Compatible Lie brackets related to elliptic curve

For the direct sum of several copies of sl_n, a family of Lie brackets compatible with the initial one is constructed. The structure constants of these brackets are expressed in terms of theta-functions associated with an elliptic curve. The structure of Casimir elements for these brackets is investigated. A generalization of this construction to the case of vector-valued theta-functions is presented. The brackets define a multi-hamiltonian structure for the elliptic sl_n-Gaudin model. A different procedure for constructing compatible Lie brackets based on the argument shift method for quadratic Poisson brackets is discussed.Comment: 18 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

Integrable matrix equations related to pairs of compatible associative algebras

We study associative multiplications in semi-simple associative algebras over C compatible with the usual one. An interesting class of such multiplications is related to the affine Dynkin diagrams of A, D, E-type. In this paper we investigate in details the multiplications of the A-type and integrable matrix ODEs and PDEs generated by them.Comment: 12 pages, Late

Bi-Hamiltonian ordinary differential equations with matrix variables

We consider a special class of Poisson brackets related to systems of ordinary differential equations with matrix variables. We investigate general properties of such brackets, present an example of a compatible pair of quadratic and linear brackets, and find the corresponding hierarchy of integrable models, which generalizes the two-component Manakov matrix system to the case of an arbitrary number of matrices

Integrable (2+1)-dimensional systems of hydrodynamic type

We describe the results that have so far been obtained in the classification problem for integrable (2+1)-dimensional systems of hydrodynamic type. The systems of Gibbons--Tsarev type are the most fundamental here. A whole class of integrable (2+1)-dimensional models is related to each such system. We present the known GT systems related to algebraic curves of genus g=0 and g=1 and also a new GT system corresponding to algebraic curves of genus g=2. We construct a wide class of integrable models generated by the simplest GT system, which was not considered previously because it is in a sense trivial.Comment: 47 pages, no figure

Non-homogeneous systems of hydrodynamic type possessing Lax representations

We consider 1+1 - dimensional non-homogeneous systems of hydrodynamic type that possess Lax representations with movable singularities. We present a construction, which provides a wide class of examples of such systems with arbitrary number of components. In the two-component case a classification is given.Comment: 22 pages, latex, minor change

From AKNS to derivative NLS hierarchies via deformations of associative products

Using deformations of associative products, derivative nonlinear Schrodinger (DNLS) hierarchies are recovered as AKNS-type hierarchies. Since the latter can also be formulated as Gelfand-Dickey-type Lax hierarchies, a recently developed method to obtain 'functional representations' can be applied. We actually consider hierarchies with dependent variables in any (possibly noncommutative) associative algebra, e.g., an algebra of matrices of functions. This also covers the case of hierarchies of coupled DNLS equations.Comment: 22 pages, 2nd version: title changed and material organized in a different way, 3rd version: introduction and first part of section 2 rewritten, taking account of previously overlooked references. To appear in J. Physics A: Math. Ge