect that the same may occur in the supersymmetric N = 2, α = −2 version of the Camassa– Holm equation. However this point needs further investigation. References [1] R. Camassa, D. Holm, Phys. Rev. Lett. 71 (1993) 1661. [2] A. Constantin, Proc. R. Soc. London Ser. A: Math. Phys. Eng. Sci. 457 (2001) 953. [3] M. Chen, S.-Q. Liu, Y. Zhang, nlin.SI/0501028. [4] H. Aratyn, J.F. Gomes, A.H. Zimerman, nlin.SI/0507062. [5] A. Hone, J. Phys. A: Math. Gen. 32 (1999) L307. [6] G. Falqui, nlin.SI/0505059. [7] B. Fuchssteiner, Physica D 95 (1996) 229. [8] M. Antonowicz, A. Fordy, Commun. Math. Phys. 124 (1989) 465. [9] J. Hunter, Y. Zheng, Physica D 79 (1994) 361. [10] J. Wess, J. Bagger, Supersymmetry and Supergravity, Princeton Univ. Press, Princeton, NJ, 1990. [11] W. Oevel, Z. Popowicz, Commun. Math. Phys. 139 (1991) 441. [12] Z. Popowicz, Phys. Lett. B 459 (1999) 150. [13] C. Laberge, P. Mathieu, Phys. Lett. B 212 (1988) 718. [14] Z. Popowicz, Phys. Lett. A 174 (1993) 411. [15] A. Sorin, P. Kersten, Phys. Lett. A 300 (2002) 397. [16] A. Hearn, Reduce 3.8 Manual, Santa Monica, CA, USA, 2004. [17] Z. Popowicz, Comput. Phys. Commun. 100 (1997) 277. [18] A. Das, Z. Popowicz, J. Math. Phys. 40 (2005) 082702.