Integrability of Dirac reduced bi-Hamiltonian equations

@article{Sole2014IntegrabilityOD,
  title={Integrability of Dirac reduced bi-Hamiltonian equations},
  author={Alberto De Sole and Victor G. Kac and Daniele Valeri},
  journal={arXiv: Mathematical Physics},
  year={2014},
  pages={13-32}
}
First, we give a brief review of the theory of the Lenard-Magri scheme for a non-local bi-Poisson structure and of the theory of Dirac reduction. These theories are used in the remainder of the paper to prove integrability of three hierarchies of bi-Hamiltonian PDE’s, obtained by Dirac reduction from some generalized Drinfeld-Sokolov hierarchies. 
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References

SHOWING 1-9 OF 9 REFERENCES

Non-local Poisson structures and applications to the theory of integrable systems

We develop a rigorous theory of non-local Poisson structures, built on the notion of a non-local Poisson vertex algebra. As an application, we find conditions that guarantee applicability of the

Dirac Reduction for Poisson Vertex Algebras

We construct an analogue of Dirac’s reduction for an arbitrary local or non-local Poisson bracket in the general setup of non-local Poisson vertex algebras. This leads to Dirac’s reduction of an

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

Some algebraic properties of differential operators

First, we study the subskewfield of rational pseudodifferential operators over a differential field K generated in the skewfield K((∂−1)) of pseudodifferential operators over K by the subalgebra K[∂]

Poisson vertex algebras in the theory of Hamiltonian equations

Abstract.We lay down the foundations of the theory of Poisson vertex algebras aimed at its applications to integrability of Hamiltonian partial differential equations. Such an equation is called

Classical $${\mathcal{W}}$$W -Algebras and Generalized Drinfeld-Sokolov Bi-Hamiltonian Systems Within the Theory of Poisson Vertex Algebras

We describe of the generalized Drinfeld-Sokolov Hamiltonian reduction for the construction of classical $${\mathcal{W}}$$W -algebras within the framework of Poisson vertex algebras. In this context,

Singular Degree of a Rational Matrix Pseudodifferential Operator

In our previous work we studied minimal fractional decompositions of a rational matrix pseudodifferential operator: H=A/B, where A and B are matrix differential operators, and B is non-degenerate of

Rational matrix pseudodifferential operators

The skewfield $$\mathcal{K }(\partial )$$K(∂) of rational pseudodifferential operators over a differential field $$\mathcal{K }$$K is the skewfield of fractions of the algebra of differential