N = 2 Supercomplexification of the Korteweg-de Vries, Sawada-Kotera and Kaup-Kupershmidt Equations

Ziemowit Popowicz

doi:10.1080/14029251.2019.1591732

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Volume 26, Issue 2, March 2019, Pages 294 - 312

N = 2 Supercomplexification of the Korteweg-de Vries, Sawada-Kotera and Kaup-Kupershmidt Equations

Authors

Ziemowit Popowicz

Institute of Theoretical Physics, University of Wrocław, Wrocław pl. M. Borna 9, 50-204 Wrocław Poland,ziemek@ift.uni.wroc.pl

Received 13 September 2018, Accepted 18 December 2018, Available Online 6 January 2021.

DOI: 10.1080/14029251.2019.1591732 How to use a DOI?
Keywords: suoersymnmetry; bi-Hamiltonian; Lax representation
Abstract: The supercomplexification is a special method of N = 2 supersymmetrization of the integrable equations in which the bosonic sector can be reduced to the complex version of these equations. The N = 2 supercomplex Korteweg-de Vries, Sawada-Kotera and Kaup-Kupershmidt equations are defined and investigated. The common attribute of the supercomplex equations is appearance of the odd Hamiltonian structures and super-fermionic conservation laws. The odd bi-Hamiltonian structure, Lax representation and superfermionic conserved currents for new N = 2 supersymmetric Korteweg-de Vries equation and for Sawada-Kotera one, are given.
Copyright: © 2019 The Authors. Published by Atlantis and Taylor & Francis
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. Introduction

Integrable Hamiltonian systems occupy an important place in diverse branches of theoretical physics as exactly solvable models of fundamental physical phenomena ranging from nonlinear hydrodynamics to string theory [7,8]. There are many approaches to investigate these systems as for example the Lax approach, the construction of the recursion operator and bi-Hamiltonian structure or checking the B¨acklund and Darboux transformations [3].

On the other hand, applications of the supersymmetry (SUSY) to the soliton theory provide a possibility of generalization of the integrable systems. The supersymmetric integrable equations [4–19] have drawn a lot of attention for a variety of reasons. In order to create a supersymmetric theory, we have to add to a system of k bosonic equations kN fermionic and k(N − 1) bosonic fields (k = 1, 2,..., N = 1, 2,...) in such a way that the final theory becomes SUSY invariant. A bonus of this method is, that the so called bosonic sector of the supersymmetrical equations when N ≥ 2, leads us to a new system of interacting fields. For example, the Virasoro algebra [9] and some of its extensions can be related to the second Hamiltonian structure of the Korteweg-de Vries (KdV) and KdV-like equations. This Hamiltonian structure is given by the set of Poisson brackets for the fundamental fields representing the Virasoro algebra. Now, starting from the supersymmetric generalization of the Virasoro algebra and the corresponding Hamiltonian structure, the N = 1, 2 supersymmetric extensions of the classical equations have been obtained [16,17,20].

There are many methods of supersymmetrization of integrable system as, for example, to start simply from the supersymmetric version of the Lax operator or consider the supersymmetric version of the Hamiltonian structure. Interestingly, during the process of supersymmetrization many unexpected, but typical supersymmetric effects occurred. In particular, the roots for the SUSY Lax operator are not uniquely defined [21], the non-local conservation laws [4] and the odd Hamiltonian structure appear [22,23].

The idea of introducing odd Hamiltonian structure is not new. Leites noticed [18] that in the superspace, one can consider both the even and odd sympletic structures, with even and odd Poisson brackets respectively. The odd brackets, also known as antibrackets, have drawn some interest in the context of BRST formalism in the Lagrangian framework [1], in the supersymmetrical quantum mechanics [30], and in the classical mechanics [14].

Becker and Becker in [2] proposed the supersymmetric KdV equation in the form

Φt=Φxxx+6(𝒟Φ)Φx,(1.1)

where Φ is a superfermionic N = 1 function. If we replace u in the KdV equation by (𝒟Φ) then we obtain the equation (1.1). However, as a result, the bi-Hamiltonian structure becomes the odd one.

The N = 2 supersymmetric generalization of KdV equation

Φt=Φxxx+6(𝒟1𝒟2Φ)Φx,(1.2)

was considered in [22]. Similarly to the Becker and Becker equation (1.1) this equation possess the odd bi-Hamiltonian structure. However the Lax representation for this equation has been not presented.

In this paper, we generalize the N = 1 substitution u ⇒ (𝒟Φ) to the N = 2 supersymmetric case, assuming that

u⇒(k1(𝒟1𝒟2Φ)+k2Φx)+i(k3(𝒟1𝒟2Φ)+k4Φx)),(1.3)

where i² = −1 and k_j, j = 1,...,4 are arbitrary constants. We investigate the Hamiltonian structure, Lax representation and conservation laws for the obtained equations after such substitution. We call such substitution BN = 2 supercomplexification. An unexpected feature of this supercomplexification is that if we directly substitute the ansatz eq.(1.3) to the conserved currents of the KdV equation, then the currents are no longer conserved currents, in contrast to the N = 1 case. As we show, such a supercomplexification leads us to the odd Hamiltonian structures and to the superfermionic conserved currents.

We investigate supercomplexifications of three equations: the KdV, Sawada-Kotera (S-K) and Kaup-Kupershmidt (K-K) equations. For all these equations, we fix the arbitrary constants in such a way that their bosonic sector could be transformed to the complex version of the KdV, S-K, K-K equations. This procedure justifies the name of supercomplexification. The odd bi-Hamiltonian structure, Lax representation and superfermionic conserved currents for new BN = 2 supersymmetric Korteweg-de Vries equation are given. For the BN = 2 supercomplex Sawada-Kotera equation the Lax representation, odd bi-Hamiltonian structure and superfermionic conserved currents are defined. The BN = 2 supercomplex Kaup-Kupershmidt equation is defined, for which the odd bi-Hamiltonian structure is presented with its superfermionic conserved currents.

All calculations used in the paper have been carried out with the help of computer program Susy2 [24].

The paper is organized as follows. In the first section, the notation used in the non-extended and in extended supersymmetry is explained. Section 2 contains description of the non-extended N = 1, BN = 1 and extended N = 2, BN = 2 supersymmetric KdV equation. Section 3 and section 4 treat the non-extended N = 1, BN = 1 and extended N = 2, BN = 2 supersymmetric Sawada-Kotera and Kaup-Kupershmidt equations. The last section is the conclusion.

2. Notation used in the supersymmetry

In the non-extended N = 1 supersymmetric theory, we deal with the odd and even variables. These variables are joined in the multiplet as Φ = ξ +θu or as ϒ = w+θζ where ξ = ξ(x,t),ξ² = 0, ζ = ζ(x,t), ζ² = 0 are odd functions while u = u(x,t), w = w(x,t) are even functions, and θ is Majorana spinor such that θ² = 0. In other words θ is the odd coordinate. The Φ is called superfermionic function while ϒ a superbosonic one.

The supersymmetric derivative 𝒟 is defined as

𝒟=∂∂θ+θ∂, 𝒟2=∂.(2.1)

The symbolic integration over the odd variables is defined as

∫dθ=0, ∫dθ θ=1.(2.2)

In the extended supersymmetry N = 2 case we deal with more complicated superfermionic or superbosonic functions which are defined by

Φ=w+θ1ξ1+θ2ξ2+θ1θ2u, ϒ=ζ1+θ1h+θ2k+θ1θ2ζ2,(2.3)

where w, u, k, h are even functions, ξ₁, ξ₂, ζ₁, ζ₂, ξi2=0, ζi2=0, ξ₂ξ₁ = −ξ₁ξ₂, ζ₂ζ₁ = −ζ₁ζ₂ are odd functions which take values in the Grassman algebra, θ₁ and θ₂ are two different Majorana spinors, odd coordinates, such that θi2=0, θ₂θ₁ = − θ₁θ₂. Φ is the superboson function while ϒ is superfermionic function.

The supersymmetric derivatives and symbolic integrations are defined as

𝒟i=∂∂θi+θi∂, 𝒟i−1=𝒟i∂−1, i=1,2,(2.4)

𝒟12=𝒟22=∂, 𝒟1𝒟2+𝒟2𝒟1=0,(2.5)

∫θ2θ1dθ1dθ2=1, ∫dθ1dθ2=0,(2.6)

where negative power ∂⁻¹ of formal integration is defined as

∂−1a=a∂−1−ax∂−2+axx∂−3∓⋯, a∂−1=∂−1a+∂−2ax+∂−3axx+⋯.(2.7)

From the formulas (2.4) follows ∫dθ₁dθ₂dx(𝒟_iΓ) = 0, i = 1, 2 for an arbitrary superfunction Γ which vanishes in ±∞.

For the supersymmetric extensions of the models discussed in what follows, the Lax operators may be regarded as the element of the algebra of super-differential operators 𝒢 . For N = 1 we have

𝒢:={∑k=−∞∞(ak+Φk𝒟)∂k},(2.8)

and for N = 2

𝒢:={∑k=−∞∞(bk+βk𝒟1+γk𝒟2+ak𝒟1𝒟2)∂k}.(2.9)

The supersymmetric algebra 𝒢 possess three invariant subspaces defined by the following projections P onto the subspaces of 𝒢 .

P≥0(𝒢)=𝒢≥0={∑k=−∞∞(fk+βk𝒟1+γk𝒟2+gk𝒟1𝒟2)∂k},(2.10)

P≥1(𝒢)=𝒢≥1={∑k=1∞(fk+βk𝒟1+γk𝒟2+gk𝒟1𝒟2)∂k+α𝒟1+δ𝒟2+h𝒟1𝒟2},(2.11)

P≥2(𝒢)=𝒢≥2={∑k=2∞(fk+βk𝒟1+γk𝒟2+gk𝒟1𝒟2)∂k+ (f1𝒟1+f2𝒟2+f3𝒟1𝒟2)∂+δ𝒟1𝒟2}.(2.12)

The subscript L_≥0, L_≥1 in what follows denotes the projection P_≥0(L), P_≥1(L).

The supersymmetric algebra 𝒢 is endowed with non-degenerate “trace form” given by the residues

tr(L)=Res(∑k<∞(ak+Φk𝒟)∂k)=∫dxdθΦ−1,(2.13)

tr(L)=Res(∑k<∞(bk+βk𝒟1+γk𝒟2+ak𝒟1𝒟2)∂k)=∫dxdθ1dθ2a−1,(2.14)

for N = 1 and N = 2 respectively. This “trace form” is used in the theory of integrable systems because from the knowledge of the Lax operator it is possible to obtain the conserved currents.

Similarly to the classical case it is possible to construct the generalized supersymmetric Lax representation ddtqL=[P≥k(Lq),L] where k = 0, 1, 2. Restriction to k = 0 yields to the supersymmetric Gelfand-Dikii hierarchy of equations. Restriction to k = 1 yields the supersymmetric nonstandard Lax representation while for k = 2 to the supersymmetric Harry Dym hierarchy.

As usual in the case of extended supersymmetry N > 1, we assume the invariance of the considered model under change the odd variables. It means that we always assume the invariance under the replacement of the supersymmetric derivatives 𝒟₁ → −𝒟₂,𝒟₂ → 𝒟₁ and denote this transformation as O₂. For example we have O₂(𝒟₁Φ) = −(𝒟₂Φ).

3. N = 1, BN = 1, N = 2 and BN = 2 susy KdV

The Korteweg-de Vries equation is defined as

ut=uxxx+6uux=JδH1δu=PδH2δu,(3.1)

J=∂, P=∂3+2∂u+2u∂,(3.2)

H1=12∫dx(uuxx+4u3), H2=12∫dx u2,

and is obtained from the Lax representation

L=∂2+u, Lt=[L,L≥03/2].(3.3)

The subscript ≥ 0 in (L^3/2)_≥0 denotes the purely differential part of L^3/2. The KdV equation is integrable, has bi-Hamiltonian structure and possesses infinite number of bosonic conserved currents. The Lax operator Eq. (3.3) generates the conserved currents as

Hn=tr(L(2n+1)/2)=tr(∂(2n+1)/2+∑i=−∞(2n−1)/2ai∂i)=∫dxa−1=∫dx hn.(3.4)

Let us recall the relationship between the Poisson bracket

{u(x),u(y)}=12[∂3+2∂u+2u∂]δ(x−y)(3.5)

and the Virasoro algebra [9,10]. Fourier expansion of

u(x)=−12c∑n=−∞∞Lne−inx+12(3.6)

where c is a constant leads us to the classical form of the Virasoro algebra

{Ln,Lm}=(n−m)Ln+m+c12δ(n+m,0).(3.7)

3.1. N = 1, BN = 1 supersymmetric KdV equation

The N = 1 supersymmetric KdV equation is obtained from the Lax representation

(L=∂2+𝒟Φ)t=[L,L>03/2]⇒Φt=(Φxx+3(𝒟Φ)Φ)x,(3.8)

ξt=(ξxx+3ξu)x, ut=(uxx+3u2+3ξxξ)x.(3.9)

This equation has been thoroughly investigated in many papers [12,15,19].

The BN = 1 supersymmetric KdV equation is generated by the Lax representation

(L=∂2+(𝒟Φ))t=[L,L≥03/2]⇒Φt=Φxxx+6(𝒟Φ)Φx,(3.10)

ξt=ξxxx+6ξxu, ut=(uxx+3u2)x.(3.11)

It has a triangular form, u_t does not contain the odd function, but it is a very interesting equation from integrability and supersymmetry point if view. This equation has been first considered by Becker and Becker [2] and it was named later as B extension of supersymmetric KdV equation [4].

This system possesses infinite number of the superfermionic conservation laws. For example,

H3.5=12∫dx dθ ΦΦx=−∫dx ξux,(3.12)

H5.5=12∫dx dθ Φ(Φxxx+4Φx(𝒟Φ))=−∫dx ξ(uxxx+6uxu).(3.13)

where lower index in H denotes the dimension of the expression. Assuming that deg(A) denotes the dimension of A, we have

degu=2, degΦ=32, deg𝒟=12, degθ=−12,degξ=32, degx=−1, deg∂=1.(3.14)

The variational derivative δδΦ of these superbosonic conservation laws will be a superfermionic function. Therefore, our bi-Hamiltonian structure is living in the superfermionic space and hence the Hamiltonians operators should be symmetrical operators.

The bi-Hamiltonian structure is easy to obtain using the formula (3.1) in which we assume u = (𝒟Φ) ⇒ Φ = (𝒟⁻¹u) from which follows

Φt=PδH2δu=𝒟−1(∂3+2∂(𝒟Φ)+2(𝒟Φ)∂)𝒟−1δH3.5δΦ=(∂2+4(𝒟Φ)+2∂−1+𝒟Φx+2Φx∂−1𝒟)δH3.5δΦ=𝒟−1∂𝒟−1δH5.5δΦ=δH5.5δΦ.(3.15)

The operator 𝒫 is connected with the Poisson bracket

{Φ(x,θ),Φ(y,θ′)}=𝒫δ(x,y)(θ−θ′),(3.16)

and is rewritten in the components as

{u(x),u(y)}=4ξxδ(x−y),{u(x),ξ(y)}=(∂2+2∂−1ux+4u)δ(x−y),{ξ(x),ξ(y)}=2ξxδ(x−y).(3.17)

3.2. N = 2 and BN = 2 supersymmetric KdV Equation

We have three different N = 2 supersymmetrical extensions of the KdV equation

Φt=(−Φxx+3Φ(𝒟1𝒟2Φ)+12(α−1)(𝒟1𝒟2Φ)2+αΦ3)x,(3.18)

where α = 1, −2, 4.

All these equations possess the bi-Hamiltonian structure [11, 21]. For example the second Hamiltonian structure for the equation (3.18) is

Φt=ΛδδΦ12∫dxdθ1dθ2(Φ𝒟1𝒟2Φ+α3Φ3),Λ=𝒟1𝒟2∂+2∂Φ+2Φ∂−𝒟1Φ𝒟1−𝒟2Φ𝒟2,(3.19)

and is connected with the Poisson bracket

{Φ(x,θ1,θ2),Φ(y,θ1′,θ2′)}=Λδ(x,y)(θ1−θ1′)(θ2−θ2′).(3.20)

This formula could be rewritten in the components

{u(x),u(y)}=(−∂3+4u∂+2ux)δ(x−y),{u(x),ξi(y)}=(3ξi+ξi,x)δ(x−y),{u(x),w(y)}=2wδ(x−y)x,{ξi(x),ξj(y)}=[δi,j(∂2−u)+εi,j(2w∂+wx)]δ(x−y),{w(x),w(y)}=δ(x−y)x.(3.21)

This bracket defines the N = 2 supersymmetric Virasoro algebra if we apply the Fourier expansion of Φ in Eq. (3.20).

These supersymmetric equations are possible to obtain from the following Lax representations [16,21]

α=4, L=−(𝒟1𝒟2+Φ)2, Lt=4[L,L≥03/2],(3.22)

α=−2, L=∂2+𝒟1Φ−𝒟2Φ, Lt=4[L,L≥03/2],(3.23)

α=1, L=∂+∂−1𝒟1𝒟2Φ, Lt=[L,L≥13].(3.24)

To this list of three equations we would like to add the fourth one integrable extension BN = 2 which has the Lax representation, bi-Hamiltonian formulation and possess the superfermionic conserved currents.

To end this let us send the formula (3) in which k_i, i = 1, 2, 3, 4 are real coefficients, to the KdV Eq. (3.1). Extracting the real and imaginary part we obtain the system of two equations on (𝒟₁𝒟₂Φ)_t, Φ_t_,_x. Solving this system of equations and verifying the integrability condition (𝒟₁𝒟₂Φ)_t_,_x = ∂_xΦ_t_,_{𝒟_1,}_𝒟₂ we obtain a system of algebraic equation on the coefficients k_i, i = 1, 2, 3, 4. There are two solutions.

The first one k₁ = k₄, k₂ = −k₃ is

Φt=Φxxx+6k4(𝒟1𝒟2Φ)Φx+3k3(𝒟1𝒟2Φ)2−3k3Φx2,(3.25)

and second solution k₁ = −k₄, k₂ = k₃ is

Φt=Φxxx−6k4(𝒟1𝒟2Φ)Φx−3k3(𝒟1𝒟2Φ)2+3k3Φx2,(3.26)

The bosonic part of these equations reduces to the KdV equation when Φ = θ₁θ₂u and k₃ = 0.

Therefore without losing on the generality we assume that BN = 2 KdV equation has the form

Φt=Φxxx+6(𝒟1𝒟2Φ)Φx.(3.27)

Below, we will call such procedure as supercomlexification of the equations.

If we replace u by (𝒟₁𝒟₂Φ)+iΦ_x in the Lax operator of KdV equation then we obtain complex operator

L=∂xx+(𝒟1𝒟2Φ)+iΦx(3.28)

which generates the equation (3.27) but not the conserved currents. We cannot use the supersymmetric version of trace form Eq. (2.13) to this operator because it does not contain supersymmetric derivatives. On the other side if we make the same substitution to the conserved currents obtained from the Lax operator of KdV equation (3.4) and make the replacement ∫dx ⇒ ∫dxdθ₁ dθ₂ in h_n

Hn=∫dxhn⇒Gn=∫dxdθ1dθ2hn(u⇒(𝒟1𝒟2Φ)+iΦx),(3.29)

then G_n = 0.

The following fourth order Lax representation, where we do not use the imaginary symbol i generates the supercomplex BN = 2 KdV equation.

L=∂4+2(∂(𝒟1𝒟2Φ)+(𝒟1𝒟2Φ)∂)∂+2(∂Φx+Φx∂)𝒟1𝒟2,(3.30)

Lt=2[L,L≥03/4]⇒Φt=Φxxx+6(𝒟1𝒟2Φ)Φx,(3.31)

The same equation is also generated by the nonstandard Lax representation

L=∂+Φ∂−1𝒟1𝒟2+∂−1((𝒟1𝒟2Φ)−Φ𝒟1𝒟2),(3.32)

Lt=[L≥13,L]⇒Φt=Φxxx+6(𝒟1𝒟2Φ)Φx.(3.33)

The equation (3.31) appeared also for the first time in [22], where the second Hamiltonian operator was constructed and was interpreted as odd version of Virasoro algebra.

In the components, the equation (3.31) is

wt=wxxx+6uwx, ut=uxxx+3(u2−wx2)xξ1,t=ξ1,xxx+6ξ1,xu+6ξ2,xwx,ξ2,t=ξ2,xxx−6ξ1,xwx+6ξ2,xu.(3.34)

We see that fermionic sector is invariant under the replacement ξ₁ ⇒ −ξ₂, ξ₂ ⇒ ξ₁ while the bosonic sector is purely even and does not contain the odd functions. This invariance is exactly the O₂ invariance. Therefore, this supersymmetric version of the KdV equation is BN = 2 extension.

Introducing new function w_x = v to the bosonic sector of the equation (3.34) we obtained

vt=(vxx+6uv)x, ut=(uxx+3(u2−v2))x.(3.35)

It is exactly the complex KdV equation.

On the other side the system of equations (3.34) is equivalent to the complex version of the Becker and Becker equation Eq. (3.10). Indeed if we assume that Φ = ρ₁ + iρ₂ in the equation (3.10) then we obtain

ρ1,t=ρ1,xxx+3[𝒟((𝒟ρ1)2−(𝒟ρ2)2)]ρ2,t=ρ2,xxx+6[𝒟((𝒟ρ1)(𝒟ρ2))].(3.36)

Next assuming that ρ₁ = ξ₁ + θu, ρ₂ = ξ₂ + θw, w_x = v we see that the previous equation reduces to the system of equations (3.34).

The Lax representation of the complex KdV equation is given by the bosonic part of the supercomplexified Lax representation Eq. (3.30)

Lb=(∂4+2(u∂+∂u)∂2(∂v+v∂)−2∂(v∂+∂v)∂∂4+2∂(u∂+∂u)),((L3/4)≥0)b=(∂3+3u∂3v−2∂v∂∂3+3∂u),Lb,t=[Lb,((L3/4)≥0)b],(3.37)

or by bosonic part of the supercomplexified Lax representation Eq. (3.32)

Lb=(∂+∂−1u∂−1v+v∂−1−2vx∂+u∂−1), Lb,t=[Lb,((L3)≥1)b].(3.38)

The symbol b in L_b denotes the bosonic part of the L operator.

There are many differences between the supersymmetric equations (3.34) and (3.1). The system (3.1) possess an infinite number of conservation laws but the conservation laws of (3.34) does not reduce to the conservation laws of (3.1).

Unfortunately if we apply the “trace form” to our Lax operator tr (Lⁿ^/4) for n = 2, 3, 5, 6 we did not obtain any conserved currents because then tr(Lⁿ^/4) = 0. We confirmed this observation by constructing an arbitrary superbosonic functions K_n = K_n(Φ,(𝒟₁Φ),(𝒟₂Φ),...) with arbitrary constants, where n = 1, 2,..., 11 denotes the weight. For example

H4=∫dxdθ1dθ2K4,K4=λ1Φ4+λ2(𝒟1𝒟2Φ)Φ2+λ3ΦxxΦ+λ4(𝒟2Φ)(𝒟1Φ)Φ,(3.39)

where λ_i, i = 1, 2, 3, 4 are arbitrary constants.

We verified that these functions are not constants of motion for our supercomplex BN = 2 KdV equation.

However, if we expand L operator as

L11/8=𝒟1+∑k=1∞(ϒ1,k+ϕ1,k𝒟1+ϕ2,k𝒟2+ϒ2,k𝒟1𝒟2)∂−k,(3.40)

where the super functions ϒ_1,_k, ϒ_2,_k, φ_1,_k, φ_2,_k are computed from the assumption that L = (L^1/8) ⁸ then it is possible to obtain the superfermionic conserved currents.

Indeed, as we checked

H3.5=tr(L7/8)=14∫dxdθ1dθ2Φ(𝒟1Φx)=−12∫dx(ξ1ux+ξ2wxx),(3.41)

is a conserved quantity. The lower index in H_a denotes the dimension of the expression ([θ1]=−12, [ξi]=32) .

Using the O₂ transformation it is possible to obtain the superpartners of L11/8 and H_3.5 as L21/8=O2(L1/8), Ĥ_3.5 = O₂(H_3.5)

H^3.5=14∫dxdθ1dθ2Φ(𝒟2Φx)=−∫dx(ξ1wxx−ξ2ux).(3.42)

It is difficult to obtain the next conserved quantity using the trace form, because we need to compute the L^1/8 up to the terms containing ∂⁻⁶, because then H = tr(LL^1/4L^1/8).

Therefore we assumed the most general form on the next current and verified that

H5.5=12∫dxdθ1dθ2Φ((𝒟1Φxxx)+4(𝒟1𝒟2Φ(𝒟1Φx)+4Φxx(𝒟2Φ))=∫dxξ1(−uxx+3(wx2+u2))x−ξ2(−wxxx−6uwx)x,(3.43)

is proper conserved quantity.

It is impossible to use the similar trick as in the BN = 1 supersymmetric KdV equation Eq. (3.15) because the transformation u = (𝒟₁𝒟₂Φ) + iΦ_x is not invertible. Indeed, if we assume that such an inverse exists, then

(𝒟1𝒟2+i∂)2=2i∂(𝒟1𝒟2+i∂)⇒𝒟1𝒟2+i∂=2i∂,

which is not true.

Assuming different forms of the Hamiltonian operator of the supercomplexified BN = 2 KdV equation, we obtained the following bi-Hamiltonian structure

Φt=12𝒟1∂−1δH5.5δΦ=ΠδH3.5δΦ,Π=2𝒟1∂+4∂−1[(𝒟1𝒟2Φ)𝒟1−Φx𝒟2]+4[(𝒟1𝒟2Φ)𝒟1−Φx𝒟2]∂−1.(3.44)

We checked that the operator Π defines the proper Hamiltonian operator and satisfies the Jacobi identity

∫dxdθ1dθ2 α ΠΠβ⋆γ+cyclic(α,β,γ)=0,(3.45)

where ΠΠβ⋆ is a Gateaux derivative [3]

ΠΠβ⋆=[ddεΠ(Φ+εΠβ)]ε=0,(3.46)

and α, β, γ are superfermionic test functions. The operator Π appeared first time in [22] and has been connected with the odd version of the Virasoro algebra.

{Φ(x,θ1,θ2),Φ(y,θ1′,θ2′)}=Πδ(x,y)(θ1−θ1′)(θ2−θ2′).(3.47)

This could be rewritten in the components as

{w(x),w(y)}={w(x),u(y)}={u(x),u(y)}=0,{w(x),ξ1(y)}=−(4wx∂−1+2∂−1wx))δ(x−y),{w(x),ξ2(y)}=−(2∂+4(∂−1u+u∂−1))δ(x−y),{ξ1(x),u(y)}=2(∂2+4u−2∂ux)δ(x−y),{ξ2(x),u(y)}=−4(wx−∂−1wxx)δ(x−y),{ξ1(x),ξ1(y)}=−{ξ2(x),ξ2(y)}=−4(ξ1∂−1+∂−1ξ1)δ(x−y),{ξ1(x),ξ2}=−4(ξ2,x∂−1+∂−1ξ2,x)δ(x−y).(3.48)

If we compare the right hand of these Poisson brackets with the right hand of Poisson brackets connected with the N = 2 supersymmetric Virasoro algebra Eq. (3.21) we see that they are in opposite statistics. Indeed the supersymmetric Poisson brackets (3.21) can be symbolically rewritten as

{B,B}⇒ℬ, {F,V}⇒ℱ, {F,F}⇒ℬ,(3.49)

while the brackets (3.48) as

{B,B}⇒0, {F,B}⇒ℬ, {F,F}⇒ℱ,(3.50)

where B denotes u or w, F denotes ξ₁ or ξ₂, and ℬ denotes some bosonic operator while ℱ denotes some fermionic operator. The brackets (3.48), according to general theory of odd Poisson brackets [27], meets the following properties

{A,B+C}={A,B}+{A,C},(3.51)

{A,BC}={A,B}C+(−1)(g(A)+1)g(B)B{A,C},(3.52)

(−1)(g(A)+1)(g(B)+1){A,{B,C}}+(−1)(g(B)+1)(g(C)+1){B,{C(A)}}+(−1)(g(C)+1)(g(A)+1){C,{A(B)}}=0,(3.53)

where A, B, C are homogeneous elements of the Poisson algebra, g(A) denotes the parity of A. The last equation is a generalized Jacobi identity.

Instead of using the recursion operator to generate the conserved currents it is possible to join the superfermionic currents with the usual conserved currents of KdV equation using the formula

H=∫01dλ∫dxdθ1dθ2Φδ(𝒟1−1h^kdv)δΦ,(3.54)

where ĥ_kdv is some conserved currents of KdV equation in which we make the replacement u ⇒ λ((𝒟₁𝒟₂Φ) + iΦ_x). If we split the conserved currents H onto the real and imaginary part as H=G+iG^ then it appears that G^=−O2(G).

Using the previous formula, we obtained next conserved current for the supercomplexified KdV equation

H7.5=∫dx dθ1 dθ2Φ(3(𝒟1Φ5x)+[20(𝒟1Φxx)(𝒟1𝒟2Φ)]x+[20(𝒟2Φxx)Φx]x+45(𝒟1Φx)[(𝒟1𝒟2Φ)2−Φx2]+10(𝒟2Φx)[9(𝒟1𝒟2Φ)Φx+2Φxxx]).(3.55)

To finish this section let us notice that substitution (3) suggests to replace r.h.s of (3) by complex chiral superfield F [13,29] as

u⇒(DF)+(DF¯),(3.56)

where

F=ξ+θ2u−12θ1θ2ξx, F¯=ξ¯+θ1u¯+12θ1θ2ξ¯x,(3.57)

D=∂∂θ1+12θ2∂x, D¯=∂∂θ2+12θ1∂x,(3.58)

u is a complex function, ξ is Grassman valued function.

Substituting formula (3.56) to the KdV equation we obtain

Ft=Fxxx+6Fx[(DF)+(DF¯)].(3.59)

The bosonic part of this equation is

ut=uxxx+6uxu+6u¯ux−6ξxξ¯x.(3.60)

But it is not the complex KdV equation when u = a + ib where a, b are real functions.

4. N = 1, BN = 1 and BN = 2 supersymmetric Sawada-Kotera equation

The Sawada-Kotera equation could be obtained from the Lax representation

L=∂3+u∂, Lt=9[L,L≥05/3],(4.1)

ut=u5x+5uxxxu+5uxxux+5uxu2.(4.2)

The bi-Hamiltonian formulation of this equation is

ut=16(∂3+2(∂u+u∂))δG6δu,(4.3)

12(2∂3+2u∂+2∂u+∂−1(2uxx+u2)+(2uxx+u2)∂−1)ut=δG12δu(4.4)

G6=∫dx(3uuxx+u3),

G12=118∫dx u(9u8x+96u4xuxx+33uxxx2+144uxx2u+153uxxux2−150ux2u2+4u5).

4.1. N = 1, BN = 1 susy Sawada-Kotera equation

The N = 1 supersymmetric extension of S-K equation is defined by the Lax operator L = (𝒟∂ +Φ) ² and its Lax representation [28]

Lt=[L,L≥05/3]⇒(4.5)

Φt=19(Φ5x+5Φxxx(𝒟Φ)+5Φxx(𝒟Φx)+5Φx(𝒟Φ)2),(4.6)

ξt=19(ξ5x+5ξxxxu+5ξxxux+5ξxu2),ut=19(u5x+5uxxxu+5uxxux+5uxu2−5ξxxxξx).

The odd bi-Hamiltonian representation for supersymmetric N = 1 extension of the Sawada-Kotera equation has been given in [23]

Φt=(𝒟∂2+2∂Φ+2Φ∂+𝒟Φ𝒟)∂−1(𝒟∂2+2∂Φ+2Φ∂+𝒟Φ𝒟)δH4δΦ,(4.7)

(∂2+(𝒟Φ)−∂−1𝒟Φx+Φx∂−1𝒟)Φt=δH10δu,(4.8)

H4=118∫dx dθ ΦΦx,

H4=154∫dx dθ Φ[−3Φ7x+12Φ5x(𝒟Φ)+28Φ4x(𝒟Φx)+3Φxxx[32(𝒟Φxx)+15(𝒟Φ)2]+Φxx[8(𝒟Φxxx)+30(𝒟Φx)(𝒟Φ)]+Φx[4(𝒟Φ4x)+30(𝒟Φxx)(𝒟Φ)]+15(𝒟Φx)2+8(𝒟Φx)3].

The BN = 1 supersymmetrical S-K equation is defined by the Lax operator L = ∂³ + (𝒟Φ)∂ and its Lax representation as

Lt=[L,L≥05/3]⇒Φt=19(Φ5x+5Φxxx(𝒟Φ)+5Φx(𝒟Φxx+5Φx(𝒟Φ)2),(4.9)

ξt=19(ξ5x+5ξxxxu+5ξxuxx+5ξxu2),ut=19(u5x+5uxxxu+5uxxux+5uxu2).

The bi-Hamiltonian formulation for the BN = 1 extension is easy to obtain using the same trick as in the case of the BN = 1 extension of KdV equation, see Eq. (3.15).

4.2. BN = 2 Supercomplex Sawada-Kotera equation

The following Lax operator

L=∂3+(k1(𝒟1𝒟2Φ)+k2Φx)∂+(−k2(𝒟1𝒟2Φ)+k1Φx)𝒟1𝒟2,(4.10)

where k₁,k₂ are arbitrary constants, generates the BN = 2 supersymmetrical Sawada-Kotera equation

Lt=9[L,L≥05/3],(4.11)

Φt=13[3Φ5x+15(𝒟1𝒟2Φxx)(k1Φx−k2(𝒟1𝒟2Φ))−10k1k2(𝒟1𝒟2Φ)3+15(𝒟1𝒟2Φ)2Φx(k12−k22)+15k1(𝒟1𝒟2Φ)(Φxx+2k2Φx2)+15k2ΦxxxΦx+5Φx3(k22−k12)].(4.12)

This equation is also possible to obtain after modification of the supercomplexification method as

u⇒k1(𝒟1𝒟2Φ)+k2Φx+i(−k2(𝒟1𝒟2Φ)+k1Φx)(4.13)

and substituting it to the Sawada-Kotera equation or to the Lax representation Eq. (4.1). However then our Lax operator is a complex operator which does not generate the conserved currents.

Introducing a new function w_x = v, it appears that it is always possible to find the linear transformation of v, u which changes the bosonic sector of the equation (4.11) to the complex version of the Sawada-Kotera equation for any arbitrary values of k₁, k₂,

vt=13[3v4x+15vxxu−5v3+15vuxx+15vu2]x,ut=13[3u4x+15uxxu+5u3−15vxxv−15v2u]x.(4.14)

In order to study the conservation laws and Hamiltonian structure of the BN = 2 supersymmetric Sawada-Kotera equation, we consider the special case k₁ = 1, k₂ = 0 for which we obtained

L=∂3+(𝒟1𝒟2Φ)∂+Φx𝒟1𝒟2,(4.15)

Φt=13[3Φ5x+15(𝒟1𝒟2Φxx)+15(𝒟1𝒟2Φ)2Φx+15(𝒟1𝒟2Φ)Φxxx−5Φx3].(4.16)

In the components, the equation Eq.(4.16) is

Φ=w+θ1ξ1+θ2ξ2+θ1θ2u,wt=13[3w5x+15wxxxu−5wx2+15wxuxx+15wxu2],ut=13[3u4x+15uxxu+5u3−15wxxxwx−15wx2u]x,ξ1,t=[ξ1,5x+5ξ1,xxxu+5ξ1,x(uxx+u2−wx2)+ξ12,xxxwx+5ξ2,x(wxxx+2wxu)],ξ2,t=[ξ2,5x+5ξ2,xxxu+5ξ2,x(uxx+u2−wx2)−5ξ1,xxxwx−5ξ1,x(wxxx+2wxu2)].(4.17)

The bosonic sector of the Eq. (4.17) does not interact with the fermionic variables. Thus we have the BN = 2 extension of the Sawada-Kotera equation.

In order to find the conserved current, we use the formula Eq. (3.54)

H=∫01dλ∫dx dθ1 dθ2Φδ(𝒟1−1h^sk)δΦ,(4.18)

where ĥ_sk is some conserved currents of the Sawada-Kotera equation in which we make a replacement u ⇒ λ((𝒟₁𝒟₂Φ) + iΦ_x).

Due to this formula we obtained the following conserved charges

H5.5=∫dxdθ1dθ2Φ[3(𝒟1Φxxx)+2(𝒟2Φx)Φx+2(𝒟1Φx)(𝒟1𝒟2Φ)],(4.19)

H7.5=∫dxdθ1dθ2Φ[(𝒟1Φ5x)+2[(𝒟2Φxx)Φx]x+2(𝒟2Φ)[(𝒟1𝒟2Φ)Φx+Φxxx]+ 2[(𝒟1Φxx)(𝒟1𝒟2Φ)]x+(𝒟1Φx)[2(𝒟1𝒟2Φxx)+(𝒟1𝒟2Φ)2−Φx2]](4.20)

H11.5=∫dxdθ1dθ2Φ(9(𝒟1Φ9x)+74 terms).(4.21)

The system of equation (4.12) could be rewritten as the bi-Hamiltonian system

Φt=16[𝒟1∂+2∂−1[(𝒟1𝒟2Φ)𝒟1−Φx𝒟2]+ 2[(𝒟1𝒟2Φ)𝒟1−Φx𝒟2]∂−1]δH5.5δΦ,(4.22)

𝒦Φt=δH11.5δΦ,(4.23)

𝒦=18(𝒟1∂4+2[(𝒟2Φx)+(𝒟1𝒟2Φ)𝒟1+Φx𝒟2]∂2+[(𝒟1𝒟2Φx)𝒟1+Φxx𝒟2+(𝒟2Φxx)+2(𝒟1Φx)𝒟1𝒟2]∂+2[(𝒟2Φxx)+(𝒟2Φx)(𝒟1𝒟2Φ)−(𝒟1Φx)Φx]+(𝒟1Φxx)𝒟1𝒟2+[(𝒟1𝒟2Φ)2−Φx2+2(𝒟1𝒟2Φxx)]𝒟1+2[(𝒟1𝒟2Φ)Φx+Φxxx]𝒟2+ 𝒟1𝒟2∂−1[(𝒟2Φx)Φx+(𝒟1Φxxx)+(𝒟1Φx)Φx]+ [(𝒟2Φx)Φx+(𝒟1Φxxx)+(𝒟1Φx)Φx]𝒟1𝒟2∂−1).

The operator 𝒦 defines a proper symplectic operator for the BN = 2 supersymmetric Sawada-Kotera equation and satisfies the condition [3]

∫dxdθ1dθ2 [α𝒦β⋆γ+β𝒦γ⋆α+γ𝒦α⋆β]=0.(4.24)

where α, β, γ are the test superfunctions and 𝒦W* is a Gateaux derivative defined as

𝒦W*=ddεK(Φ+εW)|ε=0.(4.25)

As we checked the equation (4.24) is satisfied for superfermionic test functions and also for the superbosonic test functions. For the superfermionic test functions we should assume that in the formula Eq. (4.25), ε is an anticommuting variable, W is a superfermionic function because Φ is a superbosonic function.

To finish this section let us mention that all our formulas presented here possess the O₂ superpartners.

5. BN = 1, BN = 2 supersymmetric Kaup-Kupershmidt equation

The Kaup-Kupershmidt (K-K) equation is derived from the Lax operator

L=∂3+∂u+u∂, Lt=9[L,L≥05/3],(5.1)

ut=u5x+10uxxxu+25uxxux+20uxu2=(∂xxx+∂u+u∂)δH6δu,(5.2)

[18∂xxx+90(∂u+u∂)+∂−1[144u2+36uxx]+[144u2+36uxx]∂−1]ut=δH12δu,(5.3)

H6=16∫dx(3uxxu+8u3),(5.4)

H12=∫dx(9u8xu−180uxxx2u+222uxx3+1224uxx2u2−186ux4−3360ux2u3+256u6).(5.5)

The N = 1 supersymmetric extension of the K-K equation does not exist. It follows from the observation that, if we assume the most general form on the supersymmetric extension of K-K as the polynomial in Φ, (𝒟Φ) and its derivatives, which reduces in the bosonic limit to the Kaup-Kupershmidt equation, then it is possible to construct only one conserved current. It is not enough for such a system to be integrable.

However it is possible to obtain the BN = 1 supersymmetric extension of K-K equation by simply substituting u = (𝒟Φ) to the Eq. (5.1)

Φt=Φ5x+10Φxxx(𝒟Φ)+15Φxxx(𝒟Φx)+10Φx(𝒟Φxx)+20Φx(𝒟Φ)2.(5.6)

In order to construct the BN = 2 supersymmetric extension of the Kaup-Kupershmidt equation let us consider the most general supercomplexified ansatz

u⇒k1(𝒟1𝒟2Φ)+k2Φx+i(k3(𝒟1𝒟2Φ)+k4Φx),(5.7)

where k₁, k₂, k₃, k₄ are arbitrary constants, and substitute it to the Kaup-Kupershmidt equation. As a result, we obtained k₃ = k₂, k₄ = − k₁ and

Φt=Φ5x+10(𝒟1𝒟2Φxx)(k1Φx−k2(𝒟1𝒟2Φ))+10k2ΦxxxΦx+15(𝒟1𝒟2Φx))(k1Φxx−12k2(𝒟1𝒟2Φx))+152k2Φxx2+203(k22−k12)Φx3+10(𝒟1𝒟2Φ)[2(k12−k22)(𝒟1𝒟2Φ)Φx−43k1k2(𝒟1𝒟2Φ)2+k1Φxxx+4k1k2Φx2].(5.8)

It is possible to transform the bosonic part of Φ_t to the complex Kaup-Kupershmidt equation after the identification w_x = v and after making the linear transformation of the function v, u for arbitrary values of k₁, k₂

v⇒k2k12+k22u+k1k12+k22v, u⇒k2k12+k22u−k2k12+k22v,ut=u5x+10uxxxu+25uxxux+20uxu2−10vxxxv−5vx(5vxx+8vu)−20v2ux,vt=v5x+10vxxxv+25vxxux+25vxuxx+10v(uxxx+4uxu)+5vx(5uxx+4u2−4v2).(5.9)

Without losing on the generality, we assume that k₂ = 0, k₁ = 1 and hence we consider following equation

Φt=Φ5x+10(𝒟1𝒟2Φxx)Φx+15(𝒟1𝒟2Φx)Φxx +10(𝒟1𝒟2Φ)(2(𝒟1𝒟2Φx)Φx+Φxxx)−203Φx3.(5.10)

In the components the Eq. (5.10) is

Φ=w+θ1ξ1+θ2ξ2+θ1θ2u,(5.11)

wt=13[3w5x+30wxxxu+45wxxux−20wx3+30wx(uxx−2u2)],

ut=16[6u4x+60uxxu+45ux2+40u3−60wxxwx−45wxx2−120wx2u]x,

ξ1,t=[ξ1,5x+10ξ1,xxxu+15ξ2,xxxux+10(ξ2,xwx)xx+ 10ξ1,x(ux+2u2−2wxx2)+40ξ2,xwxu],

ξ2,t=[ξ2,5x−105ξ1,xxxwx−15ξ1,xxxwxx−10ξ1,xwxxx−40ξ1,xwxu+ 10ξ2,xxxu+15ξ2,xxux+10ξ2,x(uxx+2u2−2wx2)].

In order to find the conserved current for BN = 2 supersymmetric Kaup-Kupershmidt equation, we apply the same method as used in the supersymmetric BN = 2 Sawada-Kotera equation. Therefore, we apply the formula Eq.(3.54) in which now

H=∫01dλ∫dx dθ1 dθ2Φδ(𝒟1−1h^kk)δΦ,(5.12)

where ĥ_sk is some conserved current of the Kaup-Kupershmidt equation in which we make the replacement u ⇒ λ((𝒟₁𝒟₂Φ) − iΦ_x). As a result we obtained the following conserved currents

H5.5=∫dxdθ1dθ2Φ[3(𝒟1Φxxx)+16(𝒟2Φx)Φx+16(𝒟1Φx)(𝒟1𝒟2Φ)],(5.13)

H7.5=∫dxdθ1dθ2Φ[(𝒟1𝒟2Φ5x)+8((𝒟2Φxx)Φx)x+8((𝒟1Φxx)(𝒟1𝒟2Φ)x+8(𝒟2Φx)(4𝒟1𝒟2Φ)Φx+Φxxx)+ 8(𝒟1Φxx)((𝒟1𝒟2Φxx)+2(𝒟1𝒟2Φ)2−2Φx2)](5.14)

H11.5=∫dxdθ1dθ2Φ(9(𝒟1Φ9x)+74 terms).(5.15)

Now the bi-Hamiltonian formulation is

Φt=16(𝒟1∂+∂−1[(𝒟1𝒟2Φ)𝒟1−Φx𝒟2]+[(𝒟1𝒟2Φ)𝒟1−Φx𝒟2]∂−1)δH5.5δΦ(5.16)

𝒦Φt=δH11.5δΦ(5.17)

𝒦=18[𝒟1∂4+10[𝒟2Φx+(𝒟1𝒟2Φ)𝒟1]∂2+5[(𝒟1𝒟2Φx)𝒟1+𝒟2Φxx+2(𝒟1Φx)𝒟1𝒟2]∂+4[(𝒟2Φxxx)+8(𝒟2Φx)(𝒟1𝒟2Φ)−(𝒟1Φx)Φx]+4[(𝒟1𝒟2Φxx)+4(𝒟1𝒟2Φ)2−4Φx2]𝒟1+4[8(𝒟1𝒟2Φ)Φx+Φxxx]𝒟2+2[8(𝒟2Φx)Φx+(𝒟1Φxxx)+8(𝒟1Φx)(𝒟1𝒟2Φ)]∂−1𝒟1𝒟2+2𝒟1𝒟2∂−1[8(𝒟2Φx)Φx+(𝒟1Φxxx)+8(𝒟1Φx)+(𝒟1𝒟2Φ)]]

The operator 𝒦 defines a proper symplectic operator for the BN = 2 supersymmetric Kaup-Kupershmidt equation and satisfies the condition [3]

∫dx dθ1 dθ2 [α𝒦β⋆γ+β𝒦γ⋆α+γ𝒦α⋆β]=0.(5.18)

To finish this section, let us notice that all our formulas possess the O₂ superpartners.

6. Conclusion

In this paper, the method of the BN = 2 supercomplexification has been applied to the supersymmetrization of known soliton equations. In that manner, we obtained new supersymmetric KdV equation with its odd bi-Hamiltonian and Lax representation. Also, the BN = 2 supercomplexification of the Sawada-Kotera with its Lax representation and Kaup-Kupershmidt equations have been discussed. Unfortunately, we have been not able to find Lax representation for the BN = 2 Kaup-Kupershmidt equation. The unexpected feature of the supercompexification is appearance of the odd Hamiltonians operators and superfermionic conserved currents. The O₂ invariance of the conserved currents and Hamiltonian operators has a special meaning here. It is similar to the invariance of the conserved currents in the complex soliton system. For example, plugging the function u ⇒ u+iv to some conserved current H = H(u, u_x, ...) we obtain H ⇒ H_r +iH_i where H_r and H_i are conserved too. In the N = 2 supercomplex version if H is conserved then O₂(H) is also conserved. The supersymmetric Lax operator, which generates the BN = 2 supercomplex KdV equation, generates also the superfermionic conserved currents. The bosonic part of this Lax operator generates the complex KdV equation. However, we do not know how it is possible to obtain the conserved currents of complex KdV equation using this operator. On the other hand, it seems that the supercomplexification is a general method and could be applied to wide classes of integrable equations.

Acknowledgements

I would like to thank the anonymous referees for the constructive remarks.

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Journal: Journal of Nonlinear Mathematical Physics
Volume-Issue: 26 - 2
Pages: 294 - 312
Publication Date: 2021/01/06
ISSN (Online): 1776-0852
ISSN (Print): 1402-9251
DOI: 10.1080/14029251.2019.1591732 How to use a DOI?
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TY  - JOUR
AU  - Ziemowit Popowicz
PY  - 2021
DA  - 2021/01/06
TI  - N = 2 Supercomplexification of the Korteweg-de Vries, Sawada-Kotera and Kaup-Kupershmidt Equations
JO  - Journal of Nonlinear Mathematical Physics
SP  - 294
EP  - 312
VL  - 26
IS  - 2
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2019.1591732
DO  - 10.1080/14029251.2019.1591732
ID  - Popowicz2021
ER  -

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