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. 
View PDF on arXiv
6 Citations

6 Citations

Introduction to vertex algebras, Poisson vertex algebras, and integrable Hamiltonian PDE

These lectures were given in Session 1: “Vertex algebras, W-algebras, and applications” of INdAM Intensive research period “Perspectives in Lie Theory” at the Centro di Ricerca Matematica Ennio De…

Logarithmic vertex algebras and non-local Poisson vertex algebras

. Logarithmic vertex algebras were introduced in our previous paper, motivated by logarithmic conformal field theory [BV]. Non-local Poisson vertex algebras were introduced by De Sole and Kac,…

Erratum to: Classical W-Algebras and Generalized Drinfeld–Sokolov Hierarchies for Minimal and Short Nilpotents (Commun. Math. Phys., (2014), 331, (623-676), 10.1007/s00220-014-2049-2)

There is an error in Sect. 6.2 of the original article in the computation of equations of the generalized Drinfeld–Sokolov hierarchy associated to the minimal nilpotent element f of the Lie algebra g…

Classical and Quantum $${\mathcal {W}}$$-Algebras and Applications to Hamiltonian Equations

  • A. Sole
  • Mathematics
    Springer INdAM Series
  • 2019
We start by giving an overview of the four fundamental physical theories, namely classical mechanics, quantum mechanics, classical field theory and quantum field theory, and the corresponding…

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…

Classical $${\mathcal{W}}$$W-Algebras and Generalized Drinfeld–Sokolov Hierarchies for Minimal and Short Nilpotents

We derive explicit formulas for λ-brackets of the affine classical $${\mathcal{W}}$$W -algebras attached to the minimal and short nilpotent elements of any simple Lie algebra $${\mathfrak{g}}$$g .…

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…

Jordan algebras and generalized Korteweg-de Vries equations