A. M. Vinogradov

Nonlocal trends in the geometry of differential equations: Symmetries, conservation laws, and Bcklund transformations

The theory of coverings over differential equations is exposed which is an adequate language for describing various nonlocal phenomena: nonlocal symmetries and conservation laws, Backlund transformations, prolongation structures, etc. A notion of a nonlocal cobweb is introduced which seems quite useful for dealing with nonlocal objects.

Local symmetries and conservation laws

Starting with Lies classical theory, we carefully explain the basic notions of the higher symmetries theory for arbitrary systems of partial differential equations as well as the necessary calculation procedures. Roughly speaking, we explain what analogs of ‘higher KdV equations’ are for an arbitrary system of partial differential equations and also how one can find and use them. The cohomological nature of conservation laws is shown and some basic results are exposed which allow one to calculate, in principle, all conservation laws for a given system of partial differential equations. In particular, it is shown that ‘symmetry’ and ‘conservation law’ are, in some sense, the ‘dual’ conceptions which coincides in the ‘self-dual’ case, namely, for Euler-Lagrange equations. Training examples are also given.

Nonlocal symmetries and the theory of coverings: An addendum to A. M. vinogradov's ‘local symmetries and conservation laws”

For a systemY of partial differential equations, the notion of a coveringŶ∞→Y∞ is introduced whereY∞ is infinite prolongation ofY. Then nonlocal symmetries ofY are defined as transformations ofŶ∞ which conserve the underlying contact structure. It turns out that generating functions of nonlocal symmetries are integro-differential-type operators.

The local structure of n-Poisson and n-Jacobi manifolds☆

Abstract n -Lie algebra structures on smooth function algebras given by means of multi-differential operators, are studied and their canonical forms are obtained. Necessary and sufficient conditions for the sum and the wedge product of two n -Poisson structures to be again a multi-Poisson are found. It is proven that the canonical n -vector on the dual of an n -Lie algebra g is n -Poisson iff dim g ⩽ n +1. The problem of compatibility of two n -Lie algebra structures is analyzed and the compatibility relations connecting hereditary structures of a given n -Lie algebra are obtained. ( n +1)-dimensional n -Lie algebras are classified and their “elementary particle-like” structure is discovered. Some simple applications to dynamics are discussed.

Cohomological analysis of partial differential equations and secondary calculus

From symmetries of partial differential equations to Secondary Calculus Elements of differential calculus in commutative algebras Geometry of finite-order contact structures and the classical theory of symmetries of partial differential equations Geometry of infinitely prolonged differential equations and higher symmetries

Geometry of nonlinear differential equations

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Symmetries and conservation laws of partial differential equations: Basic notions and results

-spectral sequence and some applications Introduction to Secondary Calculus Bibliography Index.

Extensions of the poisson bracket to differential forms and multi-vector fields

The paper contains a survey of certain contemporary concepts and results connected with the geometric foundations of the theory of nonlinear partial differential equations. At the base of the account is situated the geometry and analysis on jet spaces, finite and infinite.

On the structure of Hamiltonian operators in the field theory

The main notions and results which are necessary for finding higher symmetries and conservation laws for general systems of partial differential equations are given. These constitute the starting point for the subsequent papers of this volume. Some problems are also discussed.

Vacuum Einstein metrics with bidimensional Killing leaves ∗ I-Local aspects.

Abstract The Poisson bracket defined originally on the smooth function algebra of a Poisson manifold is extended to the space of all co-exact forms of this manifold. For the extended bracket analogues of the basic constructions and formulae of the standard hamiltonian formalism are given. The Poisson bracket is extended also, in a dual way, to the space of all co-exact multi-vector fields. Finally, we define the graded Lie algebra homomorphisms connecting these extended brackets and their “differentials” as well. The method used is based on the “unification” techniques introduced by the second author.