Non-Homogeneous Hydrodynamic Systems and Quasi-St\"ackel Hamiltonians

In this paper we present a novel construction of non-homogeneous hydrodynamic equations from what we call quasi-St\"ackel systems, that is non-commutatively integrable systems constructed from appropriate maximally superintegrable St\"ackel systems. We describe the relations between Poisson algebras generated by quasi-St\"ackel Hamiltonians and the corresponding Lie algebras of vector fields of non-homogeneous hydrodynamic systems. We also apply St\"ackel transform to obtain new non-homogeneous equations of considered type

Invertible coupled KdV and coupled Harry Dym hierarchies

In this paper we discuss the conditions under which the coupled KdV and coupled Harry Dym hierarchies possess inverse (negative) parts. We further investigate the structure of nonlocal parts of tensor invariants of these hierarchies, in particular, the nonlocal terms of vector fields, conserved one-forms, recursion operators, Poisson and symplectic operators. We show that the invertible cKdV hierarchies possess Poisson structures that are at most weakly nonlocal while coupled Harry Dym hierarchies have Poisson structures with nonlocalities of the third order

St\"ackel transform of Lax equations

We construct Lax pairs for a wide class of St\"ackel systems by applying the multi-parameter St\"ackel transform to Lax pairs of a suitably chosen systems from the seed class. For a given St\"ackel system, the obtained set of non-equivalent Lax pairs is parametrized by an arbitrary function

Deforming Lie algebras to Frobenius integrable non-autonomous Hamiltonian systems

Motivated by the theory of Painlev\'e equations and associated hierarchies, we study non-autonomous Hamiltonian systems that are Frobenius integrable. We establish sufficient conditions under which a given finite-dimensional Lie algebra of Hamiltonian vector fields can be deformed to a time-dependent Lie algebra of Frobenius integrable vector fields spanning the same distribution as the original algebra. The results are applied to quasi-St\"ackel systems.Comment: 14 pages, no figures. We repaired Example 5 and also made some minor amendments in the tex

Separable quantizations of St\"{a}ckel systems

In this article we prove that many Hamiltonian systems that can not be separably quantized in the classical approach of Robertson and Eisenhardt can be separably quantized if we extend the class of admissible quantizations through a suitable choice of Riemann space adapted to the Poisson geometry of the system. Actually, in this article we prove that for every quadratic in momenta St\"ackel system (defined on a 2n-dimensional Poisson manifold) for which the St\"ackel matrix consists of monomials in position coordinates there exist infinitely many quantizations - parametrized by n arbitrary functions - that turn this system into a quantum separable St\"ackel system.Comment: We added the journal reference and also intorduced several minor amendmend

Quasi-Lagrangian Systems of Newton Equations

Systems of Newton equations of the form $\ddot{q}=-{1/2}A^{-1}(q)\nabla k$ with an integral of motion quadratic in velocities are studied. These equations generalize the potential case (when A=I, the identity matrix) and they admit a curious quasi-Lagrangian formulation which differs from the standard Lagrange equations by the plus sign between terms. A theory of such quasi-Lagrangian Newton (qLN) systems having two functionally independent integrals of motion is developed with focus on two-dimensional systems. Such systems admit a bi-Hamiltonian formulation and are proved to be completely integrable by embedding into five-dimensional integrable systems. They are characterized by a linear, second-order PDE which we call the fundamental equation. Fundamental equations are classified through linear pencils of matrices associated with qLN systems. The theory is illustrated by two classes of systems: separable potential systems and driven systems. New separation variables for driven systems are found. These variables are based on sets of non-confocal conics. An effective criterion for existence of a qLN formulation of a given system is formulated and applied to dynamical systems of the Henon-Heiles type.Comment: 50 pages including 9 figures. Uses epsfig package. To appear in J. Math. Phy

Miura maps for St\"{a}ckel systems

We introduce the concept of Miura maps between parameter-dependent algebraic curves of hyperelliptic type. These Miura maps induce Miura maps between St\"{a}ckel systems defined (on the extended phase space) by the considered algebraic curves. This construction yields a new way of generating multi-Hamiltonian representations for St\"{a}ckel systems

Stabilization and Reactions of Sulfur Radical Cations: Relevance to One-Electron Oxidation of Methionine in Peptides and Proteins

Methionine is a key amino acid that has numerous roles in essential vital processes. Moreover, methionine oxidation is biologically important during conditions of oxidative stress and represents an important step in the development of some severe pathologies. Considerable work has been performed to understand the mechanisms of one-electron oxidation of the Met-residue as a function of its proteic environment. The most important recent results obtained by means of time-resolved techniques (laser flash photolysis and pulse radiolysis) on model peptides containing single or multiple Met-residues and in selected naturally occurring peptides (Met-enkephalin and ?-amyloid peptide) and proteins (thioredoxin and calmodulin) have been reviewed

St\"{a}ckel representations of stationary KdV systems

In this article we study St\"{a}ckel representations of stationary KdV systems. Using Lax formalism we prove that these systems have two different representations as separable St\"{a}ckel systems of Benenti type, related with different foliations of the stationary manifold. We do it by constructing an explicit transformation between the jet coordinates of stationary KdV systems and separation variables of the corresponding Benenti systems for arbitrary number of degrees of freedom. Moreover, on the stationary manifold, we present the explicit form of Miura map between both representations of stationary KdV systems, which also yields their bi-Hamiltonian formulation.Comment: 18 pagage