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For the rational, elliptic and trigonometric r-matrices, we exhibit the links between three "levels" of Poisson spaces: (a) Some finite-dimensional spaces of matrix-valued holomorphic functions on the complex line; (b) Spaces of spectral…
We construct a bi-Hamiltonian structure for the holomorphic spin Sutherland hierarchy based on collective spin variables. The construction relies on Poisson reduction of a bi-Hamiltonian structure on the holomorphic cotangent bundle of…
We generalize Toda--like integrable lattice systems to non--symmetric case. We show that they possess the bi--Hamiltonian structure.
This work concerns the definition and analysis of a new class of Lie systems on Poisson manifolds enjoying rich geometric features: the Lie--Hamilton systems. We devise methods to study their superposition rules, time independent constants…
In this study we work on a novel Hamiltonian system which is Liouville integrable. In the integrable Hamiltonian model, conserved currents can be represented as Binomial polynomials in which each order corresponds to the integral of motion…
A systematic construction of St\"{a}ckel systems in separated coordinates and its relation to bi-Hamiltonian formalism are considered. A general form of related hydrodynamic systems, integrable by the Hamilton-Jacobi method, is derived. One…
The presence of two compatible Hamiltonian structures is known to be one of the main, and the most natural, mechanisms of integrability. For every pair of Hamiltonian structures, there are associated conservation laws (first integrals).…
Based on the Kupershmidt deformation for any integrable bi-Hamiltonian systems presented in [4], we propose the generalized Kupershmidt deformation to construct new systems from integrable bi-Hamiltonian systems, which provides a…
We discuss a new geometric construction of port-Hamiltonian systems. Using this framework, we revisit the notion of interconnection providing it with an intrinsic description. Special emphasis on theoretical and applied examples is given…
In the literature, the existence of Darboux polynomials and additional polynomial first integrals has been considered in the case of Hamiltonian systems. In this article such problem is formulated in the more general framework of Poisson…
Pulling back sets of functions in involution by Poisson mappings and adding Casimir functions during the process allows to construct completely integrable systems. Some examples are investigated in detail.
The interconnection between self-duality, conformal invariance and Lie-Poisson structure of the two dimensional non-abelian Thirring model is investigated in the framework of the hamiltonian method.
It is shown that a linear separation relations are fundamental objects for integration by quadratures of St\"{a}ckel separable Liouville integrable systems (the so-called St\"{a}ckel systems). These relations are further employed for the…
The integrable system is introduced based on the Poisson $ rs $-matrix structure. This is a generalization of the Gaudin magnet, and in SL(2) case isomorphic to the generalized Neumann model. The separation of variables is discussed for…
Recently one integrable model with a cubic first integral of motion has been studied by Valent using some special coordinate system. We describe the bi-Hamiltonian structures and variables of separation for this system.
The hamiltonian structures for quartic oscillator are considered. All structures admitting quadratic hamiltonians are classified.
An algebraic definition of Gardner's deformations for completely integrable bi-Hamiltonian evolutionary systems is formulated. The proposed approach extends the class of deformable equations and yields new integrable evolutionary and…
The bi-Hamiltonian structure is established for the perturbation equations of KdV hierarchy and thus the perturbation equations themselves provide also examples among typical soliton equations. Besides, a more general bi-Hamiltonian…
The subject for investigation in this note is concerned with holomorphic Poisson structures on nilmanifolds with abelian complex structures. As a basic fact, we establish that on such manifolds, the Dolbeault cohomology with coefficients in…
We study the geometric structure of port-Hamiltonian systems. Starting with the intuitive understanding that port-Hamiltonian systems are "in between" certain closed Hamiltonian systems, the geometric structure of port-Hamiltonian systems…