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On a method for constructing the Lax pairs for nonlinear integrable equations

We suggest a direct algorithm for searching the Lax pairs for nonlinear integrable equations. It is effective for both continuous and discrete models. The first operator of the Lax pair corresponding to a given nonlinear equation is found immediately, coinciding with the linearization of the considered nonlinear equation. The second one is obtained as an invariant manifold to the linearized equation. A surprisingly simple relation between the second operator of the Lax pair and the recursion operator is discussed: the recursion operator can immediately be found from the Lax pair. Examples considered in the article are convincing evidence that the found Lax pairs differ from the classical ones. The examples also show that the suggested objects are true Lax pairs which allow the construction of infinite series of conservation laws and hierarchies of higher symmetries. In the case of the hyperbolic type partial differential equation our algorithm is slightly modified; in order to construct the Lax pairs from the invariant manifolds we use the cutting off conditions for the corresponding infinite Laplace sequence. The efficiency of the method is illustrated by application to some equations given in the Svinolupov-Sokolov classification list for which the Lax pairs and the recursion operators have not been found earlier.


Introduction
In the present article we suggest an algorithm for constructing the Lax pairs for nonlinear integrable models. Our scheme is based on the symmetry approach. Let us explain the algorithm with the evolutionary type PDE of the form u t = f (x, t, u, u 1 , u 2 , ..., u k ), u j = ∂ j u ∂x j . (1.1) Recall that equation is called a symmetry for the equation (1.1) if function g satisfies the following differential equation where D t and D x are the operators of the total derivatives with respect to t and x correspondingly. Note that equation (1.3) is overdetermined and for any fixed value of m one can effectively find all of its solutions of the form g = g(x, t, u, u 1 , u 2 , ..., u m ) (see [1]). An ordinary differential equation Here D t G is evaluated by means of the equation (1.1) and all the x-derivatives of the order greater than m − 1 are expressed due to the equation (1.4) and its differential consequences. Equation (1.5) generates a PDE with unknown G admitting a large class of solutions. However it is a hard problem to find these solutions since the equation is not overdetermined. Some of the invariant manifolds to (1.1) can be found from the stationary part of the symmetry (1.2) by taking g(x, t, u, u 1 , ..., u m ) = 0. We now concentrate on the linearization of the equation (1.1) around its arbitrary solution u = u(x, t) : Actually (1.6) defines a family of differential equations, depending on u. The important fact that u ranges the whole set of the solutions to (1.1) is formalized in the following way. We assume that in addition to the natural independent dynamical variables v, v 1 , v 2 , ... the variables u, u 1 , u 2 , ... are also considered as independent ones. Let us find the invariant manifold of the form to the equation (1.6). This means that the condition It is easily checked that equations (1.10) and (1.11) are consistent if and only if the function u = u(x, t) satisfies the equation (1.9). Therefore these equations constitute the Lax pair to the KdV equation. It differs from the usual one found earlier in [2]. Stress that there is no any second order invariant manifold of the form v 2 = a(u, u 1 , u 2 )v 1 + b(u, u 1 , u 2 )v for the equation (1.10). But there is a first order one v 1 = u 2 u 1 v which however does not contain λ and therefore does not generate any true Lax pair.
Recursion operator is an important attribute of the integrability theory. It gives a compact description for the hierarchies of both symmetries and conservation laws. Various methods for studying the recursion operators can be found in literature (see, for instance [3]- [11] and the references therein).
We observe that invariant manifold for the linearized equation is closely connected with the recursion operator for the original equation. Indeed examples in §6 show that equation defining the invariant manifold, which provides the second operator of the Lax pair, can be rewritten as a formal eigenvalue problem of the form Rv = λv for the recursion operator R. For instance, equation (1.11) is easily rewritten as (see §6) where the operator at the l.h.s. is nothing else but the recursion operator for the KdV hierarchy.
Therefore our scheme of constructing the Lax pairs provides an alternative tool for searching the recursion operator. On the other hand side when the recursion operator is known the invariant manifold and hence the Lax pair can be found by simple manipulations (see §6). At this point we have an intersection with the pioneering articles [1], where nonstandard Lax pairs are given for some of the KdV type equations in terms of the recursion operators. However these Lax pairs differ from those found within our scheme since they are nonlocal and do not contain any spectral parameter.
There is a great variety of approaches for searching the Lax pairs from the Zakharov-Shabat dressing [12,13] and prolongation structures by Wahlquist and Estabrook [14] to 3D consistency approach developed in [15]- [17]. We mention also approaches proposed in [18,19,1]. An advantage of our scheme is that it can be applied to any integrable model (at least in 1+1 -dimensional case), the first of the operators is easily found and the second is effectively computed. They allow finding the conservation laws, higher symmetries and invariant surfaces for the corresponding nonlinear equation. The found Lax pairs are more complicated since they are based on the differential operators of the orders greater than usual ones. This is their disadvantage. The question remains open whether the Lax pairs of this kind allow to find any new solution for the well studied models.
In the case of the hyperbolic type integrable equations the algorithm should be slightly modified. Let us explain it with the example of the sine-Gordon equation u xy = sin u. (1.12) Let us find the simplest but nontrivial, i.e. depending on a parameter, invariant manifold for the linearized equation It is the three dimensional surface in the space of the dynamical variables v, v x , v y , v xx , v yy , ... defined by the following two linear equations, with the coefficients, depending on the field variable u(x, t) and its derivatives Here λ is a complex parameter. Note that equations (1.14), (1.15) are not independent. One of them is immediately found from the other by differentiation by means of the Let us give a brief comment on the structure of the article. In § §2,3 we discuss the well-known Laplace cascade for linear and nonlinear hyperbolic type equations. In §3 also the problem of finding finite reductions of the infinite Laplace sequence is studied. The Lax pair to the sine-Gordon equation is derived from the Laplace cascade. In §4 the definition of the invariant manifold for the hyperbolic type equations is recalled. The Lax pair is construted via invariant manifolds for hyperbolic equation (4.12) found in [20]. In §5 the Lax pairs are constructed by evaluating invariant manifolds for the evolutionary type integrable equations. Beside the explanatory examples here we consider two equations (5.19) and (5.20) found in [21] as equations possessing infinite hierarchies of conserved quantities. To the best of our knowledge the Lax pairs for the equations (4.12), (5.19) and (5.20) have never been found before. In §6 we illustrate applications of the newly found Lax pairs. The Lax pair obtained in the previous sections is used to construct conservation laws for a Volterra type chain. We also show that the second operators of our Lax pairs are closely connected with the recursion operators for the associated nonlinear equations. In Appendix we give all of the computational details appeared when we evaluated the invariant manifold for the linearization of the sine-Gordon equation.

Laplace cascade for the linear hyperbolic type equations
Let us recall the main steps of the Laplace cascade method (see [22], [23]). Consider a linear second order hyperbolic type PDE of the form v xy + a(x, y)v x + b(x, y)v y + c(x, y)v = 0. (2.1) It can easily be checked that functions do not change under the linear transformation v → λ(x, y)v with arbitrary smooth factors λ(x, y) applied to equation (2.1). They are called the Laplace invariants for (2.1). We rewrite equation (2.1) as a system of two equations: When the invariant h [0] does not vanish then one can exclude v from (2.3) and obtain a linear PDE for v [1] v [1]xy + a [1] v [1]x + b [1] v [1]y + c [1] v [1] = 0 (2.4) where the coefficients are evaluated as follows for i ≥ 1 where the coefficients are given by Here we assume that is rewritten as only for i ≥ 1. However they can be prolonged for i ≤ 0 by virtue of the same formulas rewritten as follows Summarising the computations above we get a dynamical system of the form 11) which is reduced to the well-known Toda lattice ∂ ∂y where Define two linear operators where D x , D y are the operators of differentiation with respect to x, y correspondingly and D i is the shift operator acting as follows , etc. We summarize all the reasonings above as a statement. Proposition 1. The operators L i , M i commute for all i iff their coefficients satisfy the system (2.9).
The Laplace x-transformation can be interpreted in a similar way.

Laplace cascade for the nonlinear hyperbolic type equations. Formal Lax pairs
Let us explain how the Laplace cascade is adopted to nonlinear case [23] (see also [24]). Consider a second order nonlinear hyperbolic type PDE u xy = F (x, y, u, u x , u y ). (3.1) Its linearization around a solution u(x, y) derived by substituting u = u(x, y, ε) = where the coefficients a = −∂F/∂u x , b = −∂F/∂u y , c = −∂F/∂u depend explicitly on the independent variables x, y and the dynamical variables u, u x , u y . Let us assign the Laplace sequence (2.8)-(2.11) to the linearized equation (3.2). However now instead of the operators ∂/∂x, ∂/∂y in (2.8)-(2.11) we use the operators D x , D y of the total differentiation with respect to x, and y. Denote through u i ,ū i , i = 0, 1, . . . the i-th order derivatives of the variable u with respect to x and y correspondingly Evidently we have explicit expressions for the operators D x , D y acting on the class of smooth functions of x, y and a finite number of the dynamical variables u i ,ū i The Laplace invariants corresponding to the equation (3.2) are evaluated as (3.6) The linear system (2.8) in this case converts into The coefficients a [i] , b [i] , h [i] are evaluated due to the equations All of the mixed derivatives u xy , u xxy , . . . are replaced by means of the equation (3.1) and its differential consequences. Infinite-dimensional system (3.7) defines a sequence of the linear operators One can find all of the coefficients a [j] , h [j] due to the equations (3.8) as functions of the dynamical variables and therefore define completely the system (3.7). Now go back, suppose that the system evaluated above is consistent and show that its consistency defines uniquely equation (3.11). Indeed the consistency of (3.7) implies D x (a [1] ) = h [1] − h [0] equivalent to (− uy u ) x = uxuy u 2 − u which gives (3.11). Example 2. As an example with the finite system (3.7) we take the Liouville equation There is a freedom in choosing of v [2] , v [−2] . Put them equal to zero. Then the obtained system gives the Lax pair for (3.13) Remark that there are some degenerate cases where the commutativity condition of the operators (3.9) defines not exactly the initial equation (3.1) but some other equation connected with (3.1) by a Miura type transformation. Illustrate it with the following example.
Example 3. Consider the equation (3.14) The coefficients of its linearization Thus the commutativity condition of the operators (3.9) assigned to (3.14) implies the equation r xy = e r + r rx , connected with (3.14) by a very simple Miura type transformation u x + u = r.

A more symmetrical form of the Laplace sequence
Let us change the dependent variables in the system (3.7) to make formulas more symmetrical. Introduce new dependent variables Then the set of equations (3.7), (3.8) is changed to the form below, where i ≥ 0 : (3.16)

Sine-Gordon equation
More than two decades ago an important property of the Laplace invariants of the Liouville type integrable equations has been observed [24], [25]. It was proved that the hyperbolic equation (3.1) is an integrable equation of the Liouville type if and only if the set of its Laplace invariants is terminated on both sides. Mention also recent results obtained in [26]. The problem of describing the properties of the Laplace invariants characterizing the sine-Gordon type integrable PDE is discussed in [27]. Our investigation convinces that a connection between the sine-Gordon type equations and the Laplace cascade is clearly formulated in terms of the cascade eigenfunctions. Let us explain our observation with an example. We consider the sine-Gordon equation It can be shown that the Laplace invariants h [i] for the linearized equation do not vanish identically for any integer i. Thus system (3.7) provides an infinitedimensional Lax pair for (3.17). Below we show that for this case (3.7) admits a finite-dimensional reduction. Bring (3.7) to the symmetric form (3.16). Proposition 2. The system (3.16) corresponding to the sine-Gordon equation cos u is consistent with the following cutting off boundary conditions λ is a complex parameter. Sketch of proof. Look for the functions (3.20) providing the consistency of the following overdetermined system of equations obtained from (3.16) by imposing (3.19) The compatibility conditions (w [i]x ) y = (w [i]y ) x generate a system of nonlinear equations for the functions α(j), β(j) searched: Here all the given coefficients a [1] [1] are linear functions of the derivatives u x , u y : therefore we can assume that α(j), β(j) also linearly depend on the first derivatives of u Substitute expressions (3.24), (3.25) into equations (3.23) and then compare the coefficients before the independent combinations of the dynamical variables u x u y , u x , u y . As a result one gets twenty four equations for twelve functions p(j, u), q(j, u), r(j, u), s(j, u), j = 1, 0, −1, depending on u only. We write down explicitly only a part of the equations since the others are obtained from these by applying the replacement By solving the overdetermined system of equations (3.26) we find explicit expressions (3.20) for the coefficients of the constraint (3.19). Now the systems (3.21), (3.22) can be rewritten as follows It is easily checked that (3.27), (3.28) defines the Lax pair for the sine-Gordon equation (3.17). We failed to reduce it to the well-known usual one found in [28].

Invariant manifolds of the hyperbolic type PDE
Let us recall the definition of the invariant manifold of the hyperbolic type equation (3.1). Consider an equation of the form Note that G depends on x, y and a set of the dynamical variables u, u 1 ,ū 1 , . . ., where u j = ∂ j u ∂x j ,ū j = ∂ j u ∂y j . Take the differential consequences of (4.1) where G 1 , G 2 are evaluated by applying the operators D x , D y : G 1 = D x G, G 2 = D y G and subsequent replacement of the mixed derivatives by means of the equation (3.1) and its differential consequences. Equation (4.1) defines an invariant manifold for (3.1) if the following equation is satisfied Example 4. Show that equation defines an invariant manifold for the sine-Gordon equation (3.17). Here Therefore equation (4.4) holds and thus (4.5) defines an invariant manifold for (3.17).

From the Laplace cascade to invariant manifolds
Show that reduced system (3.16), (3.19) is closely connected with the invariant manifolds of the linearized equation (3.18). Indeed, equations (3.16) imply that w [2] = (D y + a [1] Simplify the equations obtained by using explicit expressions (3.20), (3.24) and find equations which define the invariant manifold for (3.18) discussed in Introduction (see (1.14), (1.15) above). This observation leads to an alternative algorithm to look for the Lax pair. Instead of the cutting off boundary conditions to the lattice (3.16) one searches an invariant manifold for the linearized equation (3.18).
By construction we have Commutators of the operators L x , L y , M satisfy the following relations x (A) and so on. Consequently any element of the Lie ring generated by the operators L x , L y , M is represented as a linear combination of the same three operators.
Linear equations (4.6), (4.7) define a manifold parametrized by y and the dynamical variables u, u 1 ,ū 1 , ... . By applying D x to the equations (4.6), (4.7) and then simplifying due to the equations (3.17), (3.18) one gets another parametrization of the manifold where the parameters xx are taken as independent ones. It is shown below that this parametrization is closely connected with the Lax pair for the potential KdV equation being a symmetry of the sine-Gordon equation.

Evaluation of the invariant manifolds and the Lax pair for the equation
In this section we construct a Lax pair to the equation found in [20]. It is known that S-integrable equation of the form (4.12) by an appropriate point transformation can be reduced either to the case f (u) = u or f (u) = sin u (see [20]). By analogy with the sine-Gordon equation considered in the previous section we look for the invariant manifold of the form v yy + av y + bv x + cv = 0 (4.13) for the linearized equation (4.14) Apply the operator D x to (4.13) and rewrite the result as Assuming that the searched functions depend only on u,u x , u y i.e. a = a(u, u x , u y ), b = b(u, u x , u y ) and c = c(u, u x , u y ) substitute these functions into (4.17), (4.18) and (4.19) and eliminate mixed derivatives of u using (4.12) from the resulting equations. Thus we obtain three equations of the following form α i (u, u x , u y )u xx u yy + β i (u, u x , u y )u xx + γ i (u, u x , u y )u yy + δ i (u, u x , u y ) = 0, i = 1, 2, 3. These relations are satisfied only if the following conditions: hold identically for all values u, u x and u y , i = 1, 2, 3. Here x )((pb − b uy c uy + bc uyuy − pu y b uy )f (u) + (−ab uy + ba uy )f ′ (u)), . Thus the problem is reduced to a system of equations (4.20).
We look for the functions a, b and c depending on the variable u y linearly: Then equation From equations α 1 = 0 and α 3 = 0 we obtain u y a uxux − a ux = 0, u y c uxux − c ux = 0.
Thus summarizing the reasonings above we can claim that equations define an invariant manifold for the linearized equation (4.14). Here λ is the spectral parameter. Derive the Lax pair for the equation (4.12) from the invariant manifold (4.23), (4.24). To this end evaluate the Laplace sequence of the form (3.16) for the linearized equation (4.14). In what follows we will need in explicit expressions for several first coefficients of the system (3.16) .
The constraints (4.23), (4.24) generate the cutting off boundary conditions of the form imposed on the infinite system (3.16). Find the coefficients α(i) and β(i) in the relation (4.26). Evidently equations (3.16) imply Set v := v [0] and simplify (4.27) by virtue of (4.26) and the relations v [1] The last equations should coincide with (4.23), (4.24). Comparison of the corresponding coefficients allows one to derive explicit formulas for the sought functions α(i) and β(i) λ . Now the infinite system of equations (3.16) is reduced to a pair of the third order systems of ordinary differential equations which can be specified as follows

Searching the Lax pairs for the evolutionary type integrable equations
Let us consider the evolutionary type integrable equations, for which the Laplace cascade is not defined. Here we use the scheme set out in the Introduction.

Korteweg-de Vries equation.
As an illustrative example we consider the Korteweg-de Vries equation Its linearization evidently has the form should be valid. Replacing in (5.4) v xxx and v t due to (5.3) and (5.2) respectively and then comparing the coefficients before the independent variables v xx , v x and v we obtain the following equations It is reasonable to assume that a = a(u, u x , u xx ), b = b(u, u x , u xx ) and c = c(u, u x , u xx ). We substitute these expressions into (5.5), (5.6) and (5.7) and then exclude all the mixed derivatives of u due to the equation (5.1). As a result we obtain three equations of the following form Since u xxxx , u 3 xxx , u 2 xxx , u xxx are independent variables then these equations split down into fifteen equations as follows hold for all values of u, u x and u xx , i = 1, 2, 3. Thus the searched coefficients a, b, c satisfy a highly overdetermined system of differential equations (5.8). Specify and analyse the system. It can be verified that the three equations β i = 0, i = 1, 2, 3 immediately imply a uxxuxxuxx = 0, β uxxuxxuxx = 0 and c uxxuxxuxx = 0 therefore Then since functions a, b and c depend only on the variables u, u x and u xx the coefficients at u xxx vanish, i.e. we get Substituting a, b and c into the equations (5.9), (5.10) and (5.11) we obtain (a 2 1 + (a 1 ) ux )u xx + a 1 a 2 + u x (a 1 ) u + b 1 = 0, Since functions a i , b i and c i , i = 1, 2 depend only on u and u x the coefficients at u xx vanish and we get the following system of equations Concentrate on the last system. It is easy to check that a 1 = 0. Assume that b 1 = 0 and c 1 = 0. From system (5.12) we find , , Here λ = c 5 is an arbitrary parameter. Proposition 3. Pair of equations (5.2), (5.14) defines the Lax pair for the KdV equation.

Potential and modified KdV equations
Concentrate on the potential KdV equation does not admit any second order invariant manifold of the necessary form However it admits a third order invariant manifold given by Here λ is an arbitrary parameter. The consistency condition of the equations (5.16), (5.17) is equivalent to the mKdV equation

Lax pairs for the KdV type equations from Svinolupov-Sokolov list
Consider the following two third order differential equations possessing infinite hierarchies of conservation laws [21]. As it is established in [20] these equations are symmetries of the equation (4.12). We have proved above in §4 that equations define an invariant manifold for the linearized equation (4.14). It is reasonable to expect that invariant manifolds for the linearizations of the symmetries (5.19) and (5.20) are closely connected with the same manifold. Indeed by applying the operators D y and D x to the equations (5.21) and respectively to (5.22) one can deduce the following two third order ordinary differential equations

Volterra type integrable chains
In this section we discuss the semi-discrete equations of the form ∂ ∂t u n = f (u n+1 , u n , u n−1 ) with the sought function u = u n (t), depending on the discrete n and continuous t. The direct method for constructing the Lax pairs through linearization can be applied to the discrete models as well. As illustrative examples we consider the modified Volterra chain dp n dt = −p 2 n (p n+1 − p n−1 ) (5.27) and the equation du n dt = 1 u n+1 − u n−1 (5.28) found in [29]. These two equations are related to each other by a very simple Miura type transformation du n dt = p n , p n = 1 u n+1 − u n−1 . The coefficients a, b, c are found from the equation Studying the equation (5.32) we assume that the variables {p k } ∞ k=−∞ , v n , v n+1 , v n−1 are independent dynamical variables. Omitting the simple but tediously long computations we give only the answer: Therefore the invariant manifold searched is of the form:

Construction of the recursion operators and conservation laws via newly found Lax pairs
It was observed that the Lax pairs for the integrable equations found above essentially differ from their classical counterparts. In this section we discuss some useful properties of the newly found Lax pairs. We show, for instance, that they provide a very convenient tool for searching the recursion operators and conservation laws for integrable models.
As illustrative examples we take the KdV and pKdV equations, the Volterra type chain (5.28) etc. We show that the equation of the invariant manifold to the linearized equation is easily transformed into the recursion operator.

Evaluation of the recursion operators for the KdV type equations
Let us start with the KdV equation (1.9). We can rewrite equation (1.11) of the invariant manifold in the following form We now multiply (6.1) from the left by the operator u x D −1 x 1 ux and obtain a formal eigenvalue problem of the form for the operator An amazing fact is that R coincides with the recursion operator for the KdV equation. Indeed, we have It can be simplified due to the relation and reduced to the form In a similar way we derive from (5.17) the recursion operator to the potential KdV equation (5.15). Indeed rewrite (5.17) as follows Simplify the expression for Rv: Apparently operator R = D 2 x + u x D −1 x u x D x coincides with the recursion operator for (5.15).

Recursion operators via the Lax pair for a Volterra type chain
We proceed with an example of the discrete model (5.28). Let us write equation (5.33) defining the invariant manifold for (5.30) as follows where D n is the shift operator acting due to the rule D n a(n) = a(n + 1). We multiply (6.5) from the left by the factor p n (D n − 1) −1 p n+1 and then get After some elementary transformations the operator R is reduced to the form R = p 2 n D n + D −1 n + 2p n p n−1 + 2p n (D n − 1) −1 (p n−1 − p n+1 ) (6.7) Operator R given by (6.7) defines the recursion operator for the chain (5.28). For instance, by applying the operator R to 1 u n+1 −u n−1 we obtain the r.h.s. of the equation constructed in [30] ∂ ∂τ as a fifth-point symmetry of the equation (5.28). Note that the whole hierarchy of the symmetries for the chain (5.28) is described in [31].

Conservation laws via the Lax pair for a Volterra type chain
Let us consider an equation of the form du n dt = 1 u n+1 − u n−1 . (6.9) Due to the relations (5.29) between u and p a pair of the equations (5.30), (5.33) define a Lax pair to the chain (6.9) as well.
Rewrite the scalar Lax pair (5.30), (5.33) in the matrix form To construct the conservation laws, we apply the method of the formal diagonalization suggested in [32,33] and developed in [34]. The first equation in (6.10) has singular point λ = ∞ (f has a pole at λ = ∞). It can be checked that the potential f is represented as follows are analytic and non-degenerate around λ = ∞, Z is a diagonal matrix of the form The change of variables ψ = βy is reducing the first system in (6.10) to the special form where The function P (λ) together with P −1 (λ) are analytic around λ = ∞: where The leading principal minors of P (λ) p n do not vanish if the variable p n satisfy the inequality p n = 0 for all n. According to the Proposition 1 in [33] the first system in (6.10) can be diagonalized, i.e. there exist formal series such that the formal change of the variables ψ = T ϕ converts the system (6.16) to the system of the diagonal form ϕ n+1 = hZϕ n . (6.20) Thus we see that the formal change of variables y = Rϕ = β −1 T ϕ, reduces the first system in investigated Lax pair (6.10) to the form (6.20). By construction R = β −1 T is a formal series of the form It follows from (6.19)-(6.21) that diagonalizable system (6.20) admits an asymptotic representation of the solution to the direct scattering problem y n (λ) = R(n, λ)e n−1 s=n 0 log h(s,λ) Z n (6.22) with the "amplitude" A = R(n, λ) and "phase" φ = n log Z + n−1 s=n 0 log h(s, λ). Turn back to the problem of diagonalization. By solving the following equation we find the formal series T and h According to the general scheme the second system of the Lax pair (6.10) is diagonalized by the same linear change of the variables y = Rϕ, where 13 = p n p 2 n+1 (p n + p n+2 ), R 22 = p n p 2 n+1 p n+2 + 3p n−1 p 2 n p n+1 + p 2 n p 2 n+1 , R 32 = −p n−1 p n (3p n−1 p 2 n p n+1 + p 2 n p 2 n+1 + p n−2 p n−1 p n p n+1 + p 2 n−2 p 2 n−1 +p n−3 p 2 n−2 p n−1 + p 2 n−1 p 2 n + 2p n−2 p 2 n−1 p n ). This change of the variables reduces the second system in (6.10) to the form dyn dt = Sy n with where S (2) 11 = p n p n+1 (p n+1 p n+2 + p n−1 p n ), S (2) 22 = p n p n+1 (p n−1 p n − p n+1 p n+2 ), S (2) 33 = −p n p n+1 (p n+1 p n+2 + p n−1 p n ). According the paper [33] the equation generates the infinite series of conservation laws for the equations (5.27). We write down in an explicit form three conservation laws from the infinite sequence obtained by the diagonalization procedure Being rewritten in terms of u n these relations give the conservation laws for the equation (6.9): These conservation laws coincide with those found earlier (see, for instance, [18], [29]).

Conclusions
There is a large set of classification methods allowing classes of integrable nonlinear PDEs and their discrete analogues to be described. For studying the analytical properties of these equations one needs the Lax pairs. Therefore the problem of creating convenient algorithms for constructing the Lax pairs is relevant. In the present article such a method is suggested. For the evolution type integrable equation the Lax pair consists of the linearized equation and the equation of its invariant manifold. In the case of the hyperbolic equations to obtain the Lax pair we use the Laplace cascade in addition to the invariant manifold. The method is applied to equations (4.12), (5.19) and (5.20) known to be integrable for which the Lax pairs have not been constructed before.
An interesting observation is connected with the Laplace cascade of the sine-Gordon equation. It is proved that in this case the cascade admits a finite-dimensional reduction which generates the Lax pair to the sine-Gordon model. We conjecture that the Laplace cascade corresponding to any hyperbolic type integrable equation admits a finite-dimensional reduction.
We considered examples showing that our method leads to true Lax pairs having useful applications. For the Lax pair of the Volterra type chain we found an asymptotic eigenfunction which allowed an infinite set of conservation laws to be constructed. It is also shown that these Lax pairs allow the recursion operators to be constructed describing infinite hierarchies of the higher symmetries and invariant manifolds for the given nonlinear equation.
of invariant manifolds for the corresponding equation (3.2). In this case equation (7.1) generates an overdetermined system of differential equations for defining the searched coefficients p(i) and q(m).
As an illustrative example we consider equation (3.18) obtained by linearizing the sine-Gordon equation. We search an invariant manifold for (3.18) in the following form v yy + av y + bv x + cv = 0. (7.2) Here a, b and c are functions depending on a finite number of the dynamical variables u, u 1 ,ū 1 , u 2 ,ū 2 , .... Apply the operator D x to (7.2) and rewrite the obtained result as follows Now apply the operator D y to this equation and simplify the result by means of the equation (3.18). We arrive at the equation Assume that a = a(u, u x , u y ), b = b(u, u x , u y ) and c = c(u, u x , u y ) and substitute these functions into (7.4), (7.5) and (7.6). Eliminating the mixed derivatives of u due to equation (3.17) we obtain three equations of the following form α i (u, u x , u y )u xx u yy + β i (u, u x , u y )u xx + γ i (u, u x , u y )u yy + δ i (u, u x , u y ) = 0, i = 1, 2, 3. Since the functions α i , β i , γ i and δ i , i = 1, 2, 3 depend only on u, u x and u y , we should have the coefficients at u xx u yy , u xx , u yy and remaining terms equal to zero, i.e. α i (u, u x , u y ) = 0, β i (u, u x , u y ) = 0, γ i (u, u x , u y ) = 0, δ i (u, u x , u y ) = 0 (7.7) for all u, u x and u y , i = 1, 2, 3. Here α 1 = ba uxuy − b uy a ux , (7.8) α 2 = bb uxuy − b uy b ux , (7.9) α 3 = bc uxuy − c ux b uy , (7.10) β 1 = bc ux − aba ux − b u u y a ux − b ux sin(u)a ux + ba uxux sin(u) + ba uux u y , (7.11) ux sin(u), (7.12) β 3 = bc uxux sin(u) + bc uux u y + bb ux cos(u) − cba ux − b ux sin(u)c ux − b u u y c ux , (7.13) γ 1 = − cos(u)b uy + b sin(u)a uyuy + bu x a uuy − b uy a u u x − b uy a uy sin(u), uy sin(u) + bc uy − cb uy , γ 3 = − b uy c u u x − b sin(u) − b uy c uy sin(u) + ba uy cos(u) + sin(u)u y b uy + bu x c uuy − a cos(u)b uy + b sin(u)c uyuy , δ 1 = (b ux a uy − ba uxuy ) cos 2 (u) − b ux cos(u) sin(u) + (ba uy u y + ba ux u x − b u u y ) cos(u) + (ba u − 2u y b + ba uuy u y − b u u y a uy ) sin(u) + (−aba uy − b ux a u u x + bu x a uux + bc uy ) sin(u) + bu x a uu u y − b u u y a u u x − aba u u x + ba uxuy − b ux a uy + bc u u x , δ 2 = (b ux b uy − bb uxuy ) cos 2 (u) + b(b ux u x + b uy u y ) cos(u) + (−b u u y b uy − b ux b u u x + bu x b uux ) sin(u) + (bb uuy u y − b 2 a uy − cb ux + bb u + bc ux ) sin(u) + bb uxuy + bc u u y − b 2 u u y u x − cb u u y − b 2 a u u x + bu x b uu u y − b ux b uy , δ 3 = (b ux c uy − bc uxuy − u y b ux ) cos 2 (u) + (ba ux − ab ux + bb uy ) cos(u) sin(u) + (bb u u x − ab u u y + ba u u y + bc ux u x + bc uy u y − u 2 y b) cos(u) + (bu x c uux − au y b + bc uuy u y − b ux c u u x ) sin(u) + (−cba uy − b u u y c uy − b 2 u x + bc u + u 2 y b u ) sin(u) + bu x c uu u y − b u u y c u u x − cba u u x + u y b ux − b ux c uy + bc uxuy .
Hence the problem of searching the equation (7.2) is reduced to the system of equations (7.7).
We now look for the functions a, b and c depending linearly on the variable u y : a = a 1 (u, u x )u y + a 2 (u, u x ), (7.14a) b = b 1 (u, u x )u y + b 2 (u, u x ), (7.14b) c = c 1 (u, u x )u y + c 2 (u, u x ). (7.14c) Then equation α 2 = 0 (see (7.9)) takes the form b 2 (b 1 ) ux − b 1 (b 2 ) ux = 0. This equation is satisfied only if at least one of the following conditions holds Let us consider the case b 2 = 0. Then function b defined by the formula (7.14b) takes the form b = b 1 (u, u x )u y . These equations hold if at least one of the following four conditions is satisfied: 1)a ux = 0, c ux = 0; 2)a ux = 0, c ux = 0, c uxux c ux = 1 u y ; 3)c ux = 0, a ux = 0, a uxux a ux = 1 u y ; 4)a ux c ux = 0, a uxux a ux = c uxux c ux .
Therefore the invariant manifold (7.2) is of the form v yy − u y cot u v y + λ u y sin u v x − λv = 0 and coincides with (4.6) while (7.3) gives (4.7).