On One Integrable Discrete System

2019

JOR

The method of constructing particular solutions to nonlinear partial differential equations based on the notion of differential constraint (or invariant manifold) is well known in the literature, see (Yanenko, 1961; Sidorov et al., 1984). The matter of the method is to add a compatible equation to a given equation and as a rule, the compatible equation is simpler. Such technique allows one to find particular solutions to a studied equation. In works (Pavlova et al., 2017; Habibullin et al., 2017, 2018; Khakimova, 2018; Habibullin et al., 2016, 2017, 2018) there was proposed a scheme for constructing the Lax pairs and recursion operators for integrable partial differential equations based on the use of similar idea. A suitable generalization is to impose a differential constraint not on the equation, but on its linearization. The resulting equation is referred to as a generalized invariant manifold. In works (Pavlova et al., 2017; Habibullin et al., 2017, 2018; Khakimova, 2018; Habibullin et al., 2016, 2017, 2018) it is shown that generalized invariant varieties allow efficient construction of Lax pairs and recursion operators of integrable equations. The research was supported by the RAS Presidium Program «Nonlinear dynamics: fundamental problems and applications».

Section: литератураmentioning

confidence: 99%

Algorithm for Constructing a Lax Pair and a Recursion Operator for Integrable Equations

2019

JOR

“…where D is the shift operator, such that Dy(n) = y(n + 1) and q (1) , q (−1) , s (1) , s (2) are some functions depending on the dynamical variables u n , u n±1 , u n±2 , . .…”

Section: Evaluation Of the Invariant Manifolds And The Recursion Opermentioning

confidence: 99%

“…. By the construction equations (2) and (3) are compatible if u = u(x, t) is a solution of the equation (1). It is remarkable that for an appropriate choice of the equation (3) the converse is also true: the compatibility of the equations (2) and (3) implies (1).…”

Section: Introductionmentioning

confidence: 99%

“…of the order m ≥ 1 compatible with the linearized equation (2) for all values of the dynamical variables u, u 1 , u 2 , ... considered here as parameters. The compatibility condition of the equations (2) and (3) coincides with the equation D t H(x, t, U, U 1 , .…”

Section: Introductionmentioning

confidence: 99%

“…It is supposed that in (4) all of the derivatives are expressed due to the equations (1) and (2) and the variables U s+m with s ≥ 0 are expressed due to the equation (3).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Direct Algorithm for Constructing Recursion Operators and Lax Pairs for Integrable Models

2018

Theor Math Phys

Self Cite

We suggested an algorithm for searching the recursion operators for nonlinear integrable equations. It was observed that the recursion operator R can be represented as a ratio of the form R = L −1 1 L 2 where the linear differential operators L 1 and L 2 are chosen in such a way that the ordinary differential equation (L 2 − λL 1 )U = 0 is consistent with the linearization of the given nonlinear integrable equation for any value of the parameter λ ∈ C. For constructing the operator L 1 we use the concept of the invariant manifold which is a generalization of the symmetry. Then for searching L 2 we take an auxiliary linear equation connected with the linearized equation by the Darboux transformation. Connection of the invariant manifold with the Lax pairs and the Dubrovin-Weierstrass equations is discussed.

On the recursion operators for integrable equations

J. Phys. A: Math. Theor.

2018

It is widely known that the recursion operator is a very important component of integrability. It allows one to describe in a compact form both hierarchies of the generalized symmetries and infinite series of the local conservation laws. In the literature, we can find several methods for constructing recursion operators, some of them use the Hamiltonian approach and the others are based on the Lax representation of the equation. In the present article we discuss on an alternative method, suggested in [5], which is connected only with the first several generalized symmetries of the given equation. Efficiency of the method is illustrated with the examples of KdV, Krichever-Novikov and Kaup-Kupershmidt equations and discrete models.