Supersymmetric Fifth Order Equations

Supersymmetric Fifth Order Equations Q. P. Liu Department of Mathematics China University of Mining and Technology (Beijing) 100083 Beijing July 20, 2009 a joint work with Kai TIAN

Outline SUSY: basics 5th order SUSY integrable systems A SUSY Sawada-Kotera equation A Novel SUSY 5th KdV

SUSY Super: anti-commutitive or fermionic variables theoretical physics: bosons & fermions

SUSY Super: SUSY: anti-commutitive or fermionic variables theoretical physics: bosons & fermions supersymmetry

SUSY Super: SUSY: Super or SUSY Integrable system: anti-commutitive or fermionic variables theoretical physics: bosons & fermions supersymmetry a coupled system

SUSY Super: SUSY: Super or SUSY Integrable system: anti-commutitive or fermionic variables theoretical physics: bosons & fermions supersymmetry a coupled system Two types of super extensions exist in literatures: fermionic or supersymmetric Martin (1959), Casalbuoni (1976), Kupershmidt (1984), Gurses,... Mainin & Radul, Mathieu, Popowicz, Aratyn,...

SUSY Super: SUSY: Super or SUSY Integrable system: anti-commutitive or fermionic variables theoretical physics: bosons & fermions supersymmetry a coupled system Two types of super extensions exist in literatures: fermionic or supersymmetric Martin (1959), Casalbuoni (1976), Kupershmidt (1984), Gurses,... Mainin & Radul, Mathieu, Popowicz, Aratyn,... Direct way to SUSY systems: Independent variables: Dependent variables x (x, θ) fermionic or bosonic functions

Example ( Kupershmidt: Phys. Letts. A (1984)) u t = 6uu x u xxx + 3ξξ xx, ξ t = 4ξ xxx + 3u x ξ + 6uξ x

Example ( Kupershmidt: Phys. Letts. A (1984)) u t = 6uu x u xxx + 3ξξ xx, ξ t = 4ξ xxx + 3u x ξ + 6uξ x L kuper = 2 + u ξ 1 ξ

Example ( Kupershmidt: Phys. Letts. A (1984)) u t = 6uu x u xxx + 3ξξ xx, ξ t = 4ξ xxx + 3u x ξ + 6uξ x L kuper = 2 + u ξ 1 ξ Example ( Manin and Radul: Commun. Math. Phys. (1985)) u t = 6uu x u xxx 3ξξ xx ξ t = ξ xxx + 3(uξ) x

Example ( Kupershmidt: Phys. Letts. A (1984)) u t = 6uu x u xxx + 3ξξ xx, ξ t = 4ξ xxx + 3u x ξ + 6uξ x L kuper = 2 + u ξ 1 ξ Example ( Manin and Radul: Commun. Math. Phys. (1985)) u t = 6uu x u xxx 3ξξ xx ξ t = ξ xxx + 3(uξ) x L MR = 2 ΦD D = θ + θ x, Φ = ξ + θu

MR-KdV has a compact Q. P. form: Liu SUSY Sawada-Kotera Example ( Kupershmidt: Phys. Letts. A (1984)) u t = 6uu x u xxx + 3ξξ xx, ξ t = 4ξ xxx + 3u x ξ + 6uξ x L kuper = 2 + u ξ 1 ξ Example ( Manin and Radul: Commun. Math. Phys. (1985)) u t = 6uu x u xxx 3ξξ xx ξ t = ξ xxx + 3(uξ) x L MR = 2 ΦD D = θ + θ x, Φ = ξ + θu

Soliton Equations and their SUSY Counterparts Equation Authors Journals (Years) sine-gordon Chaichian & Kulish PLB (1978) Ferrara PLB (1977/8)

Soliton Equations and their SUSY Counterparts Equation Authors Journals (Years) sine-gordon Chaichian & Kulish PLB (1978) Ferrara PLB (1977/8) KP Manin & Radul CMP(1985) Mulase, Rabin, Ueno,... Invent. Math.(1988)

Soliton Equations and their SUSY Counterparts Equation Authors Journals (Years) sine-gordon Chaichian & Kulish PLB (1978) Ferrara PLB (1977/8) KP Manin & Radul CMP(1985) Mulase, Rabin, Ueno,... Invent. Math.(1988) Toda Olshanetsky; CMP (1983); Ikeda LMP (1987) MKdV Mathieu JMP(1988) Sasaki & Yamanaka PTP(1988)

Soliton Equations and their SUSY Counterparts Equation Authors Journals (Years) sine-gordon Chaichian & Kulish PLB (1978) Ferrara PLB (1977/8) KP Manin & Radul CMP(1985) Mulase, Rabin, Ueno,... Invent. Math.(1988) Toda Olshanetsky; CMP (1983); Ikeda LMP (1987) MKdV Mathieu JMP(1988) Sasaki & Yamanaka PTP(1988) NLS Roelofs & Kersten JMP (1992)

Soliton Equations and their SUSY Counterparts Equation Authors Journals (Years) sine-gordon Chaichian & Kulish PLB (1978) Ferrara PLB (1977/8) KP Manin & Radul CMP(1985) Mulase, Rabin, Ueno,... Invent. Math.(1988) Toda Olshanetsky; CMP (1983); Ikeda LMP (1987) MKdV Mathieu JMP(1988) Sasaki & Yamanaka PTP(1988) NLS Roelofs & Kersten JMP (1992) HD Liu JPhys A 28 (1995) Brunelli, Das & Popowicz JMP (2003)

Soliton Equations and their SUSY Counterparts Equation Authors Journals (Years) sine-gordon Chaichian & Kulish PLB (1978) Ferrara PLB (1977/8) KP Manin & Radul CMP(1985) Mulase, Rabin, Ueno,... Invent. Math.(1988) Toda Olshanetsky; CMP (1983); Ikeda LMP (1987) MKdV Mathieu JMP(1988) Sasaki & Yamanaka PTP(1988) NLS Roelofs & Kersten JMP (1992) HD Liu JPhys A 28 (1995) Brunelli, Das & Popowicz JMP (2003) cbq Brunelli & Das PLB (1994)

Soliton Equations and their SUSY Counterparts Equation Authors Journals (Years) sine-gordon Chaichian & Kulish PLB (1978) Ferrara PLB (1977/8) KP Manin & Radul CMP(1985) Mulase, Rabin, Ueno,... Invent. Math.(1988) Toda Olshanetsky; CMP (1983); Ikeda LMP (1987) MKdV Mathieu JMP(1988) Sasaki & Yamanaka PTP(1988) NLS Roelofs & Kersten JMP (1992) HD Liu JPhys A 28 (1995) Brunelli, Das & Popowicz JMP (2003) cbq Brunelli & Das PLB (1994) SK Tian & Liu 2008

Fifth order evolution equations We consider u t + u xxxxx + auu xx + bu x u xx + cu 2 u x = 0 the general homogenous equation of degree 7.

Fifth order evolution equations We consider u t + u xxxxx + auu xx + bu x u xx + cu 2 u x = 0 the general homogenous equation of degree 7. Question: Is it integrable?

Fifth order evolution equations We consider u t + u xxxxx + auu xx + bu x u xx + cu 2 u x = 0 the general homogenous equation of degree 7. Question: Is it integrable? Yes, but only for three cases 5th KdV: u t + u xxxxx + 10uu xxx + 25u x u xx + 20u 2 u x = 0 Sawada-Kotera or Caudrey-Dodd-Gibbon u t + u xxxxx + 5uu xxx + 5u x u xx + 5u 2 u x = 0 Kaup-Kupershimdt: u t + u xxxxx + 10u x u xx + 30u 2 u x = 0 Fujimoto & Watanabe: Math. Japonica 28 (1983) Harada & Oishi: J Phys. Soc. Japan 54 (1985)

Question: susy 5th order equations? susy SK? susy KK?

Question: susy 5th order equations? susy SK? susy KK? susy 5th KdV: next flow of susy KdV φ t +φ xxxxx + [αφ xx (Dφ) + αφ x Dφ x + αφ(dφ) xx + 25 ] α2 φ(dφ) 2 = 0 Oevel & Popowicz CMP (1993) x

Question: susy 5th order equations? susy SK? susy KK? susy 5th KdV: next flow of susy KdV φ t +φ xxxxx + [αφ xx (Dφ) + αφ x Dφ x + αφ(dφ) xx + 25 ] α2 φ(dφ) 2 = 0 Oevel & Popowicz CMP (1993) susy SK? x

Question: susy 5th order equations? susy SK? susy KK? susy 5th KdV: next flow of susy KdV φ t +φ xxxxx + [αφ xx (Dφ) + αφ x Dφ x + αφ(dφ) xx + 25 ] α2 φ(dφ) 2 = 0 Oevel & Popowicz CMP (1993) susy SK? x φ t + φ xxxxx + [ 10(Dφ)φ xx + 5(Dφ) xx φ + 15(Dφ) 2 φ ] x = 0 φ = φ(x, t, θ), a fermioic variable, D = θ + θ x Castera: Nonlinearity 13 (2000)

Question: susy 5th order equations? susy SK? susy KK? susy 5th KdV: next flow of susy KdV φ t +φ xxxxx + [αφ xx (Dφ) + αφ x Dφ x + αφ(dφ) xx + 25 ] α2 φ(dφ) 2 = 0 Oevel & Popowicz CMP (1993) susy SK? x φ t + φ xxxxx + [ 10(Dφ)φ xx + 5(Dφ) xx φ + 15(Dφ) 2 φ ] x = 0 φ = φ(x, t, θ), a fermioic variable, D = θ + θ x Castera: Nonlinearity 13 (2000) with S x (D t + D 5 x )τ τ = 0 φ = 2D(ln τ) x where S x is the super Hirota derivative McArthur & Yung (1993) S x f g = (Df )g ( 1) f f (Dg)

Symmetry Approach We form the following equation Φ t = Φ xxxxx + α 1 Φ xxx (DΦ) + α 2 Φ xx (DΦ x ) + α 3 Φ x (DΦ xx ) +α 4 Φ x (DΦ) 2 + α 5 Φ(DΦ xxx ) + α 6 Φ(DΦ x )(DΦ) which is the most general equation with the degree where is fermionic field deg = 13 2 Φ(x, t, θ) = ξ(x, t) + θu(x, t)

Symmetry Approach We form the following equation Φ t = Φ xxxxx + α 1 Φ xxx (DΦ) + α 2 Φ xx (DΦ x ) + α 3 Φ x (DΦ xx ) +α 4 Φ x (DΦ) 2 + α 5 Φ(DΦ xxx ) + α 6 Φ(DΦ x )(DΦ) which is the most general equation with the degree where is fermionic field Question: integrability? deg = 13 2 Φ(x, t, θ) = ξ(x, t) + θu(x, t)

1 2 Φ t = Φ xxxxx + αφ xxx (DΦ) + 2αΦ xx (DΦ x ) + 2αΦ x (DΦ xx ) + 2 5 α2 Φ x (DΦ) 2 + αφ(dφ xxx ) + 4 5 α2 Φ(DΦ x )(DΦ) Φ t = Φ xxxxx + αφ xxx (DΦ) + αφ xx (DΦ x ) + αφ x (DΦ xx ) + 3 10 α2 Φ x (DΦ) 2 3 Φ t = Φ xxxxx + αφ xxx (DΦ) + αφ x (DΦ xx ) + 1 5 α2 Φ x (DΦ) 2 4 Φ t = Φ xxxxx + αφ xxx (DΦ) + 3 2 αφ xx(dφ x ) + αφ x (DΦ xx ) + 1 5 α2 Φ x (DΦ) 2 5 Φ t = Φ xxxxx + αφ xxx (DΦ) + αφ xx (DΦ x ) + 1 5 α2 Φ x (DΦ) 2 6 Φ t = Φ xxxxx + 2αΦ xxx (DΦ) + 3αΦ xx (DΦ x ) + αφ x (DΦ xx ) + 3 5 α2 Φ x (DΦ) 2 + 3 5 α2 Φ(DΦ x )(DΦ)

equations in components 2nd equation: { u t = u xxxxx + αu xxx + 2αu x u xx + 3α2 10 u2 u x ξ t = ξ xxxxx + αuξ xxx + αu x ξ xx + αu xx ξ x + 3α2 10 u2 ξ x 3rd one: u t = u xxxxx + αuu xxx + αu x u xx + α2 5 u2 u x ξ t = ξ xxxxx + αuξ xxx + αu xx ξ x + α2 5 u2 ξ x 4th one: { u t = u xxxxx + αuu xxx + 5α 2 u xu xx + α2 5 u2 u x ξ t = ξ xxxxx + αuξ xxx + 3α 2 u xξ xx + αu xx ξ x + α2 5 u2 ξ x

For last two, we have u t = u xxxxx + αuu xxx + αu x u xx + α2 5 u2 u x αξ xxx ξ x ξ t = ξ xxxxx + αuξ xxx + αu x ξ xx + α2 5 u2 ξ x susy SK u t = u xxxxx + 2αuu xxx + 4αu x u xx + 6α2 5 u2 u x αξ xxx ξ x + 3α2 5 uξ xxξ + 3α2 5 u xξ x ξ ξ t = ξ xxxxx + 2αuξ xxx + 3αu x ξ xx + αu xx ξ x + 3α2 5 u2 ξ x + 3α2 5 uu xξ a susy 5th KdV

SUSY SK L = 3 x + Ψ x D + U x + ΦD + V.

SUSY SK L = 3 x + Ψ x D + U x + ΦD + V. Following Gel fand and Dickey, we have L t n = [(L n 3 )+, L] where [A, B] = AB ( 1) A B BA. Not surprising: a special case of Manin & Radul (Commun. Math. Phys. (1985)). We consider: n = 5 and t 5 = t. L + L = 0

Then we find Ψ = 0, V = 1 2 (U x (DΦ)) that is L = 3 x + U x + ΦD + 1 2 (U x (DΦ)) In this case, we take B = 9(L 5 3 ) +, namely B = 9 5 x + 15U 3 x + 15ΦD 2 x + 15(U x + V ) 2 x for convenience. +15Φ x D x + (10U xx + 15V x + 5U 2 ) x +10(Φ xx + ΦU)D + 10V xx + 10UV + 5Φ(DU)

Then, the flow of equations, resulted from reads as L t = [B, L] U t + U xxxxx + 5 (UU xx + 34 U2x + 13 U3 + Φ x (DU) + 12 Φ(DUx ) + 12 ΦΦx 34 ) (DΦ)2 = 0 x Φ t + Φ xxxxx + 5 (UΦ xx + 12 Uxx Φ + 12 Ux Φx + U2 Φ + 12 Φ(DΦx ) 12 ) (DΦ)Φx = 0 x

Then, the flow of equations, resulted from reads as L t = [B, L] U t + U xxxxx + 5 (UU xx + 34 U2x + 13 U3 + Φ x (DU) + 12 Φ(DUx ) + 12 ΦΦx 34 ) (DΦ)2 = 0 x Φ t + Φ xxxxx + 5 (UΦ xx + 12 Uxx Φ + 12 Ux Φx + U2 Φ + 12 Φ(DΦx ) 12 ) (DΦ)Φx = 0 x setting Φ = 0, we will have the standard Kaup-Kupershimdt Hamiltonian system: where the Hamiltonian is given by ( ) ( Ut 0 x = Φ t x 0 ) δh H = [ 5 4 Φ(Dφ)2 (DU x )(DΦ xx ) 5 3 ΦU3 5 (DUx )(DU)Φ 4 + 5 4 (DU)Ux (DΦ) + 5(DU)U(DΦx ) + 5 2 (DU)Φx Φ ] dxdθ.

L = x 3 + U x + ΦD + 1 2 (U x (DΦ)) = (D 3 + W D + Υ)(D 3 DW + Υ), gives us a Miura-type transformation U = 2W x W 2 + (DΥ), Φ = Υ x 2ΥW,

L = 3 x + U x + ΦD + 1 2 (U x (DΦ)) = (D 3 + W D + Υ)(D 3 DW + Υ), gives us a Miura-type transformation U = 2W x W 2 + (DΥ), Φ = Υ x 2ΥW, and the modified system is given by W t + W xxxxx + 5W xxx (DΥ) 5W xxx W x 5W xxx W 2 5W 2 xx + 10Wxx (DΥx ) 20W xx W x W 5W xx W (DΥ) 5W 3 x 5W 2 x (DΥ) + 5Wx (DΥxx ) + 5Wx (DΥ)2 + 5W x W 4 5W x W (DΥ x ) + 10W (DΥ x )(DΥ) 10Υ x ΥW x + 5(DW xx )Υ x + 5(DW x )Υ xx + 5(DW x )ΥW x + 10(DW x )ΥW 2 5(DW )Υ x W x + 10(DW )Υ x W 2 + 10(DW )Υ(DΥ x ) 5(DW )ΥW xx + 30(DW )ΥW x W = 0, Υ t + Υ xxxxx + 5Υ xxx (DΥ) 5Υ xxx W 2 + 5Υ xx (DΥ x ) + 5Υ xx W xx 25Υ xx W x W + 5Υ xx W (DΥ) + 5Υ x (DΥ) 2 + 5Υ x W xxx 25Υ x W xx W 25Υ x W 2 x + 5Υx Wx (DΥ) + 10Υ x W x W 2 + 5Υ x W 4 10Υ x W 2 (DΥ) + 5Υ x W (DΥ x ) 10ΥW xxx W 20ΥW xx W x + 10ΥW xx W 2 + 30ΥW 2 x W + 20ΥWx W 3 30ΥW x W (DΥ) 10ΥW 2 (DΥ x ) 5(DW x )Υ x Υ 5(DW )Υ xx Υ 10(DW )Υ x ΥW = 0.

W = 0, Υ = φ. we have φ t + φ xxxxx + 5φ xxx (Dφ) + 5φ xx (Dφ x ) + 5φ x (Dφ) 2 = 0 Let φ = ξ(x, t) + θu(x, t), we have u t + u xxxxx + 5uu xxx + 5u x u xx + 5u 2 u x 5ξ xxx ξ x = 0 ξ t + ξ xxxxx + 5uξ xxx + 5u x ξ xx + 5u 2 ξ x = 0

W = 0, Υ = φ. we have φ t + φ xxxxx + 5φ xxx (Dφ) + 5φ xx (Dφ x ) + 5φ x (Dφ) 2 = 0 Let φ = ξ(x, t) + θu(x, t), we have u t + u xxxxx + 5uu xxx + 5u x u xx + 5u 2 u x 5ξ xxx ξ x = 0 ξ t + ξ xxxxx + 5uξ xxx + 5u x ξ xx + 5u 2 ξ x = 0 Lax operator: L = (D 3 + φ)(d 3 + φ)

Proposition sresl n 3 ImD. where sres means taking the super residue of a super pseudodifferential operator.

Proposition sresl n 3 ImD. where sres means taking the super residue of a super pseudodifferential operator. Proof: Consider Thus, Then D 3 + φ = Λ 3 L = Λ 6 sres(l n 3 ) = sresλ 2n = 1 2 sres[λ2n 1, Λ]

Proposition sresl n 3 ImD. where sres means taking the super residue of a super pseudodifferential operator. Proof: Consider Thus, Then D 3 + φ = Λ 3 L = Λ 6 sres(l n 3 ) = sresλ 2n = 1 2 sres[λ2n 1, Λ] nontrivial conserved quantities?

Proposition sresl n 3 ImD. where sres means taking the super residue of a super pseudodifferential operator. Proof: Consider Thus, Then D 3 + φ = Λ 3 L = Λ 6 sres(l n 3 ) = sresλ 2n = 1 2 sres[λ2n 1, Λ] nontrivial conserved quantities? We now turn to L n 6

It can be proved that So, the super residue of L n 6 t L n 6 = [9(L 5 3 )+, L n 6 ]. is conserved.

It can be proved that t L n 6 = [9(L 5 3 )+, L n 6 ]. So, the super residue of L n 6 is conserved. The first two nontrivial conserved quantities sresl 7 1 6 dxdθ = [2(Dφ xx ) + (Dφ) 2 6φ x φ]dxdθ 9 sresl 11 1 6 dxdθ = [6(Dφ xxxx ) + 18(Dφ xx )(Dφ) + 9(Dφ x ) 2 81 + 4(Dφ) 3 18φ xxx φ + 6φ xx φ x 36φ x φ(dφ)]dxdθ Remarks: 1 What is remarkable is that the conserved densities found in this way, unlike the supersymmetric KdV case, are local. 2 All those conserved quantities are fermioic. To our knowledge, this is the first supersymmetric integrable system whose only conserved quantities are fermioic.

Finally, we have the recursion operator R = 6 x + 6(Dφ) 4 x + 9(Dφx ) 3 x + 6φxx 2 x D + {5(Dφxx ) + 9(Dφ)2 } 2 x + {9φ xxx + 12φ x (Dφ)} x D + {(Dφ xxx ) + 9(Dφ x )(DΦ)} x + {5φ xxxx + 12φ xx (Dφ) + 6φ x (Dφ x )}D + {4(Dφ xx )(Dφ) + 4(Dφ) 3 3φ xx φ x } + {φ xxxxx + 5φ xxx (Dφ) + 5φ xx (Dφ x ) + 2φ x (Dφ xx ) + 6φ x (Dφ) 2 }D 1 {2(Dφ xx ) + 2(Dφ) 2 }D 1 φ x 4φ x (Dφ) 1 x φ x 2(Dφ)D 1 [φ xxx + 2φ x (Dφ)] 2φ x D 1 {(φ xxx + φ x (Dφ))D 1 3(Dφ)D 1 φ x 2φ x 1 x φ x + 2D 1 [φ xxx + 2φ x (Dφ)]}

Popowicz: Odd Hamiltonian structure for supersymmetric Sawada-Kotera equation. arxiv:0902.2861v1 Probably it is the first nontrivial supersymmetric model with odd bi-hamiltonian structure with the modified bosonic sector. So this model realizes in fully the idea of equal footing of fermions and bosons fields.

novel SUSY 5th KdV φ t = φ xxxxx + 2αφ xxx (Dφ) + 3αφ xx (Dφ x ) + αφ x (Dφ xx ) + 3 5 α2 φ x (Dφ) 2 + 3 5 α2 φ(dφ x )(Dφ)

novel SUSY 5th KdV φ t = φ xxxxx + 2αφ xxx (Dφ) + 3αφ xx (Dφ x ) + αφ x (Dφ xx ) + 3 5 α2 φ x (Dφ) 2 + 3 5 α2 φ(dφ x )(Dφ) Hamiltonian structure φ t = D δh 1 δφ where H 1 = 1 [ 2 φxxxx (φ) + 5φ xx (Dφ) 2 + 5φ(φ) 3] dxdθ

novel SUSY 5th KdV φ t = φ xxxxx + 2αφ xxx (Dφ) + 3αφ xx (Dφ x ) + αφ x (Dφ xx ) + 3 5 α2 φ x (Dφ) 2 + 3 5 α2 φ(dφ x )(Dφ) Hamiltonian structure φ t = D δh 1 δφ where H 1 = 1 [ 2 φxxxx (φ) + 5φ xx (Dφ) 2 + 5φ(φ) 3] dxdθ Recursion operator R = x 6 + 12(DΦ) 4 x + 6Φx 3 x D + 24(DΦx ) 3 x + {21Φxx + 18Φ(DΦ)} 2 x D +{19(DΦ xx ) + 30(DΦ) 2 } x 2 + {23Φxxx + 57Φx (DΦ) + 27Φ(DΦx )} x D +{7(DΦ x xx) + 60(DΦ x )(DΦ)} x +{11Φ xxxx + 54Φ xx (DΦ) + 48Φ x (DΦ x ) + 15Φ(DΦ xx ) + 36Φ(DΦ) 2 }D +{(DΦ xxxx ) + 27(DΦxx)(DΦ) + 20(DΦ x ) 2 + 28(DΦ) 3 5Φ xx Φ x + 18Φ x Φ(DΦ)} +{2Φ xxxxx + 20Φ xxx (DΦ) + 30Φ xx (DΦ x ) + 14Φ x (DΦ xx ) + 21Φ x (DΦ) 2 +3Φ(DΦ xxx ) + 51Φ(DΦ x )(DΦ)}D 1 +{ 2Φ xxx + 30Φ x (DΦ) 6Φ(DΦ x )}D 1 (DΦ) + 3(DΦ xxx ) x 1 {(DΦ) Φ x D 1 } 3(DΦ)D 1 {Φ xxx + ( 7Φ xx 3Φ(DΦ)) x 1 [(DΦ) Φ x D 1 ]} 2Φ x x 1 {Φ xxx + 3Φ(DΦ) + [2(DΦ xxx ) + 12(DΦ x )(DΦ) 3Φ xx Φ]D 1 +[ 6Φ xx 9Φ(DΦ)] x 1 [(DΦ) Φ x D 1 ]}

A possible second structure (bi-hamiltonian?) φ t = RD δh 0 δφ where H 0 = 1 2 φdxdθ

A possible second structure (bi-hamiltonian?) φ t = RD δh 0 δφ where H 0 = 1 2 φdxdθ A multi-linear form f 2 D x { f 2 S(D t D 5 x )f f } +5 { (D x (D 2 x f f ) f 2 )(SD 3 x f f ) (Dx (D4 x f f ) f 2 } )(SD x f f ) = 0

Thank you!