1 Introduction

\FirstPageHeading\ShortArticleName

On Miura Transformations and Volterra-Type Equations

\ArticleName

On Miura Transformations
and Volterra-Type Equations Associated
with the Adler–Bobenko–Suris Equations

\Author

Decio LEVI , Matteo PETRERA , Christian SCIMITERNA  and Ravil YAMILOV 

\AuthorNameForHeading

D. Levi, M. Petrera, C. Scimiterna and R. Yamilov

\Address

 Dipartimento di Ingegneria Elettronica,Università degli Studi Roma Tre and Sezione INFN,
 Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy \EmailDlevi@fis.uniroma3.it

\Address

 Dipartimento di Fisica E. Amaldi, Università degli Studi Roma Tre and Sezione INFN,
 Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy \EmailDpetrera@fis.uniroma3.it, scimiterna@fis.uniroma3.it

\Address

 Ufa Institute of Mathematics, 112 Chernyshevsky Str., Ufa 450077, Russia \EmailDRvlYamilov@matem.anrb.ru

\ArticleDates

Received August 29, 2008, in final form October 30, 2008; Published online November 08, 2008

\Abstract

We construct Miura transformations mapping the scalar spectral problems of the integrable lattice equations belonging to the Adler–Bobenko–Suris (ABS) list into the discrete Schrödinger spectral problem associated with Volterra-type equations. We show that the ABS equations correspond to Bäcklund transformations for some particular cases of the discrete Krichever–Novikov equation found by Yamilov (YdKN equation). This enables us to construct new generalized symmetries for the ABS equations. The same can be said about the generalizations of the ABS equations introduced by Tongas, Tsoubelis and Xenitidis. All of them generate Bäcklund transformations for the YdKN equation. The higher order generalized symmetries we construct in the present paper confirm their integrability.

\Keywords

Miura transformations; generalized symmetries; ABS lattice equations

\Classification

37K10; 37L20; 39A05

1 Introduction

The discovery of new two-dimensional integrable partial difference equations (or -lattice equations) is always a very challenging problem as, by proper continuous limits, many other results on differential-difference and partial differential equations may be obtained. Moreover many physical and biological applications involve discrete systems, see for instance [14, 26] and references therein.

The theory of nonlinear integrable differential equations got a boost when Gardner, Green, Kruskal and Miura introduced the Inverse Scattering Method for the solution of the Korteweg–de Vries equation. A summary of these results can be found in the Encyclopedia of Mathematical Physics [13]. A few techniques have been introduced to classify integrable partial differential equations. Let us just mention the classification scheme introduced by Shabat, using the formal symmetry approach (see [22] for a review). This approach has been successfully extended to the differential-difference case by Yamilov [31, 32, 21]. In the completely discrete case the situation turns out to be quite different. For instance, in the case of -lattice equations the formal symmetry technique does not work. In this framework, the first exhaustive classifications of families of lattice equations have been presented in [2] by Adler and in [3, 4] by Adler, Bobenko and Suris.

In the present paper we shall consider the Adler–Bobenko–Suris (ABS) classification of -lattice equations defined on the square lattice [3]. We refer to the papers [4, 25, 29, 18, 19, 28] for some recents results about these equations. Our main purpose is the analysis of their transformation properties. In fact, our aim is, on the one hand, to present new Miura transformations between the ABS equations and Volterra-type difference equations and on the other hand, to show that the ABS equations correspond to Bäcklund transformations for some particular cases of the discrete Krichever–Novikov equation found by Yamilov (YdKN equation) [31].

Section 2 is devoted to a short review of the integrable -lattice equations derived in [3] and to present details on their matrix and scalar spectral problems. In Section 3, by transforming the obtained scalar spectral problems into the discrete Schrödinger spectral problem associated with the Volterra lattice we will be able to connect the ABS equation with Volterra-type equations. In Section 4 we prove that the ABS equations correspond to Bäcklund transformations for certain subcases of the YdKN equation. Using this result and a master symmetry of the YdKN equation, we construct new generalized symmetries for the ABS list. Then we discuss the integrability of a class of non-autonomous ABS equations and of a generalization of the ABS equations introduced by Tongas, Tsoubelis and Xenitidis in [29]. Section 5 is devoted to some concluding remarks.

2 A short review of the ABS equations

A two-dimensional partial difference equation is a functional relation among the values of a function at different points of the lattices of indices . It involves the independent variables , and the lattice parameters ,

For the dependent variable we shall adopt the following notation throughout the paper

(1)

We consider here the ABS list of integrable lattice equations, namely those affine linear (i.e. polynomial of degree one in each argument) partial difference equations of the form

(2)

whose integrability is based on the consistency around a cube [3, 4]. The function depends explicitly on the values of at the vertices of an elementary quadrilateral, i.e. , where . The lattice parameters , may, in general, depend on the variables , , i.e. , . However, we shall discuss such non-autonomous extensions in Section 4.

The complete list of the ABS equations can be found in [3]. Their integrability holds by construction since the consistency around a cube furnishes their Lax pairs [3, 10, 23]. The ABS equations are given by the list H

and the list Q

The coefficients ’s appearing in equation (Q4) are connected to and by the relations

with , , .

Following [3] we remark that

  • Equations (Q1)–(Q3) and (H1)–(H3) are all degenerate subcases of equation (Q4) [8].

  • Parameter in equations (H3), (Q1) and (Q3) can be rescaled, so that one can assume without loss of generality that or .

  • The original ABS list contains two further equations (list A)

    Equations (A1) and (A2) can be transformed by an extended group of Möbius transformations into equations (Q1) and (Q3) respectively. Indeed, any solution of (A1) is transformed into a solution of (Q1) by and any solution of (A2) is transformed into a solution of (Q3) with by .

Some of the above equations were known before Adler, Bobenko and Suris presented their classification, see for instance [24, 15]. We finally recall that a more general classification of integrable lattice equations defined on the square has been recently carried out by Adler, Bobenko and Suris in [4]. But here we shall consider only the lists H and Q contained in [3].

2.1 Spectral problems of the ABS equations

The algorithmic procedure described in [3, 10, 23] produces a matrix Lax pair for the ABS equations, thus ensuring their integrability. It may be written as

(3)

with , where the lattice parameter plays the role of the spectral parameter. We shall use the following notation

where , , and , . The matrix can be obtained from by replacing with and shifting along direction 2 instead of 1. In Table 1 we give the entries of the matrix  for the ABS equations.

  \bsep1ex\tsep1ex
  \bsep1ex\tsep1ex H1
  \tsep1ex H2
  \bsep1ex
  \bsep1ex\tsep1ex H3
  \bsep1ex\tsep1ex Q1
  \tsep1ex Q2
  \bsep1ex
  \tsep1ex Q3
  \bsep1ex
  \tsep1ex Q4
  \bsep1ex
Table 1: Matrix for the ABS equations (in equation (Q4) , , ).

Note that and are computed by requiring that the compatibility condition between  and  produces the ABS equations (H1)–(H3) and (Q1)–(Q4). The factor can be written as

(4)

where the functions is an arbitrary normalization factor. The functions and for equations (H1)–(H3) and (Q1)–(Q4) are given in Table 2. A formula similar to (4) holds also for the factor .

The scalar Lax pairs for the ABS equations may be immediately computed from equation (3). Let us write the scalar equation just for the second component of the vector (the use of the first component would give similar results). For equations (H1)–(H3) and (Q1)–(Q3) it reads

(5)

where the explicit expressions of are given in Table 2. The corresponding scalar equation for equation (Q4) takes a different form and needs a separate analysis which will be done in a separate work.

  \bsep1ex\tsep1ex
  \bsep1ex\tsep1ex H1
  \bsep1ex\tsep1ex H2
  \bsep1ex\tsep2ex H3
  \bsep1ex\tsep2ex Q1
  \bsep1ex\tsep2ex Q2
  \bsep1ex\tsep2ex Q3
  \tsep1ex Q4
  \bsep2ex
Table 2: Scalar spectral problems for the ABS equations (in equation (Q4) , ).

3 Miura transformations for equations (H1)–(H3) and (Q1)–(Q3)

The aim of this Section is to show the existence of a Miura transformation mapping the scalar spectral problem (5) of equations (H1)–(H3) and (Q1)–(Q3) into the discrete Schrödinger spectral problem associated with the Volterra lattice [11]

(6)

where is the potential of the spectral problem and the function plays the role of the spectral parameter.

Suppose that a function is given by the linear equation

(7)

By performing the transformation and taking into account equation (7), equation (5) is mapped into the scalar spectral problem (6) with

(8)

From these results there follow some remarkable consequences: (i) There exists a Miura transformation between all equations of the set (H1)–(H3) and (Q1)–(Q3). Some results on this claim can be found in [8]; (ii) The Miura transformation (8) can be inverted by solving a linear difference equation. Therefore we can in principle use these remarks to find explicit solutions of the ABS equations in terms of the solutions of the Volterra equation.

The following statement holds.

{proposition}

The field for equations (H1)–(H3) and (Q1)–(Q3) can be expressed in terms of the potential of the spectral problem (6) through the solution of the following linear difference equations

(9)
(10)
(11)
(12)
(13)
(14)
Proof.

From equation (8) we get

Subtracting these relations and taking into account that (see equation (A.11) in [29])

one arrives at

(15)

Writing equation (15) explicitly for equations (H1)–(H3) and (Q1)–(Q3) we obtain equations (9)–(14). ∎

4 Generalized symmetries of the ABS equations

Lie symmetries of equation (2) are given by those continuous transformations which leave the equation invariant. We refer to [20, 32] for a review on symmetries of discrete equations.

From the infinitesimal point of view, Lie symmetries are obtained by requiring the infinitesimal invariant condition

(16)

where

(17)

By we mean the prolongation of the infinitesimal generator to all points appearing in .

If then we get point symmetries and the procedure to construct them from equation (16) is purely algorithmic [20]. If the obtained symmetries are called generalized symmetries. In the case of nonlinear discrete equations, the Lie point symmetries are not very common, but, if the equation is integrable, it is possible to construct an infinite family of generalized symmetries.

In correspondence with the infinitesimal generator (17) we can in principle construct a group transformation by integrating the initial boundary problem

(18)

with , where is the continuous Lie group parameter and is a solution of equation (2). This can be done effectively only in the case of point symmetries as in the generalized case we have a nonlinear differential-difference equation for which we cannot find the general solution , but, at most, we can construct particular solutions.

Equation (16) is equivalent to the request that the -derivative of the equation , written for , is identically satisfied on its solutions when the -evolution of is given by equation (18). This is also equivalent to say that the flows (in the group parameter space) given by equation (18) are compatible or commute with .

In the papers [25, 29] the three and five-point generalized symmetries have been found for all equations of the ABS list. We shall use these results to show that the ABS equations may be interpreted as Bäcklund transformations for the differential-difference YdKN equation [31]. This observation will allow us to provide an infinite class of generalized symmetries for the lattice equations belonging to the ABS list. We shall also discuss the non-autonomous case and the generalizations of the ABS equations considered in [29].

4.1 The ABS equations as Bäcklund transformations of the YdKN equation

In the following we show that the ABS equations may be seen as Bäcklund transformations of the YdKN equation. Moreover we prove that the symmetries of the ABS equations [25, 29] are subcases of the YdKN equation. For the sake of clarity we consider in a more detailed way just the case of equation (H3). Similar results can be obtained for the whole ABS list (see Proposition 4.1).

According to [25, 29] equation (H3) admits the compatible three-point generalized symmetries

(19)
(20)

Notice that under the discrete map , , equation (19) goes into equation (20), while equation (H3) is left invariant.

The compatibility between equation (H3) and equation (19) generates a Bäcklund transformation (see an explanation below) of any solution of equation (19) into its new solution

(21)

Thus equation (H3) can be rewritten as a Bäcklund transformation for the differential-difference equation (19)

(22)

Moreover, the discrete symmetry , implies the existence of the Bäcklund transformation for equation (20)

This interpretation of lattice equations as Bäcklund transformations has been discussed for the first time in the differential-difference case in [17]. Examples of Bäcklund transformations similar to equation (22) for Volterra-type equations can be found in [30, 12].

In [25, 29] generalized symmetries have been obtained for autonomous ABS equations, i.e. such that , are constants. We present here some results on the non-autonomous case when  and  depend on  and . Similar results can be found in [25].

Let the lattice parameters in equation (2) be such that is a constant and . Let us consider the following two forms of equation (2)

(23)

and a symmetry

(24)

given by equation (19). We suppose that depends on in all equations and write down the compatibility condition between equation (23) and equation (24)

(25)

As a consequence of equations (23), (24) the functions , and may be expressed in terms of the fields . Therefore, equation (25) depends explicitly only on the variables , which can be considered here as independent variables for any fixed . For all autonomous ABS equations, the compatibility condition (25) is satisfied identically for all values of these variables and of the constant parameter . In the non-autonomous case, equation (25) depends only on and . Therefore the compatibility condition is satisfied also for any .

So, equation (19) is compatible with equation (H3) also in the case when is constant, but . In a similar way, one can prove that equation (20) is the generalized symmetry of equation (H3) if is constant, but .

Let us now discuss the interpretation of the ABS equations as Bäcklund transformations. Let be a solution of equation (24), and the function given by equation (21) be a solution of equation (23), which is compatible with equation (24). equation (23) can be rewritten as the ordinary difference equation

(26)

where is constant, , is fixed, . Differentiating equation (26) with respect to  and using equation (24) together with the compatibility condition (25), one gets

where

The resulting equation is expressed in the form

(27)

There is for the ABS equations a formal condition . We suppose here that, for the functions , under consideration, for all . The function  is defined by equation (26) up to an integration function . We require that satisfies the first order ordinary differential equation given by . Then equation (27) implies that for all , i.e.  is a solution of equation (24).

So, we start with a solution of a generalized symmetry of the form (24), define a function by the difference equation (26) which is a form of corresponding ABS equation, then we specify the integration function by the ordinary differential equation , and thus obtain a new solution of equation (24). This solution depends on an integration constant and the parameter . We can construct in this way the solutions , and the last of them will depend on arbitrary constants . Using such Bäcklund transformation and starting with a simple initial solution, one can obtain, in principle, a multi-soliton solution. See [7, 9] for the construction of some examples of solutions.

The symmetries (19), (20) are Volterra-type equations, namely

(28)

where we have dropped one of the independent indexes or , since it does not vary. The Volterra equation corresponds to . An exhaustive list of differential-difference integrable equations of the form (28) has been obtained in [31] (details can be found in [32]). All three-point generalized symmetries of the ABS equations, with no explicit dependence on , have the same structure as equation (19) (see details in Section 4.4 below) and are particular cases of the YdKN equation

(29)

where

and the ’s are constants. equation (29) has been found by Yamilov in [31], discussed in [22, 5], and in most detailed form in [32]. Its continuous limit goes into the Krichever–Novikov equation [16]. This is the only integrable example of the form (28) which cannot be reduced, in general, to the Toda or Volterra equations by Miura-type transformations. Moreover, equation (29) is also related to the Landau–Lifshitz equation [27]. A generalization of equation (29) with nine arbitrary constant coefficients has been considered in [21].

By a straightforward computation we get the following result: all three-point generalized symmetries in the -direction with no explicit dependence on  for the ABS equations are particular cases of the YdKN equation. For the various equations of the ABS classification the coefficients , , read