Algebraic Properties of Quasilinear TwoDimensional Lattices connected with integrability
Abstract
In the article a classification method for nonlinear integrable equations with three independent variables is discussed based on the notion of the integrable reductions. We call the equation integrable if it admits a large class of reductions being Darboux integrable systems of hyperbolic type equations with two independent variables. The most natural and convenient object to be studied within the frame of this scheme is the class of two dimensional lattices generalizing the wellknown Toda lattice. In the present article we deal with the quasilinear lattices of the form . We specify the coefficients of the lattice assuming that there exist cutting off conditions which reduce the lattice to a Darboux integrable hyperbolic type system of the arbitrarily high order. Under some extra assumption of nondegeneracy we described the class of the lattices integrable in the sense indicated above. There are new examples in the obtained list of chains.
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1 Introduction
Integrable equations with three independent variables have a wide range of applications in physics. It suffices to recall such wellknown nonlinear models as the KP equation, the DaveyStewartson equation, the Toda lattice equation, and so on. From the point of view of integration and classification, multidimensional equations are the most complex. Different approaches to study the integrable multidimensional models are discussed, for example, in the papers [1]–[9]. It is known that the symmetry approach [10, 11], which has proved to be a very effective method for classifying integrable equations in 1 + 1 dimensions, is not so effective in the multidimensionality [12]. For studying multidimensional equations, the idea of the reduction is often used, when the researches replace the equation with a system of equations with fewer independent variables. The existence of a wide class of integrable reductions with two independent variables, as a rule, indicates the integrability of an equation with three independent variables. Among the specialists, the most popular method is the method of hydrodynamic reductions, when the presence of an infinite set of integrable systems of hydrodynamic type is taken as a sign of integrability of the equation, the general solution of each of which generates some solution of the equation under consideration (see, for example, [13, 1, 2]). The history of the method and related references can be found in the survey [3].
In our works [14, 15] we use an alternative approach. We call this equation integrable if it admits an infinite class of reductions in the form of Darbouxintegrable systems of partial differential equations of hyperbolic type with two independent variables. In solving classification problems for multidimensional equations, the apparatus of characteristic Lie algebras can be used in this formulation (a detailed exposition can be found in [17, 18]). This direction in the theory of integrability seems to us promising. Consider a nonlinear chain
(1.1) 
with three independent variables, where the sought function depends on the real , , and the integer . For the chain (1.1), the desired finitefield reductions are obtained in a natural way, in a sufficiently suitable way to break off the chain at two integer points
(1.2)  
(1.3)  
(1.4) 
Examples of such boundary conditions can be found below (see (4.29), (4.30)). The following two very significant circumstances should be noted:

specific form of the functions , , and is constructively determined by the requirement of integrability of the system in the sense of Darboux.
These two facts serve as motivation for the following definition (see also the work [14]):
Definition 1
In the present paper we investigate quasilinear chains of the following form
(1.5) 
assuming that the functions , ,
, are analytic in the domain . We also assume that the derivatives
(1.6) 
differ from zero.
The main result of this paper is the proof of the following assertion
Theorem 1
We note that equation i) was found earlier in the papers [27], [28] by Ferapontov and Shabat and Yamilov, equations ii) and iii) appear to be new. By applying additional conditions of the form to the equations i)iii), we obtain 1 + 1 dimensional integrable chains. It is easily verified that by point transformations they are reduced to the equations found earlier by Yamilov (see [29]).
Following Definition 1, we suppose that there are cutoff conditions such that by imposing them at two arbitrary points , () to the chain (1.5) we obtain a system of hyperbolic type equations
(1.7)  
that is integrable in the sense of Darboux.
We recall that the system of partial differential equations of the hyperbolic type (1.7) is Darboux integrable if it has a complete set of functionally independent and integrals. A function that depends on a finite set of dynamical variables is called a integral if it satisfies the equation , where is the operator of total derivative with respect to the variable , and the vector has the coordinates . Since the system (1.7) is autonomous, we consider only autonomous nontrivial integrals. It can be shown that the integral does not depend on . Therefore, we will consider only integrals that depend on at least one dynamic variable . We note that nowadays the Darboux integrable discrete and continuous models are intensively studied (see, [14, 17], [19][26]).
We give one more argument in favor of our Definition 1 concerning the integrability property of a twodimensional chain. The problem of finding a general solution of a Darbouxintegrable system reduces to the problem of solving a system of ordinary differential equations. Usually these ODEs are solved explicitly. On the other hand, any solution of the considered hyperbolic system (1.7) easily extends beyond the interval and generates the solution of the corresponding chain (1.5). Therefore, in this case the chain (1.5) has a large set of exact solutions.
Let us briefly explain the structure of the paper. In §2 we recall the necessary definitions and investigate the basic properties of the characteristic Lie algebra, which is the main tool in the theory of Darbouxintegrable systems. In §3 we introduce the definition of test sequences, by means of which we obtain a system of differential equations for the refinement of the functions , , . Paragraph 4 is devoted to the search for the function . Here we also give the final form of the desired chain (4.28) that is integrable in the sense of Definition 1 and the proof of the Theorem 1 is given.
2 Characteristic Lie algebras
Since the chain (1.5) is invariant under the shift of the variable , then without loss of generality we can put . In what follows we consider a system of hyperbolic equations
(2.1)  
Recall that here , , , . Suppose that the system (2.1) is Darboux integrable and that is its nontrivial integral. The latter means that the function must satisfy the equation , where is the operator of total derivative with respect to the variable . The operator acts on the class of functions of the form due to the rule , where
(2.2) 
Here is the right hand side of the lattice (1.5). Hence, the function satisfies the equation . The coefficients of the equation depend on the variables , while its solution does not depend on , therefore the function actually satisfies the system linear equations:
(2.3) 
where . It follows from (2.3) that the commutator of the operators and for also annuls . We use the explicit coordinate representation of the operator :
(2.4) 
By the special form of the function , the operator can be represented in the form:
(2.5) 
where
(2.6) 
Denote by the ring of locally analytic functions of the dynamical variables .
Consider the Lie algebra over the ring generated by the differential operators . It is clear that the operations of computing the commutator of two vector fields and multiplying the vector field by a function satisfy the following conditions:
(2.7)  
(2.8) 
where , . Consequently, the pair has the structure of the LieRinehart algebra^{1}^{1}1We thank D.V. Millionshchikov who drew our attention to this circumstance. (see [30]). We call this algebra the characteristic Lie algebra of the system of equations (2.1) along the direction . It is well known (see [20, 17]) that the function is a integral of the system (2.1) if and only if it belongs to the kernel of each operator from . Since the integral depends only on a finite number of dynamic variables, we can use the wellknown Jacobi theorem on the existence of a nontrivial solution of a system of firstorder linear differential equations with one unknown function. From this theorem it is easy to deduce that in the Darboux integrable case in the algebra there must exist a finite basis , consisting of linearly independent operators such that any element of can be represented as a linear combination , where the coefficients are analytic functions of dynamical variables defined in some open set. Moreover, from the equality it follows that . In this case, we call the algebra finitedimensional. Similarly, we can define the characteristic algebra in the direction . It is clear that the system (2.1) is Darboux integrable if and only if the characteristic algebras in both directions are finitedimensional.
For the sake of convenience, we introduce the notation . We note that in our study the operator plays a key role. Below we shall apply to smooth functions depending on dynamical variables . As was shown above, the operators and coincide on this class of functions. Therefore, the equality immediately implies . Replacing by virtue of (2.5), and collecting in the resulting relation the coefficients of the independent variables , we obtain
(2.9) 
It is clear that the operator takes the characteristic Lie algebra into itself. We describe the kernel of this mapping:
3 Method of test sequences
We call the sequence of operators in the algebra a test sequence if holds:
(3.1) 
The test sequence allows us to derive the integrability conditions for a system of hyperbolic type (2.1) (see [19, 17, 20]). Indeed, assume that (2.1) is Darboux integrable. Then among the operators there is only a finite set of linearly independent elements through which all the others are expressed. In other words, there exists an integer such that the operators are linearly independent and is expressed as follows:
(3.2) 
We apply the operator to both sides of the equality (3.2). As a result, we obtain the relation
(3.3) 
Collecting coefficients for independent operators, we obtain a system of differential equations for the coefficients . The resulting system is overdetermined, since is a function of a finite number of dynamical variables . The consistency conditions for this system define the integrability conditions for (2.1). For example, collecting the coefficients for we get the first equation of the indicated system:
(3.4) 
which is also overdetermined.
Below in this section, we use two test sequences to refine the form of the functions , , .
3.1 First test sequence
Let us define a sequence of operators in the characteristic algebra by the following recurrence formula:
(3.5) 
Above (see (2.9), the commutation relations for the first two terms of this sequence were derived:
(3.6) 
Applying the Jacobi identity and using the last formulas, we derive:
(3.7) 
We can prove by induction that (3.5) is a test sequence. Moreover, for any the following formula
(3.8) 
holds where the functions , are found due to the rule
(3.9) 
By assumption, in the algebra there exists only a finite set of linearly independent elements of the sequence (3.5). Hence, there exists a natural such that:
(3.10) 
operators are linearly independent, and three dots stand for a linear combination of the operators .
Lemma 2
The operators are linearly independent.
Proof. Let us assume the contrary. Suppose that
(3.11) 
The operators , have the form , while does not contain terms of the form and , hence the coefficients are zero. If, in addition, , then . We apply the operator to both sides of the last equality, then by (3.7) we obtain the equation
which implies: and . By virtue of the independence of the variables and , we obtain that . But this contradicts the assumption of (1.6) that . The proof is complete.
Lemma 3
If the expansion of the form (3.10) holds, then
(3.12) 
Proof. It is not difficult to show that equation (3.4) for the sequence (3.5) has the form:
(3.13) 
We simplify the relation (3.13) using formulas
(3.14)  
A simple analysis of the equation (3.13) shows that . Therefore, (3.13) is rewritten as
Collecting the coefficients in front of the independent variables , , we derive an overdetermined system of differential equations in :
(3.15)  
(3.16) 
Note that equations (3.15) do not contain function and completely coincide with the equations studied in our article [15]. Lemma 3 immediately follows from Lemma 3.2 in [15]. In what follows we use the equation (3.16) to refine the function .
3.2 Second test sequence
We construct a test sequence containing the operators , , and their multiple commutators:
(3.17) 
Elements of the sequence for are determined by the recurrence formula . Note that this is the simplest test sequence generated by iterations of the map , which contains the operator .
Lemma 4
The operators are linearly independent.
Proof. Arguing as in the proof of Lemma 1, we can verify that the operators are linearly independent. Let us prove the lemma 4 by contradiction. Let’s assume that
(3.18) 
First we derive the formulas by which the operator acts on the operators . For , they are immediately obtained from relation
Recall that . For we have
By applying the operator to both sides of (3.18), we obtain
(3.19) 
Collecting the coefficients for in the equality (3.19), we obtain the following equation:
(3.20) 
A simple analysis of the equation (3.20) shows that . Consequently, and equation (3.20) reduces to a system of three equations , and . From these equations it follows that . Otherwise, if , then , which implies that and this contradicts the requirement that essentially depends on and , hence . Then from (3.20) we have , which again leads to a contradiction.
Let’s return to the sequence (3.17). For further work, it is necessary to describe the action of the operator on all elements of this sequence. It is convenient to separate the sequence (3.17) into three subsequences , and .
Lemma 5
The action of the operator on the sequence (3.17) is given by the following formulas:
Lemma 5 is easily proved by induction.
Theorem 2
Assume that the operator is represented as a linear combination
(3.21) 
of the previous members of the sequence (3.17) and none of the operators for is a linear combination of the operators with . Then the coefficient satisfies the equation
(3.22) 
Lemma 6
Assume that all the conditions of Theorem 2 are satisfied. Suppose that the operator (the operator ) is linearly expressed in terms of the operators , . Then in this expansion the coefficient at is zero.
Proof. Let us prove the assertion by contradiction, assume that in formula
(3.23) 
the coefficient is nonzero. We apply the operator to both sides of the equation (3.23). As a result, according to Lemma 5, we get:
(3.24) 
Collecting the coefficients at , we obtain that the coefficient should satisfy the equation
According to our assumption above, does not vanish and, therefore,
(3.25) 
Since depends on a finite set of dynamical variables, according to the equation (3.25) can depend only on and . Therefore, from (3.24) we get that
The variables , are independent, so the last equation is equivalent to the system of equations , , . Consequently, depends only on , . The latter contradicts the assumption that essentially depends on . The contradiction shows that the assumption is false. The lemma is proved.
In order to prove Theorem 2, we apply the operator to both sides of the equality (3.21) and simplify due to the formulas in Lemma 5. Collecting the coefficients for , we obtain the equation (3.22).
Let us find the exact values of the coefficients of equation (3.22)
and substitute them into (3.22):
(3.26) 
A simple analysis of the equation (3.26) shows that can depend only on the variables . Consequently,
(3.27) 
Substituting (3.27) in (3.26) and collecting coefficients for independent variables, we obtain a system of equations by the coefficient :
(3.28)  
(3.29)  
(3.30)  
(3.31) 
Substituting the expression for the function , given by the formula (3.12) into the equation (3.28), we get
We integrate the last equation with respect to the variable