Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Variations on a theme of q-oscillator   We present several ideas in direction of physical interpretation of $q$- and
$f$-oscillators as a nonlinear oscillators. First we show that an arbitrary one
dimensional integrable system in action-angle variables can be naturally
represented as a classical and quantum $f$-oscillator. As an example, the
semi-relativistic oscillator as a descriptive of the Landau levels for
relativistic electron in magnetic field is solved as an $f$-oscillator. By
using dispersion relation for $q$-oscillator we solve the linear
q-Schr\"odinger equation and corresponding nonlinear complex q-Burgers
equation. The same dispersion allows us to construct integrable q-NLS model as
a deformation of cubic NLS in terms of recursion operator of NLS hierarchy.
Peculiar property of the model is to be completely integrable at any order of
expansion in deformation parameter around $q=1$. As another variation on the
theme, we consider hydrodynamic flow in bounded domain. For the flow bounded by
two concentric circles we formulate the two circle theorem and construct
solution as the q-periodic flow by non-symmetric $q$-calculus. Then we
generalize this theorem to the flow in the wedge domain bounded by two arcs.
This two circular-wedge theorem determines images of the flow by extension of
$q$-calculus to two bases: the real one, corresponding to circular arcs and the
complex one, with $q$ as a primitive root of unity. As an application, the
vortex motion in annular domain as a nonlinear oscillator in the form of
classical and quantum f-oscillator is studied. Extending idea of q-oscillator
to two bases with the golden ratio, we describe Fibonacci numbers as a special
type of $q$-numbers with matrix Binet formula. We derive the corresponding
golden quantum oscillator, nonlinear coherent states and Fock-Bargman
representation.

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The categories of this paper are: astro-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: cmp-lg
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The categories of this paper are: chao-dyn
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The categories of this paper are: nucl-th
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The categories of this paper are: nucl-ex
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The categories of this paper are: physics
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The categories of this paper are: patt-sol
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The categories of this paper are: cond-mat
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The categories of this paper are: mtrl-th
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The categories of this paper are: solv-int
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The categories of this paper are: eess
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The categories of this paper are: q-fin
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The categories of this paper are: econ
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The categories of this paper are: gr-qc
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The categories of this paper are: ao-sci