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Miura Map between Lattice KP and its Modification is Canonical

We consider the Miura map between the lattice KP hierarchy and the lattice modified KP hierarchy and prove that the map is canonical not only between the first Hamiltonian structures, but also between the second Hamiltonian structures.


Introduction
It is well-known that Miura map, a transformation between the KdV equation and MKdV equation, plays a central role in the development of soliton theory. Indeed, the celebrated Inverse Scattering Method for solving nonlinear equations starts with the Miura map [11]. This type of transformations turns out to exist in the context of other integrable equations (see [1]- [10][13]- [14] and the references there).
Kupershmidt, in a recent paper [8], considered the canonical properties of Miura maps between KP and MKP hierarchies. He shown that, both in continuous and discrete cases, Miura transformations are canonical between the first Hamiltonian structures. For the ordinary or continuous KP and its modification, Shaw and Tu [13] generalized the results of Kupershimdt and proved that the very Miura map is also canonical between the second Hamiltonian structures.
We will consider the canonical property of the Miura map between the Lattice MKP (lMKP) and the Lattice KP (lKP) hierarchies. The lKP hierarchy is a bi-Hamiltonian system and two Hamiltonian structures were constructed by using the residue calculus in [7]. For the lMKP hierarchy, the first Hamiltonian structure was also found in [8]. A slight different version of the lMKP hierarchy was proposed by Oevel and he further obtained the bi-Hamiltonian description for this hierarchy by means of r-matrix approach [12]. By introducing a parameter, we unify Kupershmidt's version of the lMKP hierarchy and Oevel's version into a single system. Our main purpose of the paper is to prove that Kupershmidt's Miura map is a canonical map not only between the first Hamiltonian structures of lKP and lMKP, but also between the second Hamiltonian structures.
The paper is organized as follows. In the next section, we introduce notations and recall the relevant formulae such as bi-Hamiltonian structures of the lKP and lMKP hierarchies. In section 3 and section 4, we show that Kupershmidt's Miura map is a canonical transformation for the first Hamiltonian structures and the second Hamiltonian structures respectively. The last section is intended to summary and discussions.

Background and Notations
To introduce the lKP and lMKP hierarchies, we consider the algebra of shift operators where u j are scalar functions of integer n. The shift operator T is given by and for arbitrary integer k, For any operator ξ = j u j T j ∈ g, the projections to various shift orders are denoted by From the shift operator T , we also have the difference operator and its formal inverse Another important notation is so called the trace, which is defined as this permits us to identify g and its dual by the metric g * : < u * , u >= tr(u * u). It can be shown that the metric is bi-invariant. The lKP hierarchy is defined by the following Lax operator and the flow equations are constructed as The lKP hierarchy (2) is a bi-Hamiltonian system. Its two Hamiltonian structures are constructed by means of the residue calculus in [7]. Recently, Oevel proposed a r−matrix setting for these Hamiltonian structures. The two Hamiltonian structures are given by the following Poisson tensors where ρ is a skew-symmetric linear map on the algebra g 0 given explicitly by and As for the lMKP hierarchy, we consider the following Lax operator and the flow equations are represented by where α is a constant. The case α = 0 and the case α = 1 were considered by Kupershmidt and Oevel respectively. In these two cases, the mlKP hierarchy (8) is a bi-Hamiltonian system. When α = 0, the first Haimltonian structure of (8) is found by Kupershimdt in the context of the residue calculus, it is not clear how to construct the second one this way. Oevel, in the case α = 1, gives the bi-Hamiltonian structures by means of r−matrix approach.
Consider the linear operator on g by direct calculations, it is found that that above r solves the modified Yang-Baxter equation only and only if α = 0 or α = 1. As we mentioned above, these are exactly the two cases studied by Kupershimdt and by Oevel. In the following, our parameter α will take the value either one or zero. The above r-matrix leads to the first Hamiltonian structure for the lMKP hierarchy.
To get the second Poisson tensor, one may use Suris's construction [15] by considering the following linear operators When α = 1, the above operators are those presented by Oevel and are lead to the second Poisson tensor for this case. It can be proved that in the case α = 0, these operators are satisfied the conditions of Suris's theorem (or the theorem 1 of Oevel [12]), therefore they also lead to a Poisson tensor, this time is for Kupershimidt' case. Unifying both Kupershmidt's case and Oevel's case, we have the following two Poisson tensors where ρ is the one defined by (5) and ∇H is parametrized as In the remaining part of this section, we introduce the Miura map between lKP hierarchy and lMKP hierarchy following Kupershmidt. With the aid of a new field w, we introduce a map between lKP and mlKP hierarchies via the conjugacy comparing the coefficients of different power of the shift operator of two sides leads to a transformation

Let us introduce the new notations
By eliminating the intermediate variable w, we reach the Miura map between the two sets of variables this is the Miura map constructed in [8]. Now we prove that if L solves the mlKP hierarchy, L = e w Le −w solves the lKP hierarchy. From q = e w (1) −w , we obtain q t = q(T − 1)w t . On the other hand, the time evolution of q can be read from the mlKP hierarchy, that is where we used (L n ) 0 = (L n ) 0 and e w (L n ) ≥1 = (L n ) ≥1 e w . Thus the Miura map (12) indeed converts the lMKP hierarchy into the lKP hierarchy.

Canonical Properties for First Hamiltonian Structures
In this section, we prove that Miura map is canonical between the first Hamiltonian structures. First we calculate the Hamiltonian matrices from the Poisson tensors (3) and (9). By substituting (6) into P 1 and (11) into P 2 , it is straightforward to get and where B The Jacobian matrix of the Miura map (12) is easily calculated as where δ i j is the standard Kronecker symbol and D n is the abbreviated notation for the Fréchet derivative given by We need to calculate the matrix operator JB lM KP 1 J † , but first it is easy to find that now the entries of the first row of the JB lM KP 1 J † are seen as which coincide with the (B lKP 1 ) 0,m . It is noticed that we have used the second formula of (17). Therefore, for the first row and the first column, two matrix operators B lKP 1 and JB lM KP 1 J † are just the same as expected. We turn our attention to other entries of matrices. We find that αR m (a m T n−m − T n−m a n + T −m a n − a m T n )R n + R m (T n a m+n − a n+m T −m )R n = R m (T n a m+n − a n+m T −m )R n , now we use the formula in [8] R n T m R m = T m R n+m , and obtain the desired the results (JB lM KP 1 J † ) mn = (B lKP 1 ) mn . Thus, Miura map is indeed canonical.

Canonical Property for Second Hamiltonian Structures
We now show that the Miura map (12) is also canonical between the second Hamiltonian structure of the lKP hierarchy and the second Hamiltonian structure of the lMKP hierarchy. As in last section, we first calculate the Hamiltonian matrix operators from the Poisson tensors (4) and (10). The calculation in the present case is a bit cumbersome although it is straightforward. For the lKP hierarchy we have For the mlKP hierarchy, we have Thus, the matrix operator JB lM KP 2 reads as With all these formulae in hand, we find that the entries of the first row of JB lM KP which hold identically. For the remaining entries, we have, (a n−l T l a m+l − a m+l T n−m−l a n−l ) R n , so we need to prove that above expression is (B lKP 2 ) mn . It is easy to see that (20) = (21) = 0 in terms of D m .

Conclusions and Discussions
We have proved that the canonical property of Miura map holds between the lKP hierarchy and the lMKP hierarchy, that is, it maps the bi-Hamiltonian structures of the lMKP hierarchy to those of the lKP hierarchy. In [8], the lattice KP hierarchy is extended and it turns out that the extended lattice KP hierarchy is isomorphic to the lattice MKP hierarchy. Since we are dealing a slight generalized version of lMKP hierarchy here (8), we have a different extended lKP hierarchy. Introducing a new field u and define the following invertible transformation It is easy to see that the first Hamiltonian matrix operator for our extended lKP hierarchy reads and the flow equations are given where the Hamiltonian H is the seam as in the lKP case. We could have a second Hamiltonian structure for the extended lKP hierarchy, but it is in a rather complicated form. So we omit it.
To conclude the paper, we point out that it seems interesting to prove the canonical property of the Miura map on the level of the Poisson tensors since that will hopefully make the proof more concise. For the Gelfand-Dickey hierarchy, such proof was given by Dickey ( [3]) and for the continuous KP hierarchy and the constrained KP hierarchy, it is provided in [13] and in [10] [14] respectively.