Poisson vertex algebras in the theory of Hamiltonian equations
By A. Barakat, A. Sole, V. Kac. 2009
150 citations recorded by the source corpus.
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 integrable if it can be included in an infinite hierarchy of compatible Hamiltonian equations, which admit an infinite sequence of linearly independent integrals of motion in involution. The construction of a hierarchy and its integrals of motion is achieved by making use of the so called Lenard scheme. We find simple conditions which guarantee that the scheme produces an infinite sequence of closed 1-forms $$\omega_j, j \in {\mathbb {Z}}_+$$, of the variational complex Ω. If these forms are exact, i.e., ωj are variational derivatives of some local functionals ∫ hj, then the latter are integrals of motion in involution of the hierarchy formed by the corresponding Hamiltonian vector fields. We show that the complex Ω is exact, provided that the algebra of functions is $$\fancyscript {V}$$ is “normal”; in particular, for arbitrary $$\fancyscript {V}$$, any closed form in Ω becomes exact if we add to $$\fancyscript {V}$$ a finite number of antiderivatives. We demonstrate on the examples of the KdV, HD and CNW hierarchies how the Lenard scheme works. We also discover a new integrable hierarchy, which we call the CNW hierarchy of HD type. Developing the ideas of Dorfman, we extend the Lenard scheme to arbitrary Dirac structures, and demonstrate its applicability on the examples of the NLS, pKdV and KN hierarchies.
Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/9b487317be2d5de8b06eb4fd23a1b942318cbf0e
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Geometry and Topology
In Mathematical Physics
In Statistical and Nonlinear Physics
Citations in the corpus
- Geometry of jet spaces and integrable systems (2010, background): “In a recently published paper [6], the authors formulate a much weaker than triviality of the first Poisson cohomology group criterion for feasibility of the Magri scheme.”