On quadratic Poisson brackets

In this paper, we present a method for constructing large families of quadratic Poisson brackets on a manifold using more elementary brackets on a different manifold. The method is then applied to several examples of completely integrable systems. One can recover several known brackets for systems such as the Toda lattice or the open discrete KP hierarchy. New brackets for a doubly periodic discrete KP hierarchy are also constructed. (C) 2005 American Institute of Physics.


Time-dependent recursion operators and symmetries
Gurses, M; Karasu, Atalay; Turhan, R (Informa UK Limited, 2002-05-01)
The recursion operators and symmetries of nonautonomous, (1 + 1) dimensional integrable evolution equations are considered. It has been previously observed hat he symmetries of he integrable evolution equations obtained through heir recursion operators do not satisfy the symmetry equations. There have been several attempts to resolve his problem. It is shown that in the case of time-dependent evolution equations or time-dependent recursion operators associativity is lost. Due to this fact such recursion ope...
Monopole equations on 8-manifolds with spin(7) holonomy
Bilge, AH; Dereli, T; Kocak, S (Springer Science and Business Media LLC, 1999-05-01)
We construct a consistent set of monopole equations on eight-manifolds with Spin(7) holonomy. These equations are elliptic and admit non-trivial solutions including all the 4-dimensional Seiberg-Witten solutions as a special case.
The path integral quantization and the construction of the S-matrix operator in the Abelian and non-Abelian Chern-Simons theories
Fainberg, VY; Pak, Namık Kemal; Shikakhwa, MS (IOP Publishing, 1997-06-07)
The covariant path integral quantization of the theory of the scalar and spinor fields interacting through the Abelian and non-Abelian Chern-Simons gauge fields in 2 + 1 dimensions is carried out using the De Witt-Fadeev-Popov method. The mathematical ill-definiteness of the path integral of theories with pure Chern-Simons' fields is remedied by the introduction of the Maxwell or Maxwell-type (in the non-Abelian case) terms, which make the resulting theories super-renormalizable and guarantees their gauge-i...
Gardner's deformations of the Boussinesq equations
Karasu, Atalay (IOP Publishing, 2006-09-15)
Using the algebraic method of Gardner's deformations for completely integrable systems, we construct recurrence relations for densities of the Hamiltonians for the Boussinesq and the Kaup-Boussinesq equations. By extending the Magri schemes for these equations, we obtain new integrable systems adjoint with respect to the initial ones and describe their Hamiltonian structures and symmetry properties.
Integrable boundary value problems for elliptic type Toda lattice in a disk
Guerses, Metin; Habibullin, Ismagil; Zheltukhın, Kostyantyn (AIP Publishing, 2007-10-01)
The concept of integrable boundary value problems for soliton equations on R and R+ is extended to regions enclosed by smooth curves. Classes of integrable boundary conditions in a disk for the Toda lattice and its reductions are found. (C) 2007 American Institute of Physics.
Citation Formats
A. U. Ö. Kişisel, “On quadratic Poisson brackets,” JOURNAL OF MATHEMATICAL PHYSICS, pp. 0–0, 2005, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/36391.