R. I. Yamilov

Symmetry Approach to the Integrability Problem

We review the results of the twenty-year development of the symmetry approach to classifying integrable models in mathematical physics. The generalized Toda chains and the related equations of the nonlinear Schrödinger type, discrete transformations, and hyperbolic systems are central in this approach. Moreover, we consider equations of the Painlevé type, master symmetries, and the problem of integrability criteria for (2+1)-dimensional models. We present the list of canonical forms for (1+1)-dimensional integrable systems. We elaborate the effective tests for integrability and the algorithms for reduction to the canonical form.

Symmetries as integrability criteria for differential difference equations

In this paper we review the results obtained by the generalized symmetry method in the case of differential difference equations during the last 20 years. Together with general theory of the method, classification results are discussed for classes of equations which include the Volterra, Toda and relativistic Toda lattice equations.

Extension of the module of invertible transformations. Classification of integrable systems

We demonstrate that for the systems of equations, which are invariant under a point group or possess conservation laws of the zeroth or first order, a nontrivial extension of the module of invertible transformations is possible. That simplifies greatly a classification of the integrable systems of equations. Here we present an exhaustive list and a classification of the second order systems of the formut=uxx+f(u, v, uxvx), −vt=vxx+g(u, v, ux,vx), which possess the conservation laws of higher order. The reduction group approach allows us to define the Lax type representations for some new equations of our list.

Construction scheme for discrete Miura transformations

A direct and elementary scheme for the construction of Miura-type transformations and discrete differential equations related to them (scalar and vector) is presented. The scheme is illustrated using as examples the Volterra and Toda models. A discrete-differential analogue of the Calogero-Degasperis equation is discussed in detail. This example is used to show how to construct conservation laws, higher symmetries, and solutions for an equation obtained with the help of the scheme.

Canonical transformations generated by shifts in nonlinear lattices

Abstract The invertible differential substitutions which conserve the standard Poisson brackets and act on Hamiltonians in an appropriate way are considered. These canonical auto-Backlund transformations proved to be a very simple and efficient tool in the theory of solitons. In particular, they allow one to prove a general involutivity theorem and to build up simple formulae for soliton-like solutions.

Explicit auto-transformations of integrable chains

A construction scheme for explicit auto-transformations of integrable discrete-differential equations (chains) is presented. These transformations are rather convenient to obtain the exact solutions for chains as well as associated partial differential systems. On the other hand they exemplify new integrable discrete mappings. Their group properties are also of great interest. The scheme is illustrated by several examples of integrable systems which contains the nonlinear Schrodinger system and the Landau-Lifshits model.

Lattice representations of integrable systems

Abstract We indicate the general connection between one-dimensional lattices with local symmetries and nonlinear integrable partial differential equations in 1 + 1 dimensions. The nonlinear chain provides a set of finite-dimensional integrable models of the corresponding PDE. The integrals of these finite-dimensional models are related in a direct way with the conserved quantities of the PDE.

On Miura Transformations and Volterra-Type Equations Associated with the Adler-Bobenko-Suris Equations

We construct Miura transformations mapping the scalar spectral problems of the integrable lattice equations belonging to the Adler-Bobenko-Suris (ABS) list into the dis- crete Schrodinger spectral problem associated with Volterra-type equations. We show that the ABS equations correspond to Backlund transformations for some particular cases of the discrete Krichever-Novikov equation found by Yamilov (YdKN equation). This enables us to construct new generalized symmetries for the ABS equations. The same can be said about the generalizations of the ABS equations introduced by Tongas, Tsoubelis and Xenitidis. All of them generate Backlund transformations for the YdKN equation. The higher order generalized symmetries we construct in the present paper confirm their integrability.

Symmetries of the Continuous and Discrete Krichever-Novikov Equation ⋆

A symmetry classification is performed for a class of differential-difference equations depending on 9 parameters. A 6-parameter subclass of these equations is an integrable discretization of the Krichever-Novikov equation. The dimension n of the Lie point symmetry algebra satisfies 1 ≤ n ≤ 5. The highest dimensions, namely n = 5 and n = 4 occur only in the integrable cases.

Darboux integrability of trapezoidal H4 and H4 families of lattice equations I: first integrals

In this paper we prove that the trapezoidal