Characteristic Algebras of Fully Discrete Hyperbolic Type Equations

The notion of the characteristic Lie algebra of the discrete hyperbolic type equation is introduced. An effective algorithm to compute the algebra for the equation given is suggested. Examples and further applications are discussed.


Introduction
It is well known that the characteristic Lie algebra introduced by A.B. Shabat in 1980, plays the crucial role in studying the hyperbolic type partial differential equations. For example, if the characteristic algebra of the equation is of finite dimension, then the equation is solved in quadratures, if the algebra is of finite growth then the equation is integrated by the inverse scattering method. More details and references can be found in [1]. Recently it has been observed by A.V. Zhiber that the characteristic algebra provides an effective tool for classifying the nonlinear hyperbolic equations. Years ago the characteristic algebra has been used to classify integrable systems of a special type [2]. However, the characteristic algebras have not yet been used to study the discrete equations, despite the fact that the discrete equations had become very popular the last decade (see, for instance, survey [3]).
In this paper we show that the characteristic algebra can be defined for any discrete equation of the hyperbolic type and it inherits most of the important properties of its continuous counterpart. However, it has essentially more complicated structure. The work was stimulated by [4], where the discrete field theory is studied and the question is posed which types of algebraic structures are associated with the finite field discrete 3D Toda chains.

Invariants and vector fields
Consider a discrete nonlinear equation of the form where t = t(u, v) is an unknown function depending on the integers u, v, and f is a smooth function of all three arguments. The following notations are used to shorten formulae: t u = t(u + 1, v), t v = t(u, v + 1), and t uv = t(u + 1, v + 1). By using these notations one can rewrite the equation (1) as follows t uv = f (t, t u , t v ). Actually, the equation (1) is a discrete analog of the partial differential equations. Particularly, the class of equations (1) contains difference schemes for the hyperbolic type PDEs on a quadrilateral grid.
The notations above are commonly accepted, but not very convenient to indicate the iterated shifts. Below we use also different ones. Introduce the shift operators D andD, which act as follows Df (u, v) = f (u + 1, v) andDf (u, v) = f (u, v + 1). For the iterated shifts we introduce the notations f j = D j (f ) andf j =D j (f ), so that t(u+1, v) = t 1 , t(u, v+1) =t 1 , t(u+2, v) = t 2 , t(u, v + 2) =t 2 and so on.
The equation (1) is supposed to be hyperbolic. It means that it can be rewritten in any of the forms: t u = g(t, t v , t uv ), t v = r(t, t uv , t u ), and t = s(t u , t uv , t v ) with some smooth functions g, r, and s.
A function F = F (v, t, t 1 ,t 1 , . . . ), depending on v and a finite number of the dynamical variables is called v-invariant, if it is a stationary "point" of the shift with respect to v so that (see also [5]) and really the function F solves the equation . Examining carefully the last equation one can find that: The v-invariant does not depend on the variables in the set {t j } ∞ j=1 .
If any v-invariant is found, then each solution of the equation (1) can be represented as a solution of the following ordinary discrete equation F (t, t u , . . . , t j ) = c(u), where c(u) is an arbitrary function of u.
Due to the Lemma 1 the equation (2) can be rewritten as The left hand side of the equation contains t v , while the right hand side does not. Hence the total derivative ofDF with respect to t v vanishes. In other words, the operator X 1 =D −1 d dtvD annihilates the v-invariant F : X 1 F = 0. In a similar way one can check that any operator of the form X j =D −j d dtvD j , where j ≥ 1, satisfies the equation X j F = 0. Really, the right hand side of the equationD j F (v, t, t 1 ,t 1 , . . . ) = F (v, t, t 1 ,t 1 , . . . ) (which immediately follows from (2)) does not depend on t v and it implies the equation X j F = 0. As a result, one gets an infinite set of equations for the function F . For each j the operator X j is a vector field of the form Consider now the Lie algebra L v of the vector fields generated by the operators X j with the usual commutator of the vector fields [X i , X j ] = X i X j − X j X i . We refer to this algebra as characteristic algebra of the equation (1). Remark 1. Note that above we started to consider the discrete equation of the form (1) conjecturing that it admits nontrivial v-invariant. The definition of the algebra was motivated by the invariant. However the characteristic Lie algebra is still correctly defined for any equation of the form (1) even if it does not admit any invariant. Proof . Suppose that the equation (2) admits a non-constant solution. Then the following system of equations

Algebraic criterion of existence of the invariants
has a non-constant solution. It is possible only if the linear envelope of the vector-fields It is worth mentioning an appropriate property of the vector fields above. If for a fixed j the operator X j is linearly expressed through the operators X 1 , X 2 , . . . , X j−1 , then any operator X k is a linear combination of these operators. Really, we are given the expression Thus, in this case the characteristic algebra is generated by the first j − 1 operators which are linearly independent. Due to the classical Jacoby theorem the system (4) has a non-constant solution only if dimension of the Lie algebra generated by the vector fields X i is no greater than N . Thus, one part of the theorem is proved.
Suppose now that the dimension of the characteristic algebra is finite and equals, say, N , show that in this case the equation (1) admits a v-invariant. Evidently, there exists a function G(t, t 1 , . . . , t N ), which is not a constant and that XG = 0 for any X in L v . Such a function is not unique, but any other solution is expressed as h(G). Due to the construction the map X →D −1 XD leaves the algebra unchanged, hence G 1 =DG is also a solution of the same system XG = 0 and therefore G 1 = h(G). In other words, one gets a discrete first order equation:DG = h(G), write its general solution in the following form: Evidently the function F found is just the v-invariant needed.

Computation of the characteristic algebra
In this section the explicit forms of the operators {X j } are given. We show that the operators are vector fields and give a convenient way to compute the coefficients of the expansion (3).
Start with the operator X 1 . Directly by definition one gets X 1 F (t, t 1 , . . . ) =D −1 ∂ ∂tv F (t v , f , f 1 , . . . ). Computing the derivative by the chain rule one obtains and finally So the operator X 1 can be represented as for j > 0 and x 0 = 1. Actually the coefficients can be computed by the following more convenient formula or x j+1 = x j D j (x 1 ). Really, To find X 2 , use the following formula After opening the parentheses and some transformation the right hand side of the last formula gets the form So the operator can be written as Continuing this way, one gets One can derive an alternative way to compute the coefficients of the vector fields X k above. Represent the operators as follows We will show that the coefficients n kj of the operators satisfy the following linear equation closely connected with the direct linearization of the initial nonlinear equation (1). In order to derive this formula, apply the operator X k to the iterated shift f j = f (t j , t j+1 , f j−1 ) and use the chain rule Comparison of the two representations (8) and (9) of the operator X k yields X k (f j ) =Dn k+1,j+1 . By replacing in (11) X with n one gets the formula (10) required. The characteristic algebra is invariant under the map X → D −1 XD. First prove the formula D −1 X 1 D = D −1 (x 1 )X 1 . To this end use the coordinate representation of the operator X 1 = ∞ i=0 x i ∂ ∂t i and the formula (6) for the coefficients. Then check that D −1 X 2 D = ρX 2 . Really, Obviously similar representations can be derived for any generator of the characteristic algebra.
The following statement turns out to be very useful for studying the characteristic algebra.
Lemma 2. Suppose that the vector field satisfies the equation then X ≡ 0.
Proof . It follows from (12) that Comparison of the coefficients before the operators ∂ ∂t j in both sides of this equation yields: x 0 = 0, x 1 = 0, . . . and so on. 5 The commutativity property of the algebra One of the unexpected properties of the characteristic Lie algebra is the commutativity of the operators X j . Consider first an auxiliary statement.
The coefficients x ki of the vector fields X k , k ≥ 1, i ≥ 0 do not depend on the variable t v .
In other words, the coefficients of the expansions Proof . For the operator X 1 one has x 1,j+1 =D −1 ∂f j ∂tv . But the function f j does not depend ont 2 =Dt v and ont 3 ,t 4 , . . . as well. Hence the coefficients do not depend ont 1 = t v . Similarly, the functions X 1 (f j ) do not depend ont 2 , so x 2,j+1 =D −1 (X 1 (f j )) do not contain t v . One can complete the proof by using induction with respect to k. Proof . Remind that X j =D −j d dtvD j , so that one can deduce Suppose for the definiteness that k = j − i ≥ 1. Then But due to the Lemma 3 the last expression vanishes that proves the theorem.
6 Characteristic algebra for the discrete Liouville equation The well known Liouville equation ∂ 2 v ∂x∂y = e v admits a discrete analog of the form [6] which can evidently be rewritten as Specify the coefficients of the expansions (5)- (7) representing the vector fields X 1 and X 2 for the discrete Liouville equation (13). Find the coefficient Similarly It can easily be proved by induction that Remind also that x 0 = 1. So that the vector field is It is a more difficult problem to find the operator X 2 , it can be proved that where the operator X −1 is defined as follows X t k , so that the operator is represented as Theorem 3. Dimension of the characteristic Lie algebra of the Liouville equation (13) equals two.
Proof . It is more easy to deal with the operators Y + = (t − 1)X 1 and Y − = tX −1 rather than the operators X 1 and X −1 . In order to prove the theorem it is enough to check the formula Denote through X the left hand side of the equation (14) and compute D(X) to apply the Lemma 2. It is shown straightforwardly that Proof . In the case of the DPKdV equation one gets ∂f ∂t = 1, ∂f ∂t 1 = −x, ∂f ∂tv = x. The formula (10) is specified Dn k+1,j+1 = n k,j − x j n k,j+1 + x jD n k+1,j .
It is more convenient to write it as j . It is clear now that the coefficients are polynomials of the finite number of the dynamical variables in the set {x i j } ∞ i,j=0 . Order the variables in this set according to the following rule: ord(x n j ) > ord(x n p ) if j > p and ord(x m j ) > ord(x n p ) if m > n. Lemma 4. For any positive k, j the coefficient n k,j for the operator X k in the DPKdV case is represented as n k,j = m k,j x k j−1 + r k,j where m k,j and r k,j are polynomials of the variables with the orders less than the order of x k j−1 , moreover, m k,j is a monomial.
Lemma can be easily proved by using the induction method. It allows to complete the proof of the theorem. The vector fields {X k } ∞ k=1 are all linearly independent. Due to the theorem the DPKdV equation do not admit any v-invariant. It is not surprising, because the equation can be integrated by the inverse scattering method or, in other words, it is S-integrable, it is well known in the case of partial differential equations that only C-integrable equations (Liouville type) admit such kind objects called x-and y-integrals.

Conclusion
The notion of the characteristic algebra for discrete equations is introduced. It is proved that the equation is Darboux integrable if and only if its characteristic algebras in both directions are of finite dimension. The notion can evidently be generalized to the systems of discrete hyperbolic equations. It would be useful to compute the algebras for the periodically closed discrete Toda equation, or for the finite field discrete Toda equations found in [4] and [8], corresponding in the continuum limit to the simple Lie algebras of the classical series A and C.