V. V. Sokolov

Lie algebras and equations of Korteweg-de Vries type

The survey contains a description of the connection between the infinite-dimensional Lie algebras of Kats-Moody and systems of differential equations generalizing the Korteweg-de Vries and sine-Gordon equations and integrable by the method of the inverse scattering problem. A survey of the theory of Kats-Moody algebras is also given.

The Symmetry Approach to Classification of Integrable Equations

In this volume each of the contributors proposes his own test to recognize integrable PDEs. We believe that, independently from the basic definition of integrability, the test must satisfy some general requirements. Namely, it has to be effective (in other words, if an equation has passed through the test, then there are almost no doubts about its integrability); sufficiently algorithmical, yet able to admit a proper realization in a symbolic computer language (like Reduce, Formac, Macsyma, MuMath, AMP, etc.); applicable to a large class of PDEs.

On construction of recursion operators from Lax representation

In this work we develop a general procedure for constructing the recursion operators for nonlinear integrable equations admitting Lax representation. Several new examples are given. In particular, we find the recursion operators for some KdV-type systems of integrable equations.

Vector-matrix generalizations of classical integrable equations

Some vector-matrix generalizations, both known and new, for well-known integrable equations are presented. All of them possess higher symmetries and conservation laws.

On the Darboux integrable hyperbolic equations

Abstract We prove that if the sequence of Laplace invariants for the linearization operator of a nonlinear hyperbolic equation is finite then the equation is Darboux integrable.

Classification of integrable polynomial vector evolution equations

Several classes of systems of evolution equations with one or two vector unknowns are considered. We also investigate systems with one vector and one scalar unknown. For these classes all equations having the simplest higher symmetry are listed.

Generalized Operator Yang-Baxter Equations, Integrable ODEs and Nonassociative Algebras

Abstract Reductions for systems of ODEs integrable via the standard factorization method (the Adler-Kostant-Symes scheme) or the generalized factorization method, developed by the authors earlier, are considered. Relationships between such reductions, operator Yang-Baxter equations, and some kinds of non-associative algebras are established.

A New Integrable Case for the Kirchhoff Equation

A new integrable case is found for the Kirchhoff equation. The additional integral of motion is a fourth-degree polynomial, the principal metric is diagonal with the eigenvalues a1 = a2 = 1 and a3 = 2, and the other two metrics are nondiagonal.

Multicomponent generalization of the hierarchy of the Landau-Lifshitz equation

We construct a second-order 2N-component integrable system (with arbitrary N) whose spectral parameter lies on a curve of genus g=1+(N-3)2N−2. The odd-order flows admit N-component reductions, which for N=3 coincide with the odd-order flows of the hierarchy of the Landau-Lifshitz equation.

Optical materials containing rare earth Ln2S3 sulfides

Abstract A short review of the results of our studies on the synthesis and crystal growth of simple and binary rare earth sulfides, on the processing of La2S3 and NaLaS2 ceramics and on the preparation of LaGaSO glasses is presented.