23 research outputs found
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 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
A geometric study of the dispersionless Boussinesq type equation
We discuss the dispersionless Boussinesq type equation, which is equivalent
to the Benney-Lax equation, being a system of equations of hydrodynamical type.
This equation was discussed in
. 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
(cosymmetries). Highly interesting are the appearances of operators that send
conservation laws and symmetries to each other but are neither Hamiltonian, nor
symplectic. These operators give rise to a noncommutative infinite-dimensional
algebra of recursion operators
Geometry of jet spaces and integrable systems
An overview of some recent results on the geometry of partial differential
equations in application to integrable systems is given. Lagrangian and
Hamiltonian formalism both in the free case (on the space of infinite jets) and
with constraints (on a PDE) are discussed. Analogs of tangent and cotangent
bundles to a differential equation are introduced and the variational Schouten
bracket is defined. General theoretical constructions are illustrated by a
series of examples.Comment: 54 pages; v2-v6 : minor correction