The D-Boussinesq equation: Hamiltonian and symplectic structures; Noether and inverse Noether operators

Using new methods of analysis of integrable systems,based on a general geometric approach to nonlinear PDE,we discuss the Dispersionless Boussinesq Equation, which is equivalent to the Benney-Lax equation,being a system of equations of hydrodynamical type. The results include: a description of local and nonlocal Hamiltonian and symplectic structures, hierarchies of symmetries, hierarchies of conservation laws, recursion operators for symmetries and generating functions of conservation laws. Highly interesting are the appearences of the Noether and Inverse Noether operators ,leading to multiple infinite hierarchies of these operators as well as recursion operators

Lagrangian Formalism Over Graded Algebras

This paper provides a description of an algebraic setting for the Lagrangian formalism over graded algebras and is intended as the necessary first step towards the noncommutative C-spectral sequence (variational bicomplex). A noncommutative version of integration procedure, the notion of adjoint operator, Green's formula, the relation between integral and differential forms, conservation laws, Euler operator, Noether's theorem is considered.Comment: 26 pages, AMS-TeX 2.1, to appear in J. Geom. Phys. (resubmitted because of a TeX-error

On the formalism of local variational differential operators

The calculus of local variational differential operators introduced by B. L. Voronov, I. V. Tyutin, and Sh. S. Shakhverdiev is studied in the context of jet super space geometry. In a coordinate-free way, we relate these operators to variational multivectors, for which we introduce and compute the variational Poisson and Schouten brackets by means of a unifying algebraic scheme. We give a geometric definition of the algebra of multilocal functionals and prove that local variational differential operators are well defined on this algebra. To achieve this, we obtain some analytical results on the calculus of variations in smooth vector bundles, which may be of independent interest. In addition, our results give a new a new efficient method for finding Hamiltonian structures of differential equations

The Monge-Ampère equation: Hamiltonian and symplectic structures, recursions, and hierarchies

Using methods of geometry and cohomology developed recently, we study the Monge-Ampère equation, arising as the first nontrivial equation in the associativity equations, or WDVV equations. We describe Hamiltonian and symplectic structures as well as recursion operators for this equation in its orginal form, thus treating the independent variables on an equal footing. Besides this we present nonlocal symmetries and generating functions (cosymmetries)

On the integrability conditions for some structures related to evolution differential equations

Using the result by D.Gessler (Differential Geom. Appl. 7 (1997) 303-324, DIPS-9/98, http://diffiety.ac.ru/preprint/98/09_98abs.htm), we show that any invariant variational bivector (resp., variational 2-form) on an evolution equation with nondegenerate right-hand side is Hamiltonian (resp., symplectic).Comment: 5 pages, AMS-LaTeX. v2: minor correction

On integrability of the Camassa-Holm equation and its invariants. A geometrical approach

Using geometrical approach exposed in arXiv:math/0304245 and arXiv:nlin/0511012, we explore the Camassa-Holm equation (both in its initial scalar form, and in the form of 2x2-system). We describe Hamiltonian and symplectic structures, recursion operators and infinite series of symmetries and conservation laws (local and nonlocal).Comment: 24 page

On complexes related with calculus of variations

We consider the variational complex on infinite jet space and the complex of variational derivatives for Lagrangians of multidimensional paths and study relations between them. The discussion of the variational (bi)complex is set up in terms of a flat connection in the jet bundle. We extend it to supercase using a particular new class of forms. We establish relation of the complex of variational derivatives and the variational complex. Certain calculus of Lagrangians of multidimensional paths is developed. It is shown how covariant Lagrangians of higher order can be used to represent characteristic classes.Comment: LaTeX2e, 36 page