Igor' Zakharovich Golubchik

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.

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.

Compatible Lie Brackets and Integrable Equations of the Principal Chiral Model Type

We consider two classes of integrable nonlinear hyperbolic systems on Lie algebras. These systems generalize the principal chiral model. Each system is related to a pair of compatible Lie brackets and has a Lax representation, which is determined by the direct sum decomposition of the Lie algebra of Laurent series into the subalgebra of Taylor series and the complementary subalgebra corresponding to the pair. New examples of compatible Lie brackets are given.

On some generalizations of the factorization method

AbstractThe classical factorization method reduces the study of a system of ordinary differential equations Ut=[U+, U] to solving algebraic equations. Here U(t) belongs to a Lie algebra

One more type of classical Yang - Baxter equation

\mathfrak{G}

One More Kind of the Classical Yang-Baxter Equation

which is the direct sum of its subalgebras

\mathbb Z

\mathfrak{G}_ +