Abstract.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.
Abstract. The main goal of the article is testing a new classification algorithm. To this end we apply it to a relevant problem of describing the integrable cases of a subclass of twodimensional lattices. By imposing the cut-off conditions u −1 = c 0 and u N +1 = c 1 we reduce the lattice u n,xy = α(u n+1 , u n , u n−1 )u n,x u n,y to a finite system of hyperbolic type PDE. Assuming that for each natural N the obtained system is integrable in the sense of Darboux we look for α. To detect the Darboux integrability of the hyperbolic type system we use an algebraic criterion of Darboux integrability which claims that the characteristic Lie rings of such a system must be of finite dimension. We prove that up to the point transformations only one lattice in the studied class passes the test. The lattice coincides with the earlier found Ferapontov-Shabat-Yamilov equation. The one-dimensional reduction x = y of this lattice passes also the symmetry integrability test.
In the paper we discuss a classification method for nonlinear integrable equations with three independent variables based on the notion of the integrable reductions. We call an 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 in the framework of this scheme is the class of two dimensional lattices generalizing the well-known Toda lattice. In the present article we study the quasilinear lattices of the form
Abstract. A method of the formal diagonalization of the discrete linear operator with a parameter is studied. In the case when the operator provides a Lax operator for a nonlinear quad system the formal diagonalization method allows one to describe effectively conservation laws and generalized symmetries for this system. Asymptotic representation of the Lax operators eigenfunctions are constructed and infinite series of conservation laws are described for the quad system connected with A (1) 3 affine Lie algebra, for the modified discrete Boussinesq system and for the discrete Tzitzeica equation. For a newly found multiquadratic discrete model conservation laws and several generalized symmetries are presented.
In the work we discuss briefly a method for constructing a formal asymptotic solution to a system of linear difference equations in the vicinity of a special value of the parameter. In the case when the system is the Lax pair for some nonlinear equation on a square graph, the found formal asymptotic solution allows us to describe the conservation laws and higher symmetries for this nonlinear equation. In the work we give a complete description of a series of conservation laws and the higher symmetries hierarchy for a discrete potential two-component Korteweg-de Vries equation.
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