277 research outputs found
Degenerate Frobenius manifolds and the bi-Hamiltonian structure of rational Lax equations
The bi-Hamiltonian structure of certain multi-component integrable systems,
generalizations of the dispersionless Toda hierarchy, is studies for systems
derived from a rational Lax function. One consequence of having a rational
rather than a polynomial Lax function is that the corresponding bi-Hamiltonian
structures are degenerate, i.e. the metric which defines the Hamiltonian
structure has vanishing determinant. Frobenius manifolds provide a natural
setting in which to study the bi-Hamiltonian structure of certain classes of
hydrodynamic systems. Some ideas on how this structure may be extanded to
include degenerate bi-Hamiltonian structures, such as those given in the first
part of the paper, are given.Comment: 28 pages, LaTe
Quadratic Poisson brackets compatible with an algebra structure
Quadratic Poisson brackets on a vector space equipped with a bilinear
multiplication are studied. A notion of a bracket compatible with the
multiplication is introduced and an effective criterion of such compatibility
is given. Among compatible brackets, a subclass of coboundary brackets is
described, and such brackets are enumerated in a number of examples.Comment: 6 page
Coadjoint Poisson actions of Poisson-Lie groups
A Poisson-Lie group acting by the coadjoint action on the dual of its Lie
algebra induces on it a non-trivial class of quadratic Poisson structures
extending the linear Poisson bracket on the coadjoint orbits
Multicomponent bi-superHamiltonian KdV systems
It is shown that a new class of classical multicomponent super KdV equations
is bi-superHamiltonian by extending the method for the verification of graded
Jacobi identity. The multicomponent extension of super mKdV equations is
obtained by using the super Miura transformation
Supersymmetric Harry Dym Type Equations
A supersymmetric version is proposed for the well known Harry Dym system. A
general class super Lax operator which leads to consistent equations is
considered.Comment: 4 pages, latex, no figure
Quadratic Poisson brackets and Drinfel'd theory for associative algebras
Quadratic Poisson brackets on associative algebras are studied. Such a
bracket compatible with the multiplication is related to a differentiation in
tensor square of the underlying algebra. Jacobi identity means that this
differentiation satisfies a classical Yang--Baxter equation. Corresponding Lie
groups are canonically equipped with a Poisson Lie structure. A way to quantize
such structures is suggested.Comment: latex, no figures
Quadratic Poisson brackets and Drinfeld theory for associative algebras
The paper is devoted to the Poisson brackets compatible with multiplication
in associative algebras. These brackets are shown to be quadratic and their
relations with the classical Yang--Baxter equation are revealed. The paper also
contains a description of Poisson Lie structures on Lie groups whose Lie
algebras are adjacent to an associative structure.Comment: 16 pages, latex, no figure
On the Miura map between the dispersionless KP and dispersionless modified KP hierarchies
We investigate the Miura map between the dispersionless KP and dispersionless
modified KP hierarchies. We show that the Miura map is canonical with respect
to their bi-Hamiltonian structures. Moreover, inspired by the works of Takasaki
and Takebe, the twistor construction of solution structure for the
dispersionless modified KP hierarchy is given.Comment: 19 pages, Latex, no figure
Compatible Poisson Structures of Toda Type Discrete Hierarchy
An algebra isomorphism between algebras of matrices and difference operators
is used to investigate the discrete integrable hierarchy. We find local and
non-local families of R-matrix solutions to the modified Yang-Baxter equation.
The three R-theoretic Poisson structures and the Suris quadratic bracket are
derived. The resulting family of bi-Poisson structures include a seminal
discrete bi-Poisson structure of Kupershmidt at a special value.Comment: 22 pages, LaTeX, v3: Minor change
On Darboux transformation of the supersymmetric sine-Gordon equation
Darboux transformation is constructed for superfields of the super
sine-Gordon equation and the superfields of the associated linear problem. The
Darboux transformation is shown to be related to the super B\"{a}cklund
transformation and is further used to obtain super soliton solutions.Comment: 9 Page
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