433,620 research outputs found
Connection problems for quantum affine KZ equations and integrable lattice models
Cherednik attached to an affine Hecke algebra module a compatible system of
difference equations, called quantum affine Knizhnik-Zamolodchikov (KZ)
equations. In case of a principal series module we construct a basis of power
series solutions of the quantum affine KZ equations. Relating the bases for
different asymptotic sectors gives rise to a Weyl group cocycle, which we
compute explicitly in terms of theta functions. For the spin representation of
the affine Hecke algebra of type C the quantum affine KZ equations become the
boundary qKZ equations associated to the Heisenberg spin-1/2 XXZ chain. We show
that in this special case the results lead to an explicit 4-parameter family of
elliptic solutions of the dynamical reflection equation associated to Baxter's
8-vertex face dynamical R-matrix. We use these solutions to define an explicit
9-parameter elliptic family of boundary quantum Knizhnik-Zamolodchikov-Bernard
(KZB) equations.Comment: 47 pages. v2: small corrections; v.3: small corrections in the proof
of Thm. 3.1
Linearizability and fake Lax pair for a consistent around the cube nonlinear non-autonomous quad-graph equation
We discuss the linearization of a non-autonomous nonlinear partial difference
equation belonging to the Boll classification of quad-graph equations
consistent around the cube. We show that its Lax pair is fake. We present its
generalized symmetries which turn out to be non-autonomous and depending on an
arbitrary function of the dependent variables defined in two lattice points.
These generalized symmetries are differential difference equations which, in
some case, admit peculiar B\"acklund transformations.Comment: arXiv admin note: text overlap with arXiv:1311.2406 by other author
Non-linear rheology of active particle suspensions: Insights from an analytical approach
We consider active suspensions in the isotropic phase subjected to a shear
flow. Using a set of extended hydrodynamic equations we derive a variety of
{\em analytical} expressions for rheological quantities such as shear viscosity
and normal stress differences. In agreement to full-blown numerical
calculations and experiments we find a shear thickening or -thinning behaviour
depending on whether the particles are contractile or extensile. Moreover, our
analytical approach predicts that the normal stress differences can change
their sign in contrast to passive suspensions.Comment: 11 pages, 10 figures, appear in PR
Control of unstable macroscopic oscillations in the dynamics of three coupled Bose condensates
We study the dynamical stability of the macroscopic quantum oscillations
characterizing a system of three coupled Bose-Einstein condensates arranged
into an open-chain geometry. The boson interaction, the hopping amplitude and
the central-well relative depth are regarded as adjustable parameters. After
deriving the stability diagrams of the system, we identify three mechanisms to
realize the transition from an unstable to stable behavior and analyze specific
configurations that, by suitably tuning the model parameters, give rise to
macroscopic effects which are expected to be accessible to experimental
observation. Also, we pinpoint a system regime that realizes a
Josephson-junction-like effect. In this regime the system configuration do not
depend on the model interaction parameters, and the population oscillation
amplitude is related to the condensate-phase difference. This fact makes
possible estimating the latter quantity, since the measure of the oscillating
amplitudes is experimentally accessible.Comment: 25 pages, 12 figure
On the classification of discrete Hirota-type equations in 3D
In the series of recent publications we have proposed a novel approach to the
classification of integrable differential/difference equations in 3D based on
the requirement that hydrodynamic reductions of the corresponding
dispersionless limits are `inherited' by the dispersive equations. In this
paper we extend this to the fully discrete case. Our only constraint is that
the initial ansatz possesses a non-degenerate dispersionless limit (this is the
case for all known Hirota-type equations). Based on the method of deformations
of hydrodynamic reductions, we classify discrete 3D integrable Hirota-type
equations within various particularly interesting subclasses. Our method can be
viewed as an alternative to the conventional multi-dimensional consistency
approach.Comment: 29 page
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