Marek Antonowicz

Constrained flows of integrable PDEs and bi-Hamiltonian structure of the Garnier system

Abstract A new reduction procedure for integrable multi-Hamiltonian PDEs is introduced. It leads to a multi-Hamiltonian description of the resulting finite dimensional dynamical systems. The bi-Hamiltonian structure of the Garnier system is studied in some detail.

How to construct finite‐dimensional bi‐Hamiltonian systems from soliton equations: Jacobi integrable potentials

A systematic method of constructing finite‐dimensional integrable systems starting from a bi‐Hamiltonian hierarchy of soliton equations is introduced. The existence of two Hamiltonian structures of the hierarchy leads to a bi‐Hamiltonian formulation of the resulting finite‐dimensional systems. The case of coupled KdV hierarchies is studied in detail. A surprising connection with separable Jacobi potentials is uncovered and described.

Restricted flows of soliton hierarchies : coupled KdV and harry dym case

Restricted flows of soliton hierarchies associated with the energy-dependent Schrodinger spectral problem are determined explicitly. A remarkable connection with separable potentials is used for proving complete integrability of the restricted flows. A previously unknown Lagrangian and Hamiltonian formulation of the Neumann system is found. Whole families of generalizations of the Neumann and Garnier systems are given.

Bi-Hamiltonian formulation of the Hénon-Heiles system and its multidimensional extensions

Abstract We prove that a (slightly) generalized Henon-Heiles system is equivalent to a fifth order Hamiltonian evolution equation with a third order Hamiltonian operator. This equivalence makes it possible to use the machinery of restricted flows of soliton hierarchies in order to (a) find natural extensions of integrable cases of the Henon-Heiles system, and (b) determine (in the KdV case) a bi-Hamiltonian formulation for the (extended) Henon-Heiles system and prove its complete integrability.

Multicomponent Schwarzian KdV hierarchies

Abstract We extend recent results of Dorfman [1] and Wilson [2] on the Schwarzian-KdV equation to the coupled KdV and Harry Dym hierarchies isospectral to an energy dependent Schrodinger operator [3, 4]. Whilst the N-component coupled KdV hierarchies posses (N + 1) differential Hamiltonian structures, their Schwarzian modifications posses (N + 1) nonlocal Hamiltonian structures whose inverses are purely differential. We present, as examples, the Schwarzian modifications associated with the dispersive water waves (DWW) and the Harry Dym equations.

THE DYNAMICAL STRUCTURE OF GRAVITATIONAL THEORIES WITH GL(4,R) CONNECTIONS

We investigate here GL (4,R)‐gauge theories of gravity based on variational principles. The components of tetrad fields eμ(α), the components of metrics g(α)(β), and the components of connections Γλ(α)(β) are taken as the gravitational potentials. Matter potentials are the components of GL (4,R)‐tensor fields φΣ. We derive the conservation laws for a general theory, that is, the Belinfante–Rosenfeld and Bianchi identities, and find minimal systems of independent variational equations. The natural GL (4,R)‐covariant Hamiltonian formulation of the theory induces a GL (3,R)‐covariant Hamiltonian formulation related to a chosen slicing of space‐time. The Hamiltonian field equations corresponding to this formulation describe the dynamics of the system. We determine 20 symplectic constraints, 20 gauge transformations, and 20 gauge variables generic for a general gravitational Lagrangian. As an example, we consider the Gl (4,R)–Einstein theory in vacuum as well as in the presence of a vector field and find the c...

New integrable nonlinearities from affine geometry

Abstract A new completely integrable nonlinear system with nontrivial spectral problem and the corresponding Backlund transformation are presented. The affine geometric origin of the system is discussed briefly.

Geometry of canonical variables in gravity theories

We show how to pass from an SL(2, C) covariant to an SU(2) covariant formulation of the theories of gravity. Our construction determines the canonical and gauge variables of the theory and establishes an appropriate framework for a hamiltonian picture.

An SU(2)-covariant dynamical formulation of the Einstein-Cartan-Dirac theory

An SU(2)-covariant canonical formulation of the coupled Einstein-Cartan-Dirac field in the framework of the Einstein-Cartan-Sciama-Kibble theory is presented. The variational field equations are divided into constraints for initial values of the canonical variables, the gravitational evolution equations and the Dirac equations written in a dynamical form. The matter canonical variables are represented by SU(2)-spinor-valued 1/2-forms on three-dimensional surfaces in spacetime. The Poisson algebra of constraints and the problem of independent degrees of freedom are investigated.