For the Harry Dym hierarchy, a non-standard Lax formulation is deduced from that of Korteweg-de Vries (KdV) equation through a reciprocal transformation. By supersymmetrizing this Lax operator, a new N=2 supersymmetric extension of the Harry Dym hierarchy is constructed, and is further shown to be linked to one of the N=2 supersymmetric KdV equations through superconformal transformation. The bosonic limit of this new N=2 supersymmetric Harry Dym equation is related to a coupled system of KdV-MKdV equations.
The classical integrable systems, alias soliton equations, admit various extensions, among which supersymmetric extensions have been studied extensively in the past three decades. Dealing with supersymmetrization of classical integrable systems, we should distinguish the extended case from the non-extended case. In the non-extended case, one just generalizes the classical theory by pairing classical bosonic fields with some fermionic fields. When all fermionic sectors vanish, the supersymmetric theory degenerates to the known classical one. However in the extended case, in addition to fermionic fields, new bosonic fields are also introduced into the supersymmetric theory. As a bonus, if we throw fermionic sectors away, certain new classical systems may appear.
As two famous prototypes in the theory of integrable systems, both the Kortewegde Vries (KdV) and the Harry Dym (HD) equations were successfully supersymmetrized in both non-extended and extended cases. Here we only mention some results on N = 2 supersymmetric KdV and HD equations, which belong to the extended case.
There are three inequivalent N = 2 supersymmetric KdV equations, which are encoded in the the following model (denoted by SKdV a )
where Φ = Φ(y, ̺ 1 , ̺ 2 , τ ) is a bosonic super field, and
denote two super derivatives. Through different approaches [7,1], SKdV a was shown to be integrable when the parameter a takes -2, 1 or 4. For these three cases, various integrable features have been revealed, including Lax representations [7,13], bi-Hamiltonian structures [12,5] and so on. By considering the most general N = 2 Lax operator of differential type, two inequivalent N = 2 supersymmetric HD equations were constructed [2], which are respectively given by
where W = W (x, θ 1 , θ 2 , t) is a bosonic super field and
) denote super derivatives. Apart from Lax formulations, few properties have been established for these two equations until now. It is noticed that the bosonic limits of them were shown to be transformed into previously known integrable systems by reciprocal transformations [14]. Very recently, via superconformal transformation both N = 2 supersymmetric HD equations were related to supersymmetric KdV equations [15], more precisely, the equation ( 2) is changed into SKdV 4 by superconformal transformation, while the equation (3) to SKdV -2 . Hence, the superconformal transformation could be seen as a generalization of reciprocal transformation for N = 2 supersymmetric systems.
Motivated by these interesting connections and the fact that there are three rather than two N = 2 supersymmetric KdV equations, we conjecture at least one more N = 2 supersymmetric HD equation would exist, which is expected to serve as an counterpart of SKdV 1 . Guided by the fact that SKdV 1 has a non-standard Lax formulation [13]
which supplies a non-standard Lax operator for classical KdV equation, this paper aims at figuring out the missing N = 2 supersymmetric HD equation by supersymmetrizing a suitable non-standard Lax operator of the classical HD hierarchy. The paper is organized as follows. In section 2, based on the well-known nonstandard Lax KdV hierarchy, a non-standard Lax formulation is constructed for the classical HD hierarchy via reciprocal transformation and gauge transformation. In section 3, we generalize the non-standard Lax operator to N = 2 super space, and construct a new N = 2 supersymmetric HD hierarchy from it. Furthermore, as we anticipated, the Lax operator of our new N = 2 supersymmetric HD equation is shown to be linked to that of SKdV 1 through superconformal transformation. In section 4, the bosonic limit of our new equation is transformed into the bosonic limit of SKdV 1 by reciprocal transformation. The last section is devoted to some conclusions.
In the theory of classical integrable systems, reciprocal transformation plays an important role to connect some evolution equations, and is a powerful tool to investigate integrable properties. For instance, the HD equation is connected to the KdV equation by this transformation [3]. On the Lax operator level we have two possibilities to connect the HD hierarchy with the KdV hierarchy because there are two different Lax operators for the KdV equation: the so-called standard and non-standard Lax operators.
Let us briefly demonstrate this connection for the standard Lax operator of the KdV equation [6]. The KdV hierarchy can be formulated as
where L is the standard Lax operator, which is given by L s = ∂ 2 y + u, and the subscript ≥0 denotes the projection to the differential part. In the case n = 1, the Lax equation ( 5) produces the KdV equation (τ ≡ τ 3 )
Applying gauge transformation on the operator L s , we obtain
Now let u = v x -v 2 , then we conclude that the operator Ls is the Lax operator of the modified KdV equation. Taking the Liouville transformation ∂ y = w(y)∂ x , the Lax operator Ls is brought into
assuming that v = wy 2w the Lax operator Ls is nothing b
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