# Poisson vertex algebras in the theory of Hamiltonian equations - Mathematical Physics

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Abstract: We lay down the foundations of the theory of Poisson vertex algebras aimed atits applications to integrability of Hamiltonian partial differentialequations. Such an equation is called integrable if it can be included in aninfinite hierarchy of compatible Hamiltonian equations, which admit an infinitesequence of linearly independent integrals of motion in involution. Theconstruction of a hierarchy and its integrals of motion is achieved by makinguse of the so called Lenard scheme. We find simple conditions which guaranteethat the scheme produces an infinite sequence of closed 1-forms \omega j, j inZ +, of the variational complex \Omega. If these forms are exact, i.e. \omega jare variational derivatives of some local functionals \int h j, then the latterare integrals of motion in involution of the hierarchy formed by thecorresponding Hamiltonian vector fields. We show that the complex \Omega isexact, provided that the algebra of functions V is -normal-; in particular, forarbitrary V, any closed form in \Omega becomes exact if we add to V a finitenumber of antiderivatives. We demonstrate on the examples of KdV, HD and CNWhierarchies how the Lenard scheme works. We also discover a new integrablehierarchy, which we call the CNW hierarchy of HD type. Developing the ideas ofDorfman, we extend the Lenard scheme to arbitrary Dirac structures, anddemonstrate its applicability on the examples of the NLS, pKdV and KNhierarchies.

Author: ** Aliaa Barakat, Alberto De Sole, Victor G. Kac**

Source: https://arxiv.org/