$R$-matrices and Hamiltonian Structures for Certain Lax Equations

  title={\$R\$-matrices and Hamiltonian Structures for Certain Lax Equations},
  author={Chao-Zhong Wu},
  journal={arXiv: Exactly Solvable and Integrable Systems},
  • Chao-Zhong Wu
  • Published 23 December 2010
  • Mathematics
  • arXiv: Exactly Solvable and Integrable Systems
In this paper a list of $R$-matrices on a certain coupled Lie algebra is obtained. With one of these $R$-matrices, we construct infinitely many bi-Hamiltonian structures for each of the two-component BKP and the Toda lattice hierarchies. We also show that, when such two hierarchies are reduced to their subhierarchies, these bi-Hamiltonian structures are reduced correspondingly. 

A Class of Infinite-dimensional Frobenius Manifolds and Their Submanifolds

We construct a class of infinite-dimensional Frobenius manifolds on the space of pairs of certain even functions meromorphic inside or outside the unit circle. Via a bi-Hamiltonian recursion

Infinite-dimensional Frobenius Manifolds Underlying the Toda Lattice Hierarchy

Bilinear Equation and Additional Symmetries for an Extension of the Kadomtsev–Petviashvili Hierarchy

An extension of the Kadomtsev-Petviashvili (KP) hierarchy was considered in [J. Geom. Phys. 106 (2016), 327--341], which possesses a class of bi-Hamiltonian structures. In this paper, we represent

Frobenius manifolds and Frobenius algebra-valued integrable systems

The notion of integrability will often extend from systems with scalar-valued fields to systems with algebra-valued fields. In such extensions the properties of, and structures on, the algebra play a



Classical r-Matrices and Compatible Poisson Structures for Lax Equations on Poisson Algebras

Abstract:Given a classical r-matrix on a Poisson algebra, we show how to construct a natural family of compatible Poisson structures for the Hamiltonian formulation of Lax equations. Examples for

Nonlinear Poisson structures andr-matrices

We introduce quadratic Poisson structures on Lie groups associated with a class of solutions of the modified Yang-Baxter equation and apply them to the Hamiltonian description of Lax systems. The

The Hamiltonian Structures of the Two-Dimensional Toda Lattice and R-Matrices

We construct the tri-Hamiltonian structure of the two-dimensional Toda hierarchy using the R-matrix theory.

Classical double, R-operators, and negative flows of integrable hierarchies

Using the classical double G of a Lie algebra gequipped with the classical R-operator, we define two sets of functions commuting with respect to the initial Lie-Poisson bracket on g* and its

Lie algebras and equations of Korteweg-de Vries type

The survey contains a description of the connection between the infinite-dimensional Lie algebras of Kats-Moody and systems of differential equations generalizing the Korteweg-de Vries and

The Extended Toda Hierarchy

We present the Lax pair formalism for certain extension of the continuous limit of the classical Toda lattice hierarchy, provide a well defined notion of tau function for its solutions, and give an

The extended bigraded Toda hierarchy

We generalize the Toda lattice hierarchy by considering N + M dependent variables. We construct roots and logarithms of the Lax operator which are uniquely defined operators with coefficients that

Infinite-dimensional Frobenius manifolds for 2 + 1 integrable systems

We introduce a structure of an infinite-dimensional Frobenius manifold on a subspace in the space of pairs of functions analytic inside/outside the unit circle with simple poles at 0/∞, respectively.