On some integrable lattice related by the Miura-type transformation to the Itoh-Narita-Bogoyavlenskii lattice

is known to be one of a representative in the class of integrable differential-difference equations of Volterra-type form v ′ i = f (v i-1 , v i , v i+1 ) [15]. As is known, equation ( 1) is related to the Volterra lattice a ′ i = a i (a i-1 -a i+1 ) by the Miura-type transformation [5]

The Volterra lattice gives natural integrable discretization of the Korteweg-de Vries (KdV) equation. A more general class of integrable differential-difference equations sharing this property is given by the Itoh-Narita-Bogoyavlenskii (INB) lattice [2,3,6]

This equation primarily known in the literature as the Bogoyavlenskii lattice was considered in [6] as natural integrable generalization of the Volterra lattice -the equation admitting bilinear Hirota’s form and soliton solutions. In [3], Itoh used Lotka-Volterra finite-dimensional dynamical systems, which can be obtained by imposing on (3) specific periodicity conditions a i+2n+1 = a i . The Volterra lattice and its generalization (3) share the property that it admit integrable hierarchy. This means that the flow defined on suitable phase space by the evolution equation ( 3) can be included into infinite set of the pairwise commuting flows. We have shown in [10,11] that corresponding integrable hierarchies for (3) and for some number of integrable lattices are directly related to the KP hierarchy.

The goal of this paper is to present some class of differential-difference equations each of which is related to the INB lattice with any n ≥ 1 by some Miura-type transformation generalizing (2). In [12,13], we showed explicit form for INB lattice hierarchy in terms of some homogeneous polynomials S l s [a]. Using these results, we give the explicit form of modified evolution equations on the field v = v i governing corresponding integrable hierarchies in terms of some rational functions Sl s [v]. In section 3, we study Darboux transformation for underlying linear equations of modified integrable hierarchies and as a result derive some discrete quadratic equation parameterized by n ≥ 1, which is shown to relate two solutions {v i } and {v i } of modified equation yielding the first flow in corresponding integrable hierarchy. We claim but do not prove in this paper that this algebraic relation is also compatible with higher flows of modified integrable hierarchy. We can interpret the quadratic equation F [v, v] = 0 as a two-dimensional equation on {v i,j }. We show that this equation can be written in the form of the relation I i+1,j = I i,j with an appropriate integral I j . In the case n = 1, this two-dimensional equation can be treated as a deautonomization of the lattice potential KdV (lpKdV) equation I i,j = (v i,j -v i+1,j+1 ) (v i+1,j -v i,j+1 ) = c. A similar relation for any n ≥ 2 gives some generalization of the lpKdV equation.

Let us consider the INB lattice (3) together with auxiliary linear equations on the KP wavefunction:

As was shown in [12,13], the linear equation governing higher flows of equation ( 3) can be written as

Coefficients of this equation are defined by some special polynomials S l s [a]. More exactly, let

y λ j +n(j-1) .

Then S l s (i) ≡ S l s (a i , . . . , a i+s+(l-1)n ). Integrable hierarchy of the flows for the INB lattice (3) is defined by the totality of evolution differential-difference equations [12,13]

for s ≥ 1.

It is known that the noninvertible substitution

serves as the Miura-type transformation between the INB lattice and its modified version [2] u

This equation can be written in the form of the following conservation law:

The latter suggests to introduce the potential 1/u i = v i-2n -v i-n+1 . Therefore, we come to the rational substitution

which is a generalization of (2). It sends solutions of the equation

to solutions of the INB equation (3). Moreover, we have the following.

Proposition 1 Integrable hierarchy of lattice ( 8) is given by evolution differentialdifference equations ‡

Therefore, we obtained integrable hierarchy (9) related to the INB lattice hierarchy via substitution (7). Since there exist a number of Miura-type transformations, which relate the INB lattice to its modifications (see, for example [2,9]) the question arises how many integrable equations related by the Miura-type transformations do exist?

Introduce the function γ i , such that

Clearly, in terms of γ i

Proposition 2 In virtue of ( 9) and (10),

Let ψ i = γ i φ i . In terms of this new wavefunction, linear equations (4) become

To derive these equations, we take into account proposition 2, namely, that

In a similar way we get the linear equation

(-1) j z s-j γ i+(s-j)n γ i Sj sn-1 (i-(j-1)n)φ i+(s-j)n (14) governing all the flows (9).

3. Darboux transformation for linear equations ( 12) and ( 13)

Let us show that linear equations ( 12) and ( 13) admit Darboux transformation

It is worth remarking that the latter does not depend on n. It is simple exercise to verify that (15) yields Darboux transformation for linear equ

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