Integrable Discrete Nonautonomous Quad-equations as Bäcklund Auto-transformations for Known Volterra and Toda Type Semidiscrete Equations
We construct integrable discrete nonautonomous quad-equations as Bäcklund auto-transformations for known Volterra and Toda type semidiscrete equations, some of which are also nonautonomous. Additional examples of this kind are found by using transformations of discrete equations which are invertible on their solutions. In this way we obtain integrable examples of different types: discrete analogs of the sine-Gordon equation, the Liouville equation and the dressing chain of Shabat. For Liouville type equations we construct general solutions, using a specific linearization. For sine-Gordon type equations we find generalized symmetries, conservation laws and pairs.
In this paper we consider discrete quad-equations
which may explicitly depend on the discrete variables . These equations are supposed to be polylinear, i.e. the functions are polynomials of the first order in each of their argument. Most of the known integrable equations of this kind have two generalized symmetries of the form
see e.g. [11, 15, 24, 23]. Recently a few examples of quad-equations with more complicated generalized symmetries have been found [1, 6, 16, 18], but such equations are out of consideration in this paper.
Almost all known equations of the form (1) possessing the symmetries (2,3) are autonomous. In the essentially more difficult nonautonomous case, we study in this paper, only a few examples are known .
The discrete equation (1) can be interpreted as a chain of Bäcklund auto-transformations for the lattice equations (2,3). Such transformations allow one to construct a new solution in the case of eq. (2) and in the case of eq. (3), starting from a given solution see a more detailed comment in .
In this paper we start from known integrable equations of the form (2) and look for the discrete equations (1) generating for them chains of the Bäcklund auto-transformations. Such problem has been solved up to now only once by one example and only in the autonomous case . Discrete equations obtained in this way may be not integrable. We select integrable cases by requiring the existence of a second symmetry of the form (3). In such way we can construct integrable discrete equations, using known integrable equations of the Volterra type presented in [25, 29] or using their nonautonomous generalizations given in .
More precisely, we are going to use differential-discrete equations of the form:
It has been shown in  that nonlinear integrable equations of this form are described by the following conditions:
where is an arbitrary constant, and are the two-periodic functions:
Up to the transformations
where and are the two-periodic functions, we have five cases: the Volterra equation with , its three modifications with , , and an equation with . The last equation is nothing but one of forms of the Toda model, as it has been shown in . Here is given by
We fix the generalized symmetry (2) in the direction in one of the following five ways:
In all these cases we find all corresponding polylinear discrete equations (1). A more general form of such symmetry is possible:
where and for all In this case we will construct some examples.
As a result we find nonautonomous integrable examples of several different types. According to their symmetry properties, eqs. (1) are the discrete analogs of the hyperbolic type equations Some examples obtained in this paper are of the sine-Gordon type. Such equations have two generalized symmetries (2,3) and are not Darboux integrable, see definitions in the next section. We also find a few Darboux integrable equations which can be called the discrete analogs of the Liouville equation. For all equations of this type, we construct general solutions. One more new interesting integrable example presented here is a discrete analog of the well-known dressing chain studied in [20, 19, 22].
In Section 2 we give some definitions and obtain theoretical results necessary for the paper. In Section 3 we enumerate all the discrete equation (1) corresponding to the differential-discrete eqs. (8-12) and also obtain some examples in the case of eq. (13). In Section 4 a discrete analog of the dressing chain is discussed. The problem of construction of the second symmetry (3) for examples obtained in Section 3 is solved in Section 5. In Section 6 some additional examples are found by using special transformations of the discrete equations invertible on their solutions.
In this section we give necessary definitions and derive some conditions for the discrete equations which allow us to make the class (1) of the discrete equations essentially more narrow.
We consider equations of the form (1) which are polylinear and nondegenerate. It is convenient to formulate definitions in terms of the function
which depends on 4 continuous complex variables and on 2 integer discrete ones . An equation of the form (1) is polylinear if
for all . So, we consider a class of polynomial equations with 16 -dependent coefficients. The nondegeneracy is defined following . If the function depends on for all , then we can rewrite the equation in the form
and we require the function to depend essentially on all its continuous variables for all . So, we have the following nondegeneracy condition in terms of and :
The discrete equation (1) is equivalent to
which can be considered as independent variables. Eqs. (17,18) must be identically satisfied for all values of the independent variables as well as for any If eqs. (16) and (2,3) are compatible, then (2,3) are the generalized symmetries of (16) and, on the other hand, eq. (16) defines chains of the Bäcklund auto-transformations for eqs. (2,3), see .
on the solutions of eq. (16). Here are the shift operators in the first and second directions, respectively:
It is easy to show that the first integrals and depend only on the independent variables and , respectively. Applying the shift operators, we can represent these first integrals as:
The Darboux integrable equations are linearizable, with linearizing transformations , and are analogs of the Liouville equation.
A discrete equation (16) is of the sine-Gordon type if it has two generalized symmetries (2,3) and is not Darboux integrable. Such equations should be integrable by the inverse scattering method. It is difficult to prove that a given equation has no first integrals (21), see  for possible difficulties. For examples obtained below, we check that fact for
Let us derive two conditions necessary for the compatibility of the discrete equation (16) and an equation of the form
Differentiating the compatibility condition (17) with respect to and applying , we obtain the relation
This is nothing but one of so-called integrability conditions obtained in . Here we just present it in the most general non-autonomous case and write it down in a form more convenient for the present paper. In the case of eq. (22) it takes the form:
Applying to eq. (16), we can easily rewrite it in one more form
where are arbitrary -dependent functions. This form corresponds to the following restrictions:
In quite similar way the relation (27) implies:
3 Bäcklund auto-transformations
We fix one of eqs. (8-12) as the generalized symmetry (2). To find corresponding discrete equation, we use in the first step more simple necessary conditions (24,27) instead of the compatibility condition (17). The relations (24,27) are equivalent to a nonlinear algebraic system of equations for eighteen functions It is interesting that not only in this system but also in case of the compatibility condition (17) the discrete variable is not changed. So we can define the dependence of on only.
Such problem is not difficult in the autonomous case. We just need to solve an algebraic system for nine unknown coefficients , using any computer algebra system like Reduce. In the nonautonomous case, we can only find from that algebraic system a set of solutions which have the form of relations between the functions and in a fixed point . Another difficulty is that there are many divisors of zero and , such that .
Nevertheless, comparing sets of solutions at and , one can choose consistent pairs of nondegenerate solutions, i.e. such that corresponding discrete equation (28) is nondegenerate, and can determine in this way the dependence of the functions on . In case of the nonautonomous equations (11) and (12), we need to solve the corresponding algebraic system twice, at and , to avoid a dependence on the function . This the way we find the discrete equations.
In case of the Toda lattice (12), we have checked that there is no polylinear and nondegenerate discrete equation (1) compatible with (12). However, in Section 6 we present an example of the sine-Gordon type, corresponding to (12), which is not polylinear. In case of the Volterra equation (8) we get a positive result:
In the case of eqs. (9-11) the problem of finding discrete equations (1) is easier. We solve an algebraic system, corresponding to eqs. (24) and (27) in a fixed point , and see that all possible nondegenerate solutions have the same structure:
Deviding the discrete equation (28) by , we obtain a very simple ansatz of the form
Now we can interpret algebraic systems of equations for , corresponding to the conditions (24,27) and (17), as systems of ordinary difference equations for the functions and then we can specify eq. (33) with no difficulties.
The resulting discrete equation in the case of (9) is not interesting:
It can be linearized by the point transformation
Eq. (13) with for any can be transformed into
Introducing such that
Excluding we are led to the discrete equation
4 Discrete analog of the dressing chain
Eq. (42) can be constructed by two Miura transformations into the Korteweg-de Vries equation as well as eq. (41) is constructed by two discrete Miura transformations into the Volterra equation which also is called the discrete Korteweg-de Vries equation. There is in  an pair for eq. (41) which is the direct analog of an pair for (42) constructed in . In this section we present a generalization of eq. (41) together with its pair.
That generalization is a particular case of eq. (40) corresponding to with :
and we lose e.g. the case . In the case when and for all , we can introduce a new function : where . The last product does not depend on , as is two-periodic with respect to . This function satisfies an equation of the form (41), i.e. we get nothing new in this case. So, only the case when or for some is interesting, and we will show examples of this kind in next sections.
An pair for eq. (43) reads:
and these matrices satisfy the standard relation In partilucar case (41), this pair coincides with one of . A hierarchy of conservation laws for eq. (41) has been constructed in  by using that pair of .
and, for , are given by the following functions and :
Equations of the sine-Gordon type possess two hierarchies of generalized symmetries and conservation laws. Eq. (43) probably has, in general, only one hierarchy of conservation laws and generalized symmetries, see a comment below. Nevertheless, it has the pair and is integrable for this reason. It deserves further study as a direct discrete analogue of the dressing chain (42).
The discrete equation (41) has the generalized symmetry (37) with in the direction . However, in the case when we can show that if the second generalized symmetry of the form (3) exists, then the following relation must take place:
In this case, using rescaling we can transform this equation into:
5 Second generalized symmetry
Among discrete equations, we consider in this paper, there may be integrable equations which do not have two generalized symmetries of the form (2,3). Nevertheless, we use here the existence of two generalized symmetries of such form as an integrability criterion. It is constructive and allows us not only to check an equation for integrability, but also to solve some classification problems, as it is demonstrated below, see also .
For some of discrete equations found in Section 3, we select in this section cases in which a generalized symmetry of the form (3) exists. The symmetry must be nondegenerate, i.e. its right hand side must differ from zero for any
When searching for generalized symmetries, we use a scheme developed in [13, 15, 5]. To check the Darboux integrability of an equation, we use results of [8, 6]. In both cases some special annihilation operators introduced in  play an important role.
5.1 Volterra case
where is defined by (7). Nonautonomous integrable equations of this kind are known, see . Examples of  and eq. (50) are essentially different, as corresponding generalized symmetries strongly differ from each other.
where are arbitrary constants. Eq. (51) with is a representative of the well-known complete list of integrable Volterra type equations presented in [25, 29]. Eq. (51) with is its master symmetry found in . It generates generalized symmetries for eq. (51) with and, therefore, for the discrete equation (50), for instance:
In the second case, in eq. (32), hence , and this equation is of the form:
Its generalized symmetry of the form (3) reads:
In the next section, a general solution of eq. (53) will be constructed.
There may be very few points of degeneration of a symmetry, for instance, if
then and may be a nonzero constant in all the other points . Such examples probably have not been considered in the literature up to now. Equations of this kind seem to be very close to the integrable ones and, in our opinion, deserve further study.
5.2 General solution
In this section we present, by an example, a scheme of the construction of a general solution for the Darboux integrable discrete equation. We use an observation of  that, in many cases, first integrals of the first order (i.e. such that or in (21)) can be rewritten as a linear equation. Using this fact, corresponding discrete equation can be equivalently rewritten as a nonautonomous linear equation.
where is an arbitrary -dependent function of integration.
We express the function