Natural coordinates for a class of Benenti systems

We present explicit formulas for the coordinates in which the Hamiltonians of the Benenti systems with flat metrics take natural form and the metrics in question are represented by constant diagonal matrices.Comment: LaTeX 2e, 8 p., no figures; extended version with enlarged bibliograph

From bi-Hamiltonian geometry to separation of variables: stationary Harry-Dym and the KdV dressing chain

Separability theory of one-Casimir Poisson pencils, written down in arbitrary coordinates, is presented. Separation of variables for stationary Harry-Dym and the KdV dressing chain illustrates the theory.Comment: LaTex 14 pages, Proceedings of the Special Session on Integrable Systems of the First Joint Meeting of the American Mathematical Society and the Hong Kong Mathematical Society, to appear in J. Nonl. Math. Phy

On reciprocal equivalence of St\"ackel systems

In this paper we ivestigate St\"ackel transforms between different classes of parameter-dependent St\"ackel separable systems of the same dimension. We show that the set of all St\"ackel systems of the same dimension splits to equivalence classes so that all members within the same class can be connected by a single St\"ackel transform. We also give an explicit formula relating solutions of two St\"ackel-related systems. These results show in particular that any two geodesic St\"ackel systems are St\"ackel equivalent in the sense that it is possible to transform one into another by a single St\"ackel transform. We also simplify proofs of some known statements about multiparameter St\"ackel transform.Comment: We clarified arguments in Section 3, Studies in Applied Mathematics 201

Construction of coupled Harry Dym hierarchy and its solutions from St\"ackel systems

In this paper we show how to construct the coupled (multicomponent) Harry Dym (cHD) hierarchy from classical St\"ackel separable systems. Both nonlocal and purely differential parts of hierarchies are obtained. We also construct various classes of solutions of cHD hierarchy from solutions of corresponding St\"ackel systems.Comment: 16 page

Quantized W-algebra of sl(2,1) and quantum parafermions of U_q(sl(2))

In this paper, we establish the connection between the quantized W-algebra of ${\frak sl}(2,1)$ and quantum parafermions of $U_q(\hat {\frak sl}(2))$ that a shifted product of the two quantum parafermions of $U_q(\hat {\frak sl}(2))$ generates the quantized W-algebra of ${\frak sl}(2,1)$

Separable Hamiltonian equations on Riemann manifolds and related integrable hydrodynamic systems

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 Casimir bi-Hamiltonian case is studed in details and in this case, a systematic construction of related hydrodynamic systems in arbitrary coordinates is presented, using a cofactor method and soliton symmetry constraints.Comment: to appear in Journal of Geometry and Physic

Non-Hamiltonian systems separable by Hamilton-Jacobi method

We show that with every separable calssical Stackel system of Benenti type on a Riemannian space one can associate, by a proper deformation of the metric tensor, a multi-parameter family of non-Hamiltonian systems on the same space, sharing the same trajectories and related to the seed system by appropriate reciprocal transformations. These system are known as bi-cofactor systems and are integrable in quadratures as the seed Hamiltonian system is. We show that with each class of bi-cofactor systems a pair of separation curves can be related. We also investigate conditions under which a given flat bi-cofactor system can be deformed to a family of geodesically equivalent flat bi-cofactor systems.Comment: 20 pages, LaTeX, no figure

Gauge transformation and reciprocal link for (2+1)-dimensional integrable field systems

Appropriate restrictions of Lax operators which allows to construction of (2+1)-dimensional integrable field systems, coming from centrally extended algebra of pseudo-differential operators, are reviewed. The gauge transformation and the reciprocal link between three classes of Lax hierarchies are established.Comment: to appear in J. Nonl. Math. Phys., 12 page