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-component Camassa-Holm equation and the 2-component Hunter-Saxton equation. To our knowledge, most of them are new.


Introduction
For a classical rigid body with a fixed point, the configuration space is the group SO(3) of rotations 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 on G, 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.
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 M c 1 ,c 2 ,c 3 ,c 4 ,c 5 ,c 6 in (2.1) with six-parameters given 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 Yes Yes As a byproduct, we obtain some supersymmetric or bi-superhamiltonian generalizations of some

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 −1 m, m * lifted from the Lie algebra G to its dual space G * , where m = Av ∈ G * . In the following, we take G to be the N = 1 generalized Neveu-Schwarz algebra [34]. Let V be a Z 2 graded vector space, i.e., V = V B ⊕ V F . An element v of V B (resp., V F ) is said to be even (resp., odd). The super commutator of a pair of elements v, w ∈ V is defined by Let D s S 1 be the group of orientation preserving Sobolev H s diffeomorphisms of the circle and T id D s S 1 the corresponding Lie algebra of vector fields, denoted by Vect 34]). The GNS algebra G is an algebra V B ⊕ V F with the commutation relation given by where φ, χ, α and β are fermionic functions, and f , g, a and b are bosonic functions, and Here Let us denote to be the regular part of the dual space G * to G, under the following pair By the definition, using integration by parts we have So the coadjoint action on G * reg is given by On G, let us introduce an inner product M c 1 ,c 2 ,c 3 ,c 4 ,c 5 ,c 6 given by which is a generalization of that in [11,16]. By the Definition 2.1, the Euler equation A direct computation shows that the inertia operator A : G → G * has the form 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 bisuperhamiltonian.
3 Bihamiltonian Euler equations on G * reg Unless otherwise stated, in the following we use "(bi)hamiltonian" to denote "(bi)-superhamiltonian". In this section we want to study bihamiltonian Euler equations on G *

Bihamiltonian Euler equations on
In this case, we have By setting φ = η x and α = µ x , then and the Euler equation becomes We are now in a position to state our main theorem.
Proof . Direct computation gives Under the special freezing point Using (3.2), we have The system (3.5) becomes which is the desired system (3.3) due to (3.2) and (3.4). We thus complete the proof of the theorem.
Let us remark that when we choose ς 1 = 1 4 and ς 2 = ς 3 = 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.

Supersymmetric Euler equations on G * reg
In this section, we want to discuss a class of supersymmetric Euler equations on G * associated to a special metric M c 1 ,c 2 ,c 1 ,c 2 ,c 5 ,−c 5 . Moreover, we present a class of supersymmetric and bihamiltonian Euler equations.

Supersymmetric Euler equations on
In this case, we have By setting φ = η x and α = µ x , we obtain Let us define a superderivative D by D = ∂ θ + θ∂ x and introduce two superfields where θ is an odd coordinate. A direct computation gives

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.