Euler Equations Related to the Generalized Neveu-Schwarz Algebra

In this paper, we study supersymmetric or bi-superhamiltonian Euler equations related to the generalized Neveu-Schwarz algebra. As an application, we obtain several supersymmetric or bi-superhamiltonian generalizations of some well-known integrable systems including the coupled KdV equation, the 2-c...

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spelling	irk-123456789-1492052019-02-20T01:25:45Z Euler Equations Related to the Generalized Neveu-Schwarz Algebra Zuo, D. In this paper, we study supersymmetric or bi-superhamiltonian Euler equations related to the generalized Neveu-Schwarz algebra. As an application, we obtain several supersymmetric or bi-superhamiltonian generalizations of some well-known integrable systems including the coupled KdV equation, the 2-component Camassa-Holm equation and the 2-component Hunter-Saxton equation. To our knowledge, most of them are new. 2013 Article Euler Equations Related to the Generalized Neveu-Schwarz Algebra / D. Zuo // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 39 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 37K10; 35Q51 DOI: http://dx.doi.org/10.3842/SIGMA.2013.045 http://dspace.nbuv.gov.ua/handle/123456789/149205 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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language	English
description	In this paper, we study supersymmetric or bi-superhamiltonian Euler equations related to the generalized Neveu-Schwarz algebra. As an application, we obtain several supersymmetric or bi-superhamiltonian generalizations of some well-known integrable systems including the coupled KdV equation, the 2-component Camassa-Holm equation and the 2-component Hunter-Saxton equation. To our knowledge, most of them are new.
format	Article
author	Zuo, D.
spellingShingle	Zuo, D. Euler Equations Related to the Generalized Neveu-Schwarz Algebra Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Zuo, D.
author_sort	Zuo, D.
title	Euler Equations Related to the Generalized Neveu-Schwarz Algebra
title_short	Euler Equations Related to the Generalized Neveu-Schwarz Algebra
title_full	Euler Equations Related to the Generalized Neveu-Schwarz Algebra
title_fullStr	Euler Equations Related to the Generalized Neveu-Schwarz Algebra
title_full_unstemmed	Euler Equations Related to the Generalized Neveu-Schwarz Algebra
title_sort	euler equations related to the generalized neveu-schwarz algebra
publisher	Інститут математики НАН України
publishDate	2013
url	http://dspace.nbuv.gov.ua/handle/123456789/149205
citation_txt	Euler Equations Related to the Generalized Neveu-Schwarz Algebra / D. Zuo // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 39 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT zuod eulerequationsrelatedtothegeneralizedneveuschwarzalgebra
first_indexed	2025-07-12T21:38:29Z
last_indexed	2025-07-12T21:38:29Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 9 (2013), 045, 12 pages Euler Equations Related to the Generalized Neveu–Schwarz Algebra Dafeng ZUO †‡ † School of Mathematical Science, University of Science and Technology of China, Hefei 230026, P.R. China E-mail: dfzuo@ustc.edu.cn ‡ Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences, P.R. China Received March 11, 2013, in final form June 12, 2013; Published online June 16, 2013 http://dx.doi.org/10.3842/SIGMA.2013.045 Abstract. In this paper, we study supersymmetric or bi-superhamiltonian Euler equations related to the generalized Neveu–Schwarz algebra. As an application, we obtain several supersymmetric or bi-superhamiltonian generalizations of some well-known integrable sys- tems including the coupled KdV equation, the 2-component Camassa–Holm equation and the 2-component Hunter–Saxton equation. To our knowledge, most of them are new. Key words: supersymmetric; bi-superhamiltonian; Euler equations; generalized Neveu– Schwarz algebra 2010 Mathematics Subject Classification: 37K10; 35Q51 1 Introduction For a classical rigid body with a fixed point, the configuration space is the group SO(3) of rota- tions of three-dimensional Euclidean space. In 1765, L. Euler proposed the equations of motion of the rigid body describing as geodesics in SO(3), where SO(3) is provided with a left-invariant metric. In essence, the Euler theory of a rigid theory is fully described by this invariance. Let G be an arbitrary (possibly infinite-dimensional) Lie group and G the corresponding Lie algebra and G∗ the dual of G. V.I. Arnold in [3] suggested a general framework for Euler equations onG, which can be regarded as a configuration space of some physical systems. In this framework Euler equations describe geodesic flows w.r.t. suitable one-side invariant Riemannian metrics on G and can be given to a variety of conservative dynamical systems in mathematical physics, for instance, see [2, 4, 7, 8, 9, 11, 12, 14, 15, 16, 18, 19, 20, 21, 22, 24, 25, 28, 30, 32, 33, 35, 37, 39] and references therein. Since V. Ovsienko and B. Khesin in [35] interpreted the Kuper–KdV equation [23] as a geodesic flow equation on the superconformal group w.r.t. an L2-type metric, it has been attracted a lot of interest in studying super (fermionic or supersymmetric) anologue of Arnold’s approach, which has some different characteristic flavors, for instance [2, 11, 16, 23, 24, 25, 36, 38]. In this paper, we are interested in Euler equations related to the N = 1 generalized Neveu– Schwarz (GNS in brief) algebra G, which was introduced by P. Marcel, V. Ovsienko and C. Roger in [29] as a generalization of the N = 1 Neveu–Schwarz algebra and the extended Virasoro algebra. In [16], P. Guha and P.J. Ovler have studied the Euler equations related to the GNS algebra G and obtained fermionic versions of the 2-component Camassa–Holm equation and the Ito equation in some special metrics. Our motivations are twofold. One is to study the Euler equation related to G for a more general metric Mc1,c2,c3,c4,c5,c6 in (2.1) with six-parameters given dfzuo@ustc.edu.cn http://dx.doi.org/10.3842/SIGMA.2013.045 2 D. Zuo by 〈F̂ , Ĝ〉 = ∫ S1 ( c1fg + c2fxgx + c3φ∂ −1χ+ c4φxχ+ c5ab+ c6α∂ −1β ) dx+ ~σ · ~τ , which can be regarded as a super-version of Sobolev-metrics in the super space. The other is to study the condition under which Euler equations are supersymmetric or bi-superhamiltonian. Our main results is to show that � the Euler equation is bi-superhamiltonian supersymmetric when the metric is � Mc1,c2, 1 4 c1,c2,c5,−c5 Yes No (if c1 6= 0) Mc1,c2,c1,c2,c5,−c5 only find a superhamiltonian Yes structure (if c1 6= 0) M0,c2,0,c2,c5,−c5 Yes Yes As a byproduct, we obtain some supersymmetric or bi-superhamiltonian generalizations of some well-known integrable systems including the coupled KdV equation, the 2-component Camassa– Holm equation and the 2-component Hunter–Saxton equation. This paper is organized as follows. In Section 2, we calculate the Euler equation on G∗reg and discuss their Hamiltonian properties. In Section 3, we study bi-superhamiltonian Euler equations. Section 4 is devoted to describe supersymmetric Euler equations, also including a class of both supersymmetric and bi-superhamiltonian Euler equations. A few concluding remarks are given in the last section. 2 Euler equations related to the GNS algebra To be self-contained, let us recall the Anorld’s approach [4, 20, 21]. Let G be an arbitrary Lie group and G the corresponding Lie algebra and G∗ the dual of G. Firstly let us fix a energy quadratic form E(v) = 1 2〈v,Av〉 ∗ on G and consider right translations of this quadratic form on G. Then the energy quadratic form defines a right-invariant Riemannian metric on G. The geodesic flow on G w.r.t. this energy metric represents the extremals of the least action principle, i.e., the actual motions of our physical system. For a rigid body, one has to consider left translations. We next identify G and its dual G∗ with the help of E(·). This identification A : G → G∗, called an inertia operator, allows us to rewrite the Euler equation on G∗. It turns out that the Euler equation on G∗ is Hamiltonian w.r.t. a canonical Lie–Poisson structure on G∗. Notice that in some cases it turns out to be not only Hamiltonian, but also bihamiltonian. Moreover, the corresponding Hamiltonian function is −E(m) = −1 2〈A −1m,m〉∗ lifted from the Lie algebra G to its dual space G∗, where m = Av ∈ G∗. Definition 2.1 ([4, 20]). The Euler equation on G∗, corresponding to the right-invariant metric −E(m) = −1 2〈A −1m,m〉∗ on G, is given by the following explicit formula dm dt = −ad∗A−1mm, as an evolution of a point m ∈ G∗. In the following, we take G to be the N = 1 generalized Neveu–Schwarz algebra [34]. Let V be a Z2 graded vector space, i.e., V = VB ⊕ VF . An element v of VB (resp., VF ) is said to be even (resp., odd). The super commutator of a pair of elements v, w ∈ V is defined by [v, w] = vw − (−1)\|v\|\|w\|wv. Euler Equations Related to the Generalized Neveu–Schwarz Algebra 3 Let Ds ( S1 ) be the group of orientation preserving Sobolev Hs diffeomorphisms of the cir- cle and TidDs ( S1 ) the corresponding Lie algebra of vector fields, denoted by Vects ( S1 ) ={ f(x) d dx \|f(x) ∈ Hs ( S1 )} . We denote VB = Vects ( S1 ) ⊕ C∞ ( S1 ) ⊕ R3, VF = C∞ ( S1 ) ⊕ C∞ ( S1 ) . Definition 2.2 ([34]). The GNS algebra G is an algebra VB⊕VF with the commutation relation given by [F̂ , Ĝ] = (( fgx − fxg + 1 2 φχ ) d dx , ( fχx − 1 2 fxχ− gφx + 1 2 gxφ ) dx− 1 2 , fbx − axg + 1 2 φβ + 1 2 αχ, ( fβx + 1 2 fxβ − 1 2 axχ− gαx − 1 2 gxα+ 1 2 bxφ ) dx 1 2 , ~ω ) , where φ, χ, α and β are fermionic functions, and f , g, a and b are bosonic functions, and F̂ = ( f(x, t) d dx , φ(x, t)dx− 1 2 , a(x, t), α(x, t)dx 1 2 , ~σ ) ∈ G and Ĝ = ( g(x, t) d dx , χ(x, t)dx− 1 2 , b(x, t), β(x, t)dx 1 2 , ~τ ) ∈ G and ~σ, ~τ ∈ R3 and ~ω = (ω1, ω2, ω3) ∈ R3. Here ω1(F̂ , Ĝ) = ∫ S1 (fxgxx + φxχx)dx, ω2(F̂ , Ĝ) = ∫ S1 (fxxb− gxxa− φxβ + χxα)dx, ω3(F̂ , Ĝ) = ∫ S1 (2abx + 2αβ)dx. Let us denote G∗reg = C∞ ( S1 ) ⊕ C∞ ( S1 ) ⊕ C∞ ( S1 ) ⊕ C∞ ( S1 ) ⊕ R3 to be the regular part of the dual space G∗ to G, under the following pair 〈Û , F̂ 〉∗ = ∫ S1 (uf + ψφ+ va+ γα)dx+ ~ς · ~σ, where Û = ( u(x, t)dx2, ψ(x, t)dx 3 2 , v(x, t)dx, γ(x, t)dx 1 2 , ~ς ) ∈ G∗ and ~ς = (ς1, ς2, ς3) ∈ R3. By the definition, using integration by parts we have 〈ad∗ F̂ (Û), Ĝ〉∗ = −〈Û , [F̂ , Ĝ]〉∗ = − ∫ S1 ( u ( fgx − fxg + 1 2 φχ ) + ψ ( fχx − 1 2 fxχ− gφx + 1 2 gxφ ) + v ( fbx − axg + 1 2 φβ + 1 2 αχ ) + γ ( fβx + 1 2 fxβ − 1 2 axχ− gαx − 1 2 gxα+ 1 2 bxφ )) dx− ~ς · ~ω = ∫ S1 ( 2ufx + uxf − ς1fxxx + ς2axx + 3 2 ψφx + 1 2 ψxφ+ 1 2 γαx − 1 2 γxα+ vax ) gdx + ∫ S1 ( ς1φxx − ς2αx − 1 2 uφ− 1 2 vα+ 3 2 fxψ + fψx + 1 2 γax ) χdx + ∫ S1 ( (vf)x + 1 2 (γφ)x − ς2fxx + 2ς3ax ) bdx + ∫ S1 ( γxf + 1 2 γfx − 1 2 vφ+ ς2φx − 2ς3α ) βdx. 4 D. Zuo So the coadjoint action on G∗reg is given by ad∗ F̂ (Û) = (( 2ufx + uxf − ς1fxxx + ς2axx + 3 2 ψφx + 1 2 ψxφ+ 1 2 γαx − 1 2 γxα+ vax ) dx2,( ς1φxx − ς2αx − 1 2 uφ− 1 2 vα+ 3 2 fxψ + fψx + 1 2 γax ) dx 3 2 ,( (vf)x+ 1 2 (γφ)x− ς2fxx+ 2ς3ax ) dx, ( γxf+ 1 2 γfx− 1 2 vφ+ ς2φx− 2ς3α ) dx 1 2 , 0 ) . On G, let us introduce an inner product Mc1,c2,c3,c4,c5,c6 given by 〈F̂ , Ĝ〉 = ∫ S1 ( c1fg + c2fxgx + c3φ∂ −1χ+ c4φxχ+ c5ab+ c6α∂ −1β ) dx+ ~σ · ~τ , (2.1) which is a generalization of that in [11, 16]. By the Definition 2.1, the Euler equation on G∗reg for Mc1,c2,c3,c4,c5,c6 is dÛ dt = −ad∗A−1Û Û (2.2) as an evolution of a point Û= ( u(x, t)dx2, ψ(x, t)dx 3 2 , v(x, t), γ(x, t)dx 1 2 , ~ς ) ∈G∗, whereA : G→G∗ is an inertia operator defined by 〈F̂ , Ĝ〉 = 〈A(F̂ ), Ĝ〉∗. A direct computation shows that the inertia operator A : G → G∗ has the form A(F̂ ) = ( Λ(f)dx2,Θ(φ)dx 3 2 , c5adx, c6∂ −1αdx 1 2 , ~σ ) , where Λ(f) = c1f − c2fxx and Θ(φ) = c4φx − c3∂−1φ. Thus we have Proposition 2.3. The Euler equation (2.2) on G∗reg for Mc1,c2,c3,c4,c5,c6 reads ut = ς1fxxx − ς2axx − 2ufx − uxf − vax − 3 2 ψφx − 1 2 ψxφ− 1 2 γαx, ψt = 1 2 uφ+ 1 2 vα− ς1φxx + ς2αx − 3 2 fxψ − fψx − 1 2 γax, vt = ς2fxx − 2ς3ax − (vf)x − 1 2 (γφ)x, (2.3) γt = 1 2 vφ− γxf − 1 2 γfx − ς2φx + 2ς3α, where u = Λ(f) = c1f − c2fxx, ψ = Θ(φ) = c4φx − c3∂−1φ, v = c5a and γ = c6∂ −1α. Let us remark that the system (2.3) has been obtained in [16] with minor typos. But they didn’t discuss the condition under which the Euler equation (2.3) is supersymmetric or bi- superhamiltonian. According to Definition 2.1, the Euler equation (2.3) has a natural Hamiltonian description [4, 20, 21]. Let Fi : G∗ → R, i = 1, 2, be two arbitrary smooth functionals. The dual space G∗ carries a canonical Lie–Poisson bracket {F1, F2}2(Û) = 〈 Û , [ δF1 δÛ , δF2 δÛ ]〉∗ , Euler Equations Related to the Generalized Neveu–Schwarz Algebra 5 where Û ∈ G∗ and δFi δÛ = ( δFi δu , δFi δψ , δFi δv , δFi δγ , δFi δ~ς ) ∈ G, i = 1, 2. The induced superhamiltonian operator is given by J2 =  ς1∂ 3 − u∂ − ∂u −ψ∂ − 1 2∂ψ, −ς2∂ 2 − v∂ −γ∂ + 1 2∂γ −∂ψ − 1 2ψ∂ 1 2u− ς1∂ 2 −1 2γ∂ 1 2v + ς2∂ ς2∂ 2 − ∂v −1 2∂γ −2ς3∂ 0 −∂γ + 1 2γ∂ 1 2v − ς2∂ 0 2ς3  . (2.4) Proposition 2.4. The Euler equation (2.3) could be written as d dt (u, ψ, v, γ)T = J2 ( δH1 δu , δH1 δψ , δH1 δv , δH1 δγ )T (2.5) with the Hamiltonian H1 = 1 2 ∫ S1(uf + ψφ + va + γα)dx, where (·)T means the transpose of vectors. Proof. Indeed, for a functional F [u, ψ, v, γ], the variational derivatives δF δu , δF δψ , δF δv and δF δγ are defined by d dε \|ε=0 F [u+ εδu, ψ + εδψ, v + εδv, γ + εδγ] = ∫ ( δu δF δu + δψ δF δψ + δv δF δv + δγ δF δγ ) dx. (2.6) By using (2.6), we have δH1 δf = Λ(f), δH1 δφ = −Θ(φ), δH1 δa = v, δH1 δα = −γ. It follows from the definition of u, ψ and γ that δH1 δu = Λ−1 δH1 δu = f, δH1 δψ = −Θ−1 δH1 δφ = φ, δH1 δv = a, δH1 δγ = −∂ δH1 δα = α. (2.7) Hence, (2.5) could be easily verified by using (2.4) and (2.7). � 3 Bihamiltonian Euler equations on G∗reg Unless otherwise stated, in the following we use “(bi)hamiltonian” to denote “(bi)-superhamil- tonian”. In this section we want to study bihamiltonian Euler equations on G∗reg w.r.t. the metric Mc1,c2, 1 4 c1,c2,c5,−c5 and propose some new bihamiltonian and fermionic extensions of well- known integrable systems including coupled the KdV equation, the 2-CH equation and the 2-HS equation. 3.1 The frozen Lie–Poisson bracket on G∗ reg For the purpose of discussing possible bihamiltonian Euler equations, we introduce a frozen Lie–Poisson bracket on G∗reg defined by {F1, F2}1(Û) = 〈 Û0, [ δF1 δÛ , δF2 δÛ ]〉∗ , 6 D. Zuo for a fixed point Û0 ∈ G∗. The corresponding Hamiltonian equation is given by dÛ dt = −ad∗δH2 δÛ Û0 (3.1) for a functional H2 : G∗reg → R. If we could find a functional H2 and a suitable point Û0 ∈ G∗reg such that the system (3.1) coincides with (2.3). This means that the Euler equation (2.3) is bihamiltonian and could be written as d dt (u, ψ, v, γ)T = J1 ( δH2 δu , δH2 δψ , δH2 δv , δH2 δγ )T = J2 ( δH1 δu , δH1 δψ , δH1 δv , δH1 δγ )T with Hamiltonian operators J2 in (2.4) and J1 = J2\|Û=Û0 . Moreover, according to Proposi- tion 5.3 in [20], { , }1 and { , }2 are compatible for every freezing point Û0. 3.2 Bihamiltonian Euler equations on G∗ reg w.r.t. Mc1,c2, 1 4 c1,c2,c5,−c5 In this case, we have c1 = 4 c3, c2 = c4, c6 = −c5. By setting φ = ηx and α = µx, then u = Λ(f) = c1f − c2fxx, ψ = Π(η) = c2ηxx − 1 4 c1η, v = c5a, γ = −c5µ (3.2) and the Euler equation becomes ut = ς1fxxx − ς2axx − 2ufx − uxf − 3 2 ψηxx − 1 2 ψxηx − vax − 1 2 γµxx, ψt = 1 2 uηx − ς1ηxxx + ς2µxx − 3 2 fxψ − fψx + 1 2 (vµ)x, vt = ς2fxx − 2ς3ax − (vf)x − 1 2 (γηx)x, (3.3) γt = 1 2 vηx − γxf − 1 2 γfx − ς2ηxx + 2ς3µx. We are now in a position to state our main theorem. Theorem 3.1. The system (3.3) is bihamiltonian on G∗reg with a freezing point Û0 = ( c1 2 dx 2, 0, 0, 0, ( c2, 0, c5 2 )) ∈ G∗reg and a Hamiltionian functional H2 = ∫ S1 ( − ς1 2 ffxx + c1 2 f3 − c2 4 f2fxx − c2 2 fηxηxx − 3c1 8 fηηx + ς1 2 ηηxx − ς2afx + 1 2 avf + ς3a 2 + 1 2 aγηx + ς3µµx + ς2µηxx + 1 2 γµxf ) dx. Proof. Direct computation gives δH2 δf = ς2ax − ς1fxx + 3c1 2 f2 − c2ffxx − c2 2 f2x − c2 2 ηxηxx − 3c1 8 ηηxx + 1 2 av + 1 2 γµx, δH2 δη = 3c2 2 fxηxx + c2fηxxx + c2 2 fxxηx − 3c1 4 fηx − 3c1 8 fxη + ς1ηxx − ς2µxx + 1 2 (aγ)x, (3.4) δH2 δa = 2ς3a+ vf + 1 2 γηx − ς2fx, Euler Equations Related to the Generalized Neveu–Schwarz Algebra 7 δH2 δµ = γxf + 1 2 γfx − 1 2 vηx + ς2ηxx − 2ς3µx. Under the special freezing point Û0 = (c1 2 dx2, 0, 0, 0, ( c2, 0, c5 2 )) ∈ G∗reg, the system (3.1) reads ut = c2 ( δH2 δu ) xxx − c1 ( δH2 δu ) x , vt = −c5 ( δH2 δv ) x , ψt = c1 4 δH2 δψ − c2 ( δH2 δψ ) xx , γt = c5 δH2 δγ . (3.5) Using (3.2), we have δH2 δu = Λ−1 ( δH2 δf ) , δH2 δψ = Π−1 ( δH2 δη ) , c5 δH2 δv = δH2 δa , c5 δH2 δγ = −δH2 δµ . The system (3.5) becomes ut = − ( δH2 δf ) x , ψt = −δH2 δη , vt = − ( δH2 δa ) x , γt = −δH2 δµ , which is the desired system (3.3) due to (3.2) and (3.4). We thus complete the proof of the theorem. � 3.3 Examples Example 3.2 (an L2-type metric M1,0, 1 4 ,0,1,−1). The systems (3.3) reduces to ft = ς1fxxx − ς2axx − 3ffx + 3 8 ηηxx − aax + 1 2 µµxx, ηt = 4ς1ηxxx − 3fηx − 3 2 fxη − 4ς2µxx − 2(aµ)x, at = ς2fxx − 2ς3ax − (af)x + 1 2 (µηx)x, (3.6) µt = ς2ηxx − 2ς3µx − 1 2 aηx − µxf − 1 2 µfx. We call this system (3.6) to be a Kuper-2KdV equation. Especially, (1) if we set η = µ = 0, we have ft = ς1fxxx − ς2axx − 3ffx − aax, at = ς2fxx − 2ς3ax − (af)x, which is a two-component generalization of the KdV equation with three parameters including the Ito equation in [17] for ς1 6= 0, ς2 = ς3 = 0; (2) if we set ς1 = 1 2 , ς2 = 0, a = 0 and µ = 0, we have ft = 1 2 fxxx − 3ffx + 3 8 ηηxx, ηt = 2ηxxx − 3fηx − 3 2 fxη, which is the Kuper–KdV equation in [23]. 8 D. Zuo Let us remark that when we choose ς1 = 1 4 and ς2 = ς3 = 0, up to a rescaling, the Kuper–2KdV equation (3.6) is the super-Ito equation (equation (4.14b) in [1]) proposed by M. Antonowicz and A.P. Fordy, which has three Hamiltonian structures. According to our terminologies, we would like to call it the Kuper–Ito equation. Example 3.3 (an H1-type metric M1,1, 1 4 ,1,1,−1). The systems (3.3) reduces to ft − fxxt = ς1fxxx − ς2axx − 3ffx + 2fxfxx + ffxxx + 3 8 ηηxx + 1 2 ηxηxxx − aax + 1 2 µµxx, ηxxt − 1 4 ηt = 3 4 fηx + 3 8 fxη − fηxxx − 1 2 fxxηx − 3 2 fxηxx − ς1ηxxx + ς2µxx − 1 2 (aµ)x, (3.7) at = ς2fxx − 2ς3ax − (af)x + 1 2 (µηx)x, µt = ς2ηxx − 2ς3µx − 1 2 aηx − µxf − 1 2 µfx. We call this system (3.7) to be a Kuper–2CH equation. Especially, (1) if we set ς1 = ς2 = ς3 = 0 and η = µ = 0, we have ft − fxxt = 2fxfxx + ffxxx − 3ffx − aax, at = −(af)x, which is the 2-CH equation in [6, 13]; (2) if by setting ς1 = ς2 = ς3 = 0, a = 0 and µ = 0, the system (3.7) becomes ft − fxxt = ffxxx + 2fxfxx − 3ffx + 3 8 ηηxx + 1 2 ηxηxxx, ηxxt − 1 4 ηt = 3 4 fηx + 3 8 fxη − fηxxx − 1 2 fxxηx − 3 2 fxηxx, which is the Kuper–CH equation in [10, 38]. 4 Supersymmetric Euler equations on G∗reg In this section, we want to discuss a class of supersymmetric Euler equations on G∗ associa- ted to a special metric Mc1,c2,c1,c2,c5,−c5 . Moreover, we present a class of supersymmetric and bihamiltonian Euler equations. 4.1 Supersymmetric Euler equations on G∗ reg w.r.t. Mc1,c2,c1,c2,c5,−c5 In this case, we have c1 = c3, c2 = c4, c6 = −c5. By setting φ = ηx and α = µx, we obtain u = c1f − c2fxx, ψ = c2ηxx − c1η, v = c5a, γ = −c5µ. Let us define a superderivative D by D = ∂θ + θ∂x and introduce two superfields Φ = η + θf, Ω = µ+ θa, where θ is an odd coordinate. A direct computation gives Euler Equations Related to the Generalized Neveu–Schwarz Algebra 9 Theorem 4.1. The Euler equation (2.3) on G∗reg w.r.t. Mc1,c2,c1,c2,c5,−c5 is invariant under the supersymmetric transformation δf = θηx, δη = θf, δa = θµx, δµ = θa and could be rewritten as c1Φt − c2D4Φt = ς1D 6Φ− 3 2 c1 ( ΦD3Φ +DΦD2Φ ) + c2 ( DΦD6Φ + 1 2 D2ΦD5Φ + 3 2 D3ΦD4Φ ) − ς2D4Ω + 1 2 c5 ( DΩD2Ω + ΩD3Ω ) , (4.1) c5Ωt = ς2D 4Φ− 2ς3D 2Ω− 1 2 c5 ( DΩD2Φ + 2D2ΩDΦ + ΩD3Φ ) . 4.2 Examples Example 4.2 (another L2-type metric M1,0,1,0,1,−1). The system (4.1) reduces to Φt = ς1D 6Φ− 3 2 ( ΦD3Φ +DΦD2Φ ) − ς2D4Ω + 1 2 ( DΩD2Ω + ΩD3Ω ) , Ωt = ς2D 4Φ− 2ς3D 2Ω− 1 2 ( DΩD2Φ + 2D2ΩDΦ + ΩD3Φ ) . (4.2) We call this system (4.2) to be a super-2KdV equation. Especially, (1) if we set η = µ = 0, we recover the two-component KdV equation again; (2) but if we choose ς1 = 1 2 , ς2 = ς3 = 0 and Ω = 0, the system (4.2) becomes Φt = 1 2 D6Φ− 3 2 ( ΦD3Φ +DΦD2Φ ) , equivalently in componentwise forms, ft = 1 2 fxxx − 3ffx + 3 2 ηηxx, ηt = 1 2 ηxxx − 3 2 (fη)x, which is the super-KdV equation in [31]. Example 4.3 (another H1-type metric M1,1,1,1,1,−1). The system (4.1) reduces to Φt −D4Φt = ς1D 6Φ− 3 2 (ΦD3Φ +DΦD2Φ) + ( DΦD6Φ + 1 2 D2ΦD5Φ + 3 2 D3ΦD4Φ ) − ς2D4Ω + 1 2 ( DΩD2Ω + ΩD3Ω ) , (4.3) Ωt = ς2D 4Φ− 2ς3D 2Ω− 1 2 ( DΩD2Φ + 2D2ΩDΦ + ΩD3Φ ) . We call this system (4.3) to be a super-2CH equation. Especially, (1) if we set ς1 = ς2 = ς3 = 0 and η = µ = 0, we obtain the 2-CH equation in [6, 13] again; (2) but if by setting ς1 = ς2 = ς3 = 0 and Ω = 0, the system (4.3) becomes Φt −D4Φt = ( DΦD6Φ + 1 2 D2ΦD5Φ + 3 2 D3ΦD4Φ ) − 3 2 ( ΦD3Φ +DΦD2Φ ) , which is the super-CH equation in [11]. 10 D. Zuo 4.3 Supersymmetric and bihamiltonian Euler equations w.r.t. M0,c2,0,c2,c5,−c5 on G∗ reg Let us combine with Theorem 3.1 and Theorem 4.1, we have Theorem 4.4. The Euler equation on G∗reg w.r.t. the metric M0,c2,0,c2,c5,−c5 is supersymmetric and bihamiltonian. Example 4.5 (an Ḣ1-type metric M0,1,0,1,1,−1). The systems (4.1) reduces to −D4Φt = ς1D 6Φ + ( DΦD6Φ + 1 2 D2ΦD5Φ + 3 2 D3ΦD4Φ ) − ς2D4Ω + 1 2 ( DΩD2Ω + ΩD3Ω ) , (4.4) Ωt = ς2D 4Φ− 2ς3D 2Ω− 1 2 ( DΩD2Φ + 2D2ΩDΦ + ΩD3Φ ) . We call this system (4.4) to be a super-2HS equation. Especially, (i) if we set ς1 = ς2 = ς3 = 0 and η = µ = 0, we have −fxxt = 2fxfxx + ffxxx − aax, at = −(af)x, which is a 2-HS equation in [39]; (ii) if by setting ς1 = ς2 = ς3 = 0 and Ω = 0, the system (4.4) becomes −D4Φt = DΦD6Φ + 1 2 D2ΦD5Φ + 3 2 D3ΦD4Φ, which is the super-HS equation in [5, 24]. 5 Concluding remarks We have described Euler equations associated to the GNS algebra and shown that under which conditions there are superymmetric or bihamiltonian. Here we only obtain some sufficient conditions but not necessary conditions. As an application, we have naturally presented several generalizations of some well-known integrable systems including the Ito equation, the 2-CH equation and the 2-HS equation. It is well-known that the Virasoro algebra, the extended Virasoro algebra and the Neveu–Schwarz algebras are subalgebras of the GNS algebra. Thus our result could be regarded as a generalization of that related to those subalgebras, see for instances [2, 4, 7, 8, 9, 11, 12, 14, 15, 16, 18, 19, 20, 21, 22, 24, 25, 28, 30, 32, 33, 35, 37, 39] and references therein. In the past twenty years, in this subject it has grown in many different directions, please see [21] and references therein. Finally let us point out that in this paper all super-Hamiltonian operators are even. Recently, in [5, 26, 27], the classical Harry–Dym equation is supersymmetrized in two ways, either by even supersymmetric Hamiltonian operators or by odd supersymmetric Hamiltonian operators. Notice that the HS equation is one of a member of negative Harry–Dym hierarchy. It would be interesting to investigate whether the above point of view has an extension to the odd supersymmetric integrable system, for instance, the odd HS equation. 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Mc1,c2,14c1,c2,c5,-c5 3.3 Examples 4 Supersymmetric Euler equations on Greg* 4.1 Supersymmetric Euler equations on Greg* w.r.t. Mc1,c2,c1,c2,c5,-c5 4.2 Examples 4.3 Supersymmetric and bihamiltonian Euler equations w.r.t. M0,c2,0,c2, c5,-c5 on Greg* 5 Concluding remarks References

Euler Equations Related to the Generalized Neveu-Schwarz Algebra

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