Blazej M. Szablikowski

Meromorphic Lax representations of (1+1)-dimensional multi-Hamiltonian dispersionless systems

Rational Lax hierarchies introduced by Krichever are generalized. A systematic construction of infinite multi-Hamiltonian hierarchies and related conserved quantities is presented. The method is based on the classical R-matrix approach applied to Poisson algebras. A proof that Poisson operators constructed near different points of Laurent expansion of Lax functions are equal is given. All results are illustrated by several examples.

Novikov Algebras and a Classification of Multicomponent Camassa–Holm Equations

A class of multicomponent integrable systems associated with Novikov algebras, which interpolate between Korteweg–de Vries (KdV) and Camassa–Holm-type equations, is obtained. The construction is based on the classification of low-dimensional Novikov algebras by Bai and Meng. These multicomponent bi-Hamiltonian systems obtained by this construction may be interpreted as Euler equations on the centrally extended Lie algebras associated with the Novikov algebras. The related bilinear forms generating cocycles of first, second, and third order are classified. Several examples, including known integrable equations, are presented.

Integrable discrete systems on R and related dispersionless systems

A general framework for integrable discrete systems on R, in particular, containing lattice soliton systems and their q-deformed analogs, is presented. The concept of regular grain structures on R, generated by discrete one-parameter groups of diffeomorphisms, in terms of which one can define algebra of shift operators is introduced. Two integrable hierarchies of discrete chains together with bi-Hamiltonian structures and their continuous limits are constructed. The inverse problem based on the deformation quantization scheme is considered.

The R-matrix approach to integrable systems on time scales

A general unifying framework for integrable soliton-like systems on time scales is introduced. The R-matrix formalism is applied to the algebra of δ-differential operators in terms of which one can construct an infinite hierarchy of commuting vector fields. The theory is illustrated by two infinite-field integrable hierarchies on time scales which are Δ-differential counterparts of KP and mKP. The difference counterparts of AKNS and Kaup–Broer soliton systems are constructed as related finite-field restrictions.

Classical r-matrix like approach to Frobenius manifolds, WDVV equations and flat metrics

A general scheme for the construction of flat pencils of contravariant metrics and Frobenius manifolds as well as related solutions to Witten–Dijkgraaf–Verlinde–Verlinde associativity equations is formulated. The advantage is taken from the Rota–Baxter identity and some relation being counterpart of the modified Yang–Baxter identity from the classical r-matrix formalism. The scheme for the construction of Frobenius manifolds is illustrated on the algebras of formal Laurent series and meromorphic functions on Riemann sphere.

Integrable quantum Stäckel systems

Abstract The Stackel separability of a Hamiltonian system is well known to ensure existence of a complete set of Poisson commuting integrals of motion quadratic in the momenta. We consider a class of Stackel separable systems where the entries of the Stackel matrix are monomials in the separation variables. We show that the only systems in this class for which the integrals of motion arising from the Stackel construction keep commuting after quantization are, up to natural equivalence transformations, the so-called Benenti systems. Moreover, it turns out that the latter are the only quantum separable systems in the class under study.

Construction and separability of nonlinear soliton integrable couplings

Abstract The paper is motivated by recent works of several authors, initiated by articles of Ma and Zhu [W. X. Ma, Z. N. Zhu, Constructing nonlinear discrete integrable Hamiltonian couplings, Comput. Math. Appl. 60 (2010) 2601] and Ma [W. X. Ma, Nonlinear continuous integrable Hamiltonian couplings, Appl. Math. Comput. 217 (2011) 7238], where new class of soliton systems, being nonlinear integrable couplings, was introduced. Here, we present a general construction of such class of systems and we develop the decoupling procedure, separating them into copies of underlying original equations.

On deformations of standard R-matrices for integrable infinite-dimensional systems ∗

Simple deformations, with a parameter ϵ, of classical R-matrices which follow from decomposition of appropriate Lie algebras, are considered. As a result nonstandard Lax representations for some well-known integrable systems are presented and integrable evolution equations are constructed.