2,010 research outputs found
Bihamiltonian geometry and separation of variables for Toda lattices
We discuss the bihamiltonian geometry of the Toda lattice (periodic and
open). Using some recent results on the separation of variables for
bihamiltonian manifolds, we show that these systems can be explicitly
integrated via the classical Hamilton-Jacobi method in the so-called
Darboux-Nijenhuis coordinates.Comment: 12 pages, Latex with amsmath and amssymb. Report of talks given at
NEEDS9
WDVV equations
The paper aims to point out a novel geometric characterization of the WDVV equations of 2D topological field theory
Separation of Coupled Systems of Schrodinger Equations by Darboux transformations
Darboux transformations in one independent variable have found numerous
applications in various field of mathematics and physics. In this paper we show
that the extension of these transformations to two dimensions can be used to
decouple systems of Schrodinger equations and provide explicit representation
for three classes of such systems. We show also that there is an elegant
relationship between these transformations and analytic complex matrix
functions.Comment: 14 page
Self-assembly of multi-component fluorescent molecular logic gates in micelles
A recent strategy for developing supramolecular
logic gates in water is based on combinations
of molecules via self-assembly with surfactants, which
eliminates the need for time-consuming synthesis. The
self-assembly of surfactants and lumophores and receptors
can result in interesting properties providing cooperative
e ffects useful for molecular information processing
and other potential applications such as drug delivery
systems. This article highlights some of the recent advancements
in supramolecular information processing
using microheterogeneous media including micelles in
aqueous solution.peer-reviewe
Generalized Lenard Chains, Separation of Variables and Superintegrability
We show that the notion of generalized Lenard chains naturally allows
formulation of the theory of multi-separable and superintegrable systems in the
context of bi-Hamiltonian geometry. We prove that the existence of generalized
Lenard chains generated by a Hamiltonian function defined on a four-dimensional
\omega N manifold guarantees the separation of variables. As an application, we
construct such chains for the H\'enon-Heiles systems and for the classical
Smorodinsky-Winternitz systems. New bi-Hamiltonian structures for the Kepler
potential are found.Comment: 14 pages Revte
Quasi-BiHamiltonian Systems and Separability
Two quasi--biHamiltonian systems with three and four degrees of freedom are
presented. These systems are shown to be separable in terms of Nijenhuis
coordinates. Moreover the most general Pfaffian quasi-biHamiltonian system with
an arbitrary number of degrees of freedom is constructed (in terms of Nijenhuis
coordinates) and its separability is proved.Comment: 10 pages, AMS-LaTeX 1.1, to appear in J. Phys. A: Math. Gen. (May
1997
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Optimisation of chaotically perturbed acoustic limit cycles
In an acoustic cavity with a heat source, the thermal energy of the heat source can be converted into acoustic energy, which may generate a loud oscillation.
If uncontrolled, these acoustic oscillations, also known as thermoacoustic instabilities, can cause mechanical vibrations, fatigue and structural failure.
The objective of manufacturers is to design stable thermoacoustic configurations.
In this paper, we propose a method to optimise a chaotically perturbed limit cycle in the bistable region of a subcritical bifurcation.
In this situation, traditional stability and sensitivity methods, such as eigenvalue and Floquet analysis, break down.
First, we propose covariant Lyapunov analysis and shadowing methods as tools to calculate the stability and sensitivity of chaotically perturbed acoustic limit cycles.
Second, covariant Lyapunov vector analysis is applied to an acoustic system with a heat source. The acoustic velocity at the heat source is chaotically perturbed to qualitatively mimic the effect of the turbulent hydrodynamic field. It is shown that the tangent space of the acoustic attractor is hyperbolic, which has a practical implication: the sensitivities of time--averaged cost functionals exist and can be robustly calculated by a shadowing method.
Third, we calculate the sensitivities of the time--averaged acoustic energy and Rayleigh index to small changes to the heat--source intensity and time delay. By embedding the sensitivities into a gradient--update routine, we suppress an existing chaotic acoustic oscillation by optimal design of the heat source.
The analysis and methods proposed enable the reduction of chaotic oscillations in thermoacoustic systems by optimal passive control. Because the theoretical framework is general, the techniques presented can be used in other unsteady deterministic multi-physics problems with virtually no modification
Lax-Nijenhuis operators for integrable systems
Abstract The relationship between Lax and bihamiltonian formulations of dynamical systems on finite-or infinite-dimensional phase spaces is investigated. The LaxNijenhuis equation is introduced and it is shown that every operator that satisfies that equation satisfies the Lenard recursion relations, while the converse holds for an operator with a simple spectrum. Explicit higher-order Hamiltonian structures for the Toda system, a second Hmailtonian structure of the Euler equation for a rigid body in n-dimensional space, and the quadratic Adler-Gelfand-Dickey structure for the KdV hierarchy are derived using the Lax-Nijenhuis equation. RĂ©sumĂ© OnĂ©tudie la relation entre formalisme de Lax et formalisme bihamiltonien sur des espaces de phases de dimension finie ou infinie. On introduit l'Ă©quation de Lax-Nijenhuis et l'on montre que tout opĂ©rateur qui satisfait cetteĂ©quation satisfait les relations de rĂ©currence de Lenard, tandis que la rĂ©ciproque est valable pour un opĂ©rateurĂ spectre simple. On calcule des structures hamiltoniennes d'ordre supĂ©rieur pour le systĂšme de Toda, une deuxiĂšme structure hamiltonienne pour leĆ equations d'Euler d'un corps solide dans l'espaceĂ n dimensions, et la deuxiĂšme structure de Adler-Gelfand-Dickey pour la hiĂ©rarchie KdV en utilisant l'Ă©quation de Lax-Nijenhuis. PACS numbers: 02.20 +b, 02.40 +
Versal deformations of a Dirac type differential operator
If we are given a smooth differential operator in the variable its normal form, as is well known, is the simplest form
obtainable by means of the \mbox{Diff}(S^1)-group action on the space of all
such operators. A versal deformation of this operator is a normal form for some
parametric infinitesimal family including the operator. Our study is devoted to
analysis of versal deformations of a Dirac type differential operator using the
theory of induced \mbox{Diff}(S^1)-actions endowed with centrally extended
Lie-Poisson brackets. After constructing a general expression for tranversal
deformations of a Dirac type differential operator, we interpret it via the
Lie-algebraic theory of induced \mbox{Diff}(S^1)-actions on a special Poisson
manifold and determine its generic moment mapping. Using a Marsden-Weinstein
reduction with respect to certain Casimir generated distributions, we describe
a wide class of versally deformed Dirac type differential operators depending
on complex parameters
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