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

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. Résumé Onétudie la relation entre formalisme de Lax et formalisme bihamiltonien sur des espaces de phases de dimension finie ou infinie. On introduit l'équation de Lax-Nijenhuis et l'on montre que tout opérateur qui satisfait cetteéquation satisfait les relations de récurrence de Lenard, tandis que la réciproque est valable pour un opérateurà spectre simple. On calcule des structures hamiltoniennes d'ordre supérieur pour le système de Toda, une deuxième structure hamiltonienne pour leś equations d'Euler d'un corps solide dans l'espaceà n dimensions, et la deuxième structure de Adler-Gelfand-Dickey pour la hiérarchie KdV en utilisant l'équation 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 $x\in {\mathbb R}/2\pi {\mathbb Z},$ 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