A supersymmetric Sawada-Kotera equation
aa r X i v : . [ n li n . S I] D ec A supersymmetric Sawada-Kotera equation
Kai Tian and Q. P. Liu ∗ Department of Mathematics,China University of Mining and Technology,Beijing 100083, P.R. China
Abstract
A new supersymmetric equation is proposed for the Sawada-Kotera equation. Theintegrability of this equation is shown by the existence of Lax representation and infiniteconserved quantities and a recursion operator.
PACS : 02.30.Ik; 05.45.Yv
Key Words : integrability; Lax representation; recursion operator; supersymmetry
The following fifth-order evolution equation u t + u xxxxx + 5 uu xxx + 5 u x u xx + 5 u u x = 0 (1)is a well-known system in soliton theory. It was proposed by Sawada and Kotera, alsoby Caudrey, Dodd and Gibbon independently, more than thirty years ago [1][2], so it isreferred as Sawada-Kotera (SK) equation or Caudrey-Dodd-Gibbon-Sawada-Kotera equationin literature. Now there are a large number of papers about it and thus its various propertiesare established. For example, its B¨acklund transformation and Lax representation were givenin [3][4], its bi-Hamiltonian structure was worked out by Fuchssteiner and Oevel [5], and aDarboux transformation was derived for this system [6][7], to mention just a few (see also[8][9]).Soliton equations or integrable systems have supersymmetric analogues. Indeed, manyequations such as KdV, KP, and NLS equations were embedded into their supersymmetriccounterparts and it turns out that these supersymmetric systems have also remarkable prop-erties. Thus, it is interesting to work out supersymmetric extensions for a given integrableequation. ∗ Email: [email protected] Tel: 86 10 62339015 φ t + φ xxxxx + (cid:2) D φ ) φ xx + 5( D φ ) xx φ + 15( D φ ) φ (cid:3) x = 0where φ = φ ( x, t, θ ) is a fermioic super variable depending on usual temporal variable t and super spatial variables x and θ . D = ∂ θ + θ∂ x is the super derivative. Rewriting theequation in components, it is easy to see that this system does reduce to the SK equationwhen the fermionic variable is absent. However, apart from the fact that the system can beput into a Hirota’s bilinear form, not much is known for its integrability. We will give analternative supersymmetric extension for the SK equation and will show the evidence for theintegrability of our system.The paper is organized as follows. In section two, by considering a Lax operator andits factorization, we construct the supersymmetric SK (sSK) equation. In section three, wewill show that our sSK equation has an interesting property, namely, it does not have theusual bosonic conserved quantities since those, resulted from the super residues of a frac-tional power for Lax operator, are trivial. Evermore, there are infinite fermionic conservedquantities. In the section four, we construct a recursion operator for our sSK equation. Lastsection contains a brief summary of our new findings and presents some interesting openproblems. The main purpose of this section is to construct a supersymmetric analogy for the SKequation. To this end, we will work with the algebra of super-pseudodifferential operatorson a (1 |
1) superspace with coordinates ( x, θ ). We start with the following general Laxoperator L = ∂ x + Ψ ∂ x D + U ∂ x + Φ D + V. (2)By the standard fractional power method [12], we have an integrable hierarchy of equationsgiven by ∂L∂t n = [( L n ) + , L ] (3)where we are using the standard notations: [ A, B ] = AB − ( − | A || B | BA is the supercom-mutator and the subscript + means taking the projection to the differential part for a givensuper-pseudodifferential operator. It is remarked that the system (3) is a kind of even ordergeneralized SKdV hierarchies considered in [11].In the following, we will consider the particular t flow. Our interest here is to find aminimal supersymmetric extension for the SK equation, so we have to do reductions for thegeneral Lax operator (2). To this end, we impose L + L ∗ = 02here ∗ means taking formal adjoint. Then we findΨ = 0 , V = 12 ( U x − ( D Φ))that is L = ∂ x + U ∂ x + Φ D + 12 ( U x − ( D Φ))a Lax operator with two field variables. In this case, we take B = 9( L ) + , namely B = 9 ∂ x + 15 U ∂ x + 15Φ D ∂ x + 15( U x + V ) ∂ x +15Φ x D ∂ x + (10 U xx + 15 V x + 5 U ) ∂ x +10(Φ xx + Φ U ) D + 10 V xx + 10 U V + 5Φ( D U )for convenience. Then, the flow of equations, resulted from ∂L∂t = [ B, L ]reads as U t + U xxxxx + 5 (cid:18) U U xx + 34 U x + 13 U + Φ x ( D U ) + 12 Φ( D U x ) + 12 ΦΦ x −
34 ( D Φ) (cid:19) x = 0 (4a)Φ t + Φ xxxxx + 5 (cid:18) U Φ xx + 12 U xx Φ + 12 U x Φ x + U Φ + 12 Φ( D Φ x ) −
12 ( D Φ)Φ x (cid:19) x = 0(4b)where we identify t with t for simplicity. Remarks :1. It is interesting to note that the above system has an obvious reduction. Indeed, settingΦ = 0, we will have the standard Kaup-Kupershimdt (KK) equation. Therefore, wemay consider it as a supersymmetric extension of the KK equation.2. The coupled system (4a-4b) admits the following simple Hamiltonian structure (cid:18) U t Φ t (cid:19) = (cid:18) ∂ x ∂ x (cid:19) δ H where the Hamiltonian is given by H = Z (cid:20)
54 Φ( D φ ) − ( D U x )( D Φ xx ) −
53 Φ U −
54 ( D U x )( D U )Φ+ 54 ( D U ) U x ( D Φ) + 5( D U ) U ( D Φ x ) + 52 ( D U )Φ x Φ (cid:21) d x d θ.
3t this point, it is not clear how this system (4a-4b) is related to the SK equation. Tofind a supersymmetric SK equation from it, we now consider the factorization of the Laxoperator in the following way L = ∂ x + U ∂ x + Φ D + 12 ( U x − ( D Φ))= ( D + W D + Υ)( D − D W + Υ) , (5)which gives us a Miura-type transformation U = − W x − W + ( D Υ) , Φ = − Υ x − W, and the modified system corresponding to this factorization is given by W t + W xxxxx + 5 W xxx ( D Υ) − W xxx W x − W xxx W − W xx + 10 W xx ( D Υ x ) − W xx W x W − W xx W ( D Υ) − W x − W x ( D Υ) + 5 W x ( D Υ xx ) + 5 W x ( D Υ) + 5 W x W − W x W ( D Υ x ) + 10 W ( D Υ x )( D Υ) − x Υ W x + 5( D W xx )Υ x + 5( D W x )Υ xx + 5( D W x )Υ W x + 10( D W x )Υ W − D W )Υ x W x + 10( D W )Υ x W + 10( D W )Υ( D Υ x ) − D W )Υ W xx + 30( D W )Υ W x W = 0 , Υ t + Υ xxxxx + 5Υ xxx ( D Υ) − xxx W + 5Υ xx ( D Υ x ) + 5Υ xx W xx − xx W x W + 5Υ xx W ( D Υ) + 5Υ x ( D Υ) + 5Υ x W xxx − x W xx W − x W x + 5Υ x W x ( D Υ)+ 10Υ x W x W + 5Υ x W − x W ( D Υ) + 5Υ x W ( D Υ x ) − W xxx W − W xx W x + 10Υ W xx W + 30Υ W x W + 20Υ W x W − W x W ( D Υ) − W ( D Υ x ) − D W x )Υ x Υ − D W )Υ xx Υ − D W )Υ x Υ W = 0 . Although this modification does indeed have a complicated form, the remarkable fact isthat it allows a simple reduction. What we need to do is simply putting W to zero, namely W = 0 , Υ = φ. In this case, we have φ t + φ xxxxx + 5 φ xxx ( Dφ ) + 5 φ xx ( Dφ x ) + 5 φ x ( Dφ ) = 0 (6)this equation is our supersymmetric SK equation. To see the connection with the originalSK equation (1), we let φ = θu ( x, t ) + ξ ( x, t ) and write the equation (6) out in components u t + u xxxxx + 5 uu xxx + 5 u x u xx + 5 u u x − ξ xxx ξ x = 0 , (7a) ξ t + ξ xxxxx + 5 uξ xxx + 5 u x ξ xx + 5 u ξ x = 0 . (7b)It is now obvious that the system reduces to the SK equation when ξ = 0. Therefore, oursystem (6) does qualify as a supersymmetric SK equation.4ur system (6) is integrable in the sense that it has a Lax representation. In fact, thefactorization (5) implies that the reduced Lax operator has the following appealing form L = ( D + φ )( D + φ ) (8)or L = ∂ x + ( D φ ) ∂ x − φ x D + ( D φ x ) In general, an integrable system has infinite number of conserved quantities. Since the sSKequation has a simple Lax operator (8), it is natural to take advantage of the fractional powermethod of Gel’fand and Dickey [12] to find conserved quantities. In the present situation,we have to work with the super residue of a pesudodifferential operator.The obvious choice in this case is to consider the operators L n and their super residues.Then, we have the Proposition 1 sres L n ∈ Im D . where sres means taking the super residue of a super pseudodifferential operator. Proof: As observed already in [13], there exists a unique odd operator Λ = D + O(1), whosecoefficients are all differential polynomials of φ , such that( D + φ ) = Λ , thus, the Lax operator (8) is written as L = ( D + φ )( D + φ ) = Λ . From it we havesres L n = sresΛ n = 12 sres { Λ n − Λ + ΛΛ n − } = 12 sres[Λ k − , Λ] ∈ Im D . This completes the proof.
Remark:
The triviality of L n implies that the Lax operator could not generate any Hamil-tonian structures for the equation (6).To find nontrivial conserved quantities, we now turn to L n rather than L n . It is easy toprove that ∂∂t L = [9( L ) + , L ]thus ∂∂t L n = [9( L ) + , L n ] . L n is conserved.By direct calculation, we obtain the first two nontrivial conserved quantities Z sres L d x d θ = − Z [2( D φ xx ) + ( D φ ) − φ x φ ]d x d θ Z sres L d x d θ = − Z [6( D φ xxxx ) + 18( D φ xx )( D φ ) + 9( D φ x ) + 4( D φ ) − φ xxx φ + 6 φ xx φ x − φ x φ ( D φ )]d x d θ Remarks:
1. What is remarkable is that the conserved quantities found in this way, unlike thesupersymmetric KdV case [15][14], are local.2. All those conserved quantities are fermioic. To our knowledge, this is the first super-symmetric integrable system whose only conserved quantities are fermioic.
An integrable system often appears as a particular flow of hierarchy equations and an im-portant ingredient in this aspect is the existence of recursion operators. In this section, wededuce the recursion operator for the sSK equation (6) following the method proposed in[16]. We first notice that the sSK hierarchy can be written as ∂∂t n L = [( L n ) + , L ] (9)where L is given by (8). It is easy to see that the flow equations are nontrivial only if n isan integer satisfying n = 0 mod 3 and n = 1 mod 2 . Therefore the next flow which is achieved by applying recursion operator to (9) should be ∂∂t n +6 L = [( L n +63 ) + , L ] . (10)But h ( L n +63 ) + , L i = h(cid:0) L ( L n ) + + L ( L n ) − (cid:1) + , L i = (cid:2) L ( L n ) + , L (cid:3) + h(cid:0) L ( L n ) − (cid:1) + , L i = L (cid:2) ( L n ) + , L (cid:3) + [ R n , L ]= L ∂∂t n L + [ R n , L ]6here R n = (cid:0) L ( L n ) − (cid:1) + (11)is a differential operator of O ( ∂ x D ), that is, R n = ( α∂ x + β∂ x + γ∂ x + δ∂ x + ξ∂ x + η ) D + a∂ x + b∂ x + c∂ x + d∂ x + e∂ x + f. Therefore, ∂∂t n +6 L = L ∂∂t n L + [ R n , L ] . (12)Next we may determine the coefficients in R n . Using (12), we obtain a = 13 ( D − φ n ) , b = 2( D φ n ) ,c = 449 ( D φ n,x ) + 53 ( D φ )( D − φ n ) + 49 ( ∂ − x φ x φ n ) ,d = 559 ( D φ n,xx ) + 199 ( D φ )( D φ n ) + 59 φ x φ n + 109 ( D φ x )( D − φ n ) e = 127 { D φ n,xxx ) + 74( D φ )( D φ n,x ) , − φ x φ n,x + 79( D φ x )( D φ n )+ 27 φ xx φ n + [23( D φ xx ) + 4( D φ ) ]( D − φ n ) + 16( D φ )( ∂ − x φ x φ n )+ 2 D − [( φ xxx + φ x ( D φ ))( D − φ n ) − D φ )( D − φ x φ n ) − φ x ( ∂ − x φ x φ n ) + 2 D − ( φ xxx φ n + 2 φ x ( D φ ) φ n )] } ,f = 127 { D φ n,xxxx ) + 32( D φ )( D φ n,xx ) − φ x φ n,xx + 54( D φ x )( D φ n,x )+ 16 φ xx φ n,x + [30( D φ xx ) + 4( D φ ) ]( D φ n ) + [8 φ xxx + 4 φ x ( D φ )] φ n + [10( D φ xxx ) + 10( D φ x )( D φ )]( D − φ n ) − φ x ( D − φ x φ n )+ 12( D φ x )( ∂ − x φ x φ n ) } ,α = 0 , β = − φ n , γ = 53 φ n,x ,δ = − { φ n,xx + 5 φ n ( D φ ) + 5 φ x ( D − φ n ) − D − φ x φ n ) } ,ξ = − { φ n,xxx + 16 φ x ( D φ n ) + 3 φ n ( D φ x ) + 14 φ n,x ( D φ ) + 5 φ xx ( D − φ n ) } ,η = − { φ n,xxxx + 32( D φ ) φ n,xx + 28 φ x ( D φ n,x ) + 26( D φ x ) φ n,x + 28 φ xx ( D φ n ) + [2( D φ xx ) + 4( D φ ) ] φ n + [10 φ xxx + 10 φ x ( D φ )]( D − φ n ) − D φ )( D − φ x φ n ) + 12 φ x ( ∂ − x φ x φ n ) − D − [ φ xxx φ n + 2 φ x ( D φ ) φ n ] } . where we used the shorthand notation φ n = ∂φ/∂t n .7inally, we have the recursion operator R = ∂ x + 6( D φ ) ∂ x + 9( D φ x ) ∂ x + 6 φ xx ∂ x D + { D φ xx ) + 9( D φ ) } ∂ x + { φ xxx + 12 φ x ( D φ ) } ∂ x D + { ( D φ xxx ) + 9( D φ x )( D Φ) } ∂ x + { φ xxxx + 12 φ xx ( D φ ) + 6 φ x ( D φ x ) }D + { D φ xx )( D φ ) + 4( D φ ) − φ xx φ x } + { φ xxxxx + 5 φ xxx ( D φ ) + 5 φ xx ( D φ x ) + 2 φ x ( D φ xx ) + 6 φ x ( D φ ) }D − − { D φ xx ) + 2( D φ ) }D − φ x − φ x ( D φ ) ∂ − x φ x − D φ ) D − [ φ xxx + 2 φ x ( D φ )] − φ x D − { ( φ xxx + φ x ( D φ )) D − − D φ ) D − φ x − φ x ∂ − x φ x + 2 D − [ φ xxx + 2 φ x ( D φ )] } Remark:
When calculating the coefficients of R n , one should solve a system of differentialequations. Due to nonlocality (those underlined terms), there is certain ambiguity and toavoid it, we used the t -flow φ t = φ xxxxxxx + 7 φ xxxxx ( D φ ) + 14 φ xxxx ( D φ x ) + 14 φ xxx ( D φ xx )+14 φ xxx ( D φ ) + 7 φ xx ( D φ xxx ) + 28 φ xx ( D φ x )( D φ )+14 φ x ( D φ xx )( D φ ) + 7 φ x ( D φ x ) + 283 φ x ( D φ ) . Summarizing, we find a supersymmetric SK equation which has Lax representation. We alsoobtain infinite conserved quantities and a recursion operator for this new proposed system.These imply that the system is integrable. It is interesting to establish other properties forit, such as B¨acklund transformation, Hirota bilinear form, etc..
Acknowledgements
The calculations were done with the assistance of SUSY2 packageof Popowicz [17]. We would like to thank him for helpful discussion about his package.The comments of anonymous referee has been very useful. The work is supported in partby National Natural Science Foundation of China under the grant numbers 10671206 and10731080.
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