7,220 research outputs found
Integrable ODEs on Associative Algebras
In this paper we give definitions of basic concepts such as symmetries, first
integrals, Hamiltonian and recursion operators suitable for ordinary
differential equations on associative algebras, and in particular for matrix
differential equations. We choose existence of hierarchies of first integrals
and/or symmetries as a criterion for integrability and justify it by examples.
Using our componentless approach we have solved a number of classification
problems for integrable equations on free associative algebras. Also, in the
simplest case, we have listed all possible Hamiltonian operators of low order.Comment: 19 pages, LaTe
Double Poisson brackets on free associative algebras
We discuss double Poisson structures in sense of M. Van den Bergh on free
associative algebras focusing on the case of quadratic Poisson brackets. We
establish their relations with an associative version of Young-Baxter
equations, we study a bi-hamiltonian property of the linear-quadratic pencil of
the double Poisson structure and propose a classification of the quadratic
double Poisson brackets in the case of the algebra with two free generators.
Many new examples of quadratic double Poisson brackets are proposed.Comment: 19 pages, late
Classification of integrable Vlasov-type equations
Classification of integrable Vlasov-type equations is reduced to a functional
equation for a generating function. A general solution of this functional
equation is found in terms of hypergeometric functions.Comment: latex, 15 pages, to appear in Theoretical and Mathematical Physic
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