348 research outputs found
The Discrete AKNS-D Hierarchy
In this paper, we consider the discrete AKNS-D hierarchy, find the
construction of the hierarchy, prove the bilinear identity and give the
construction of the -functions of this hierarchy.Comment: 11 page
Askey-Wilson Type Functions, With Bound States
The two linearly independent solutions of the three-term recurrence relation
of the associated Askey-Wilson polynomials, found by Ismail and Rahman in [22],
are slightly modified so as to make it transparent that these functions satisfy
a beautiful symmetry property. It essentially means that the geometric and the
spectral parameters are interchangeable in these functions. We call the
resulting functions the Askey-Wilson functions. Then, we show that by adding
bound states (with arbitrary weights) at specific points outside of the
continuous spectrum of some instances of the Askey-Wilson difference operator,
we can generate functions that satisfy a doubly infinite three-term recursion
relation and are also eigenfunctions of -difference operators of arbitrary
orders. Our result provides a discrete analogue of the solutions of the purely
differential version of the bispectral problem that were discovered in the
pioneering work [8] of Duistermaat and Gr\"unbaum.Comment: 42 pages, Section 3 moved to the end, minor correction
KP Trigonometric Solitons and an Adelic Flag Manifold
We show that the trigonometric solitons of the KP hierarchy enjoy a differential-difference bispectral property, which becomes transparent when translated on two suitable spaces of pairs of matrices satisfying certain rank one conditions. The result can be seen as a non-self-dual illustration of Wilson's fundamental idea [Invent. Math. 133 (1998), 1-41] for understanding the (self-dual) bispectral property of the rational solutions of the KP hierarchy. It also gives a bispectral interpretation of a (dynamical) duality between the hyperbolic Calogero-Moser system and the rational Ruijsenaars-Schneider system, which was first observed by Ruijsenaars [Comm. Math. Phys. 115 (1988), 127-165]
QUAGMIRE v1.3: a quasi-geostrophic model for investigating rotating fluids experiments
QUAGMIRE is a quasi-geostrophic numerical model for performing fast, high-resolution simulations of multi-layer rotating annulus laboratory experiments on a desktop personal computer. The model uses a hybrid finite-difference/spectral approach to numerically integrate the coupled nonlinear partial differential equations of motion in cylindrical geometry in each layer. Version 1.3 implements the special case of two fluid layers of equal resting depths. The flow is forced either by a differentially rotating lid, or by relaxation to specified streamfunction or potential vorticity fields, or both. Dissipation is achieved through Ekman layer pumping and suction at the horizontal boundaries, including the internal interface. The effects of weak interfacial tension are included, as well as the linear topographic beta-effect and the quadratic centripetal beta-effect. Stochastic forcing may optionally be activated, to represent approximately the effects of random unresolved features. A leapfrog time stepping scheme is used, with a Robert filter. Flows simulated by the model agree well with those observed in the corresponding laboratory experiments
Matrix biorthogonal polynomials on the unit circle and non-Abelian Ablowitz-Ladik hierarchy
Adler and van Moerbeke \cite{AVM} described a reduction of 2D-Toda hierarchy
called Toeplitz lattice. This hierarchy turns out to be equivalent to the one
originally described by Ablowitz and Ladik \cite{AL} using semidiscrete
zero-curvature equations. In this paper we obtain the original semidiscrete
zero-curvature equations starting directly from the Toeplitz lattice and we
generalize these computations to the matrix case. This generalization lead us
to the semidiscrete zero-curvature equations for the non-abelian (or
multicomponent) version of Ablowitz-Ladik equations \cite{GI}. In this way we
extend the link between biorthogonal polynomials on the unit circle and
Ablowitz-Ladik hierarchy to the matrix case.Comment: 23 pages, accepted on publication on J. Phys. A., electronic link:
http://stacks.iop.org/1751-8121/42/36521
Control of an atom laser using feedback
A generalised method of using feedback to control Bose-Einstein condensates
is introduced. The condensates are modelled by the Gross-Pitaevskii equation,
so only semiclassical fluctations can be suppressed, and back-action from the
measurement is ignored. We show that for any available control, a feedback
scheme can be found to reduce the energy while the appropriate moment is still
dynamic. We demonstrate these schemes by considering a condensate trapped in a
harmonic potential that can be modulated in strength and position. The
formalism of our feedback scheme also allows the inclusion of certain types of
non-linear controls. If the non-linear interaction between the atoms can be
controlled via a Feshbach resonance, we show that the feedback process can
operate with a much higher efficiency.Comment: 6 pages, 7 figure
Measuring the quantum statistics of an atom laser beam
We propose and analyse a scheme for measuring the quadrature statistics of an
atom laser beam using extant optical homodyning and Raman atom laser
techniques. Reversal of the normal Raman atom laser outcoupling scheme is used
to map the quantum statistics of an incoupled beam to an optical probe beam. A
multimode model of the spatial propagation dynamics shows that the Raman
incoupler gives a clear signal of de Broglie wave quadrature squeezing for both
pulsed and continuous inputs. Finally, we show that experimental realisations
of the scheme may be tested with existing methods via measurements of Glauber's
intensity correlation function.Comment: 4 pages, 3 figure
On the duality between the hyperbolic Sutherland and the rational Ruijsenaars-Schneider models
We consider two families of commuting Hamiltonians on the cotangent bundle of
the group GL(n,C), and show that upon an appropriate single symplectic
reduction they descend to the spectral invariants of the hyperbolic Sutherland
and of the rational Ruijsenaars-Schneider Lax matrices, respectively. The
duality symplectomorphism between these two integrable models, that was
constructed by Ruijsenaars using direct methods, can be then interpreted
geometrically simply as a gauge transformation connecting two cross sections of
the orbits of the reduction group.Comment: 16 pages, v2: comments and references added at the end of the tex
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