1,606 research outputs found

### On the variational noncommutative Poisson geometry

We outline the notions and concepts of the calculus of variational
multivectors within the Poisson formalism over the spaces of infinite jets of
mappings from commutative (non)graded smooth manifolds to the factors of
noncommutative associative algebras over the equivalence under cyclic
permutations of the letters in the associative words. We state the basic
properties of the variational Schouten bracket and derive an interesting
criterion for (non)commutative differential operators to be Hamiltonian (and
thus determine the (non)commutative Poisson structures). We place the
noncommutative jet-bundle construction at hand in the context of the quantum
string theory.Comment: Proc. Int. workshop SQS'11 `Supersymmetry and Quantum Symmetries'
(July 18-23, 2011; JINR Dubna, Russia), 4 page

### An integrable shallow water equation with peaked solitons

We derive a new completely integrable dispersive shallow water equation that
is biHamiltonian and thus possesses an infinite number of conservation laws in
involution. The equation is obtained by using an asymptotic expansion directly
in the Hamiltonian for Euler's equations in the shallow water regime. The
soliton solution for this equation has a limiting form that has a discontinuity
in the first derivative at its peak.Comment: LaTeX file. Figure available from authors upon reques

### Non-classical symmetries and the singular manifold method: A further two examples

This paper discusses two equations with the conditional Painleve property.
The usefulness of the singular manifold method as a tool for determining the
non-classical symmetries that reduce the equations to ordinary differential
equations with the Painleve property is confirmed once moreComment: 9 pages (latex), to appear in Journal of Physics

### On Non-Commutative Integrable Burgers Equations

We construct the recursion operators for the non-commutative Burgers
equations using their Lax operators. We investigate the existence of any
integrable mixed version of left- and right-handed Burgers equations on higher
symmetry grounds.Comment: 8 page

### Rational Approximate Symmetries of KdV Equation

We construct one-parameter deformation of the Dorfman Hamiltonian operator
for the Riemann hierarchy using the quasi-Miura transformation from topological
field theory. In this way, one can get the approximately rational symmetries of
KdV equation and then investigate its bi-Hamiltonian structure.Comment: 14 pages, no figure

### Symmetries of Differential Equations via Cartan's Method of Equivalence

We formulate a method of computing invariant 1-forms and structure equations
of symmetry pseudo-groups of differential equations based on Cartan's method of
equivalence and the moving coframe method introduced by Fels and Olver. Our
apparoach does not require a preliminary computation of infinitesimal defining
systems, their analysis and integration, and uses differentiation and linear
algebra operations only. Examples of its applications are given.Comment: 15 pages, LaTeX 2.0

### On the geometry of lambda-symmetries, and PDEs reduction

We give a geometrical characterization of $\lambda$-prolongations of vector
fields, and hence of $\lambda$-symmetries of ODEs. This allows an extension to
the case of PDEs and systems of PDEs; in this context the central object is a
horizontal one-form $\mu$, and we speak of $\mu$-prolongations of vector fields
and $\mu$-symmetries of PDEs. We show that these are as good as standard
symmetries in providing symmetry reduction of PDEs and systems, and explicit
invariant solutions

### The Hunter-Saxton equation: remarkable structures of symmetries and conserved densities

In this paper, we present extraordinary algebraic and geometrical structures
for the Hunter-Saxton equation: infinitely many commuting and non-commuting
$x,t$-independent higher order symmetries and conserved densities. Using a
recursive relation, we explicitly generate infinitely many higher order
conserved densities dependent on arbitrary parameters. We find three Nijenhuis
recursion operators resulting from Hamiltonian pairs, of which two are new.
They generate three hierarchies of commuting local symmetries. Finally, we give
a local recursion operator depending on an arbitrary parameter.
As a by-product, we classify all anti-symmetric operators of a definite form
that are compatible with the Hamiltonian operator $D_x^{-1}$

### Jet Bundles in Quantum Field Theory: The BRST-BV method

The geometric interpretation of the Batalin-Vilkovisky antibracket as the
Schouten bracket of functional multivectors is examined in detail. The
identification is achieved by the process of repeated contraction of even
functional multivectors with fermionic functional 1-forms. The classical master
equation may then be considered as a generalisation of the Jacobi identity for
Poisson brackets, and the cohomology of a nilpotent even functional multivector
is identified with the BRST cohomology. As an example, the BRST-BV formulation
of gauge fixing in theories with gauge symmetries is reformulated in the jet
bundle formalism. (Hopefully this version will be TeXable)Comment: 26 page

### The Moyal bracket and the dispersionless limit of the KP hierarchy

A new Lax equation is introduced for the KP hierarchy which avoids the use of
pseudo-differential operators, as used in the Sato approach. This Lax equation
is closer to that used in the study of the dispersionless KP hierarchy, and is
obtained by replacing the Poisson bracket with the Moyal bracket. The
dispersionless limit, underwhich the Moyal bracket collapses to the Poisson
bracket, is particularly simple.Comment: 9 pages, LaTe

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