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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-Schrödinger 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.