1,668 research outputs found
Non-local Hamiltonian structures and applications to the theory of integrable systems I
We develop a rigorous theory of non-local Hamiltonian structures, built on
the notion of a non-local Poisson vertex algebra. As an application, we find
conditions that guarantee applicability of the Lenard-Magri scheme of
integrability to a pair of compatible non-local Hamiltonian structures.Comment: 55 page
Non-local Poisson structures and applications to the theory of integrable systems II
We develop further the Lenard-Magri scheme of integrability for a pair of
compatible non-local Poisson structures, which we discussed in Part I. We apply
this scheme to several such pairs, proving thereby integrability of various
evolution equations, as well as hyperbolic equations. Some of these equations
may be new.Comment: 55 page
Poisson vertex algebras in the theory of Hamiltonian equations
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
Z_+, of the variational complex \Omega. If these forms are exact, i.e. \omega_j
are variational derivatives of some local functionals \int h_j, then the latter
are integrals of motion in involution of the hierarchy formed by the
corresponding Hamiltonian vector fields. We show that the complex \Omega is
exact, provided that the algebra of functions V is "normal"; in particular, for
arbitrary V, any closed form in \Omega becomes exact if we add to V a finite
number of antiderivatives. We demonstrate on the examples of 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.Comment: 95 page
A new approach to the Lenard-Magri scheme of integrability
We develop a new approach to the Lenard-Magri scheme of integrability of
bi-Hamiltonian PDE's, when one of the Poisson structures is a strongly
skew-adjoint differential operator.Comment: 20 page
Adler-Gelfand-Dickey approach to classical W-algebras within the theory of Poisson vertex algebras
We put the Adler-Gelfand-Dickey approach to classical W-algebras in the
framework of Poisson vertex algebras. We show how to recover the bi-Poisson
structure of the KP hierarchy, together with its generalizations and reduction
to the N-th KdV hierarchy, using the formal distribution calculus and the
lambda-bracket formalism. We apply the Lenard-Magri scheme to prove
integrability of the corresponding hierarchies. We also give a simple proof of
a theorem of Kupershmidt and Wilson in this framework. Based on this approach,
we generalize all these results to the matrix case. In particular, we find
(non-local) bi-Poisson structures of the matrix KP and the matrix N-th KdV
hierarchies, and we prove integrability of the N-th matrix KdV hierarchy.Comment: 47 pages. In version 2 we fixed the proof of Corollary 4.15 (which is
now Theorem 4.14), and we added some reference
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 arbitrary non-local Poisson
structure. We apply this construction to an example of a generalized
Drinfeld-Sokolov hierarchy.Comment: 31 pages. Corrected some typos and added missing equations in Section
Classical W-algebras for gl_N and associated integrable Hamiltonian hierarchies
We apply the new method for constructing integrable Hamiltonian hierarchies
of Lax type equations developed in our previous paper, to show that all
W-algebras W(gl_N,f) carry such a hierarchy. As an application, we show that
all vector constrained KP hierarchies and their matrix generalizations are
obtained from these hierarchies by Dirac reduction, which provides the former
with a bi-Poisson structure.Comment: 48 pages. Minor revisions and a correction to formulas (7.25) and
(7.48
- …