Mathieu's Equation and Its Generalizations: Overview of Stability Charts and Their Features

Kovacic, Ivana; Rand, Richard H.; Sah, Si Mohamed

doi:10.1115/1.4039144

Cited by 195 publications

(149 citation statements)

References 39 publications

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“…It is known that, in the presence of the axion dark matter, the propagation of photons is governed by the Mathieu equation [10]. The properties of the Mathieu equation are well studied in mathematics [11][12][13]. It is known that the system becomes unstable for specific parameter regions.…”

Section: Introductionmentioning

confidence: 99%

Stability of axion dark matter-photon conversion

2020

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It is known that a coherently oscillating axion field is a candidate of the dark matter. In the presence of the oscillating axion, the photon can be resonantly produced through the parametric amplification. In the universe, there also exist cosmological magnetic fields which are coherent electromagnetic fields. In the presence of magnetic fields, an axion can be converted into a photon, and vice versa. Thus, it is interesting to investigate what happens for the axion-photon system in the presence of both the axion dark matter and the magnetic fields. This system can be regarded as a coupled system of the axion and the photon whose equations contain the Mathieu type terms.We find that the instability condition is changed in the presence of magnetic fields in contrast to the conventional Mathieu equation. The positions of bifurcation points between stable and unstable are shifted and new instability bands appear. This is because the resonantly amplified axion can be converted to photon, and vice versa.

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Section: Introductionmentioning

confidence: 99%

Stability of axion dark matter-photon conversion

2020

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“…It should be noted that the multiple timescale technique (see Ref. [29]) yields identical answers. The numerically obtained picture of solutions of the Mathieu equation are shown in Fig.…”

Section: Properties Of Mathieu Equationmentioning

confidence: 93%

On the properties of a class of higher-order Mathieu equations originating from a parametric quantum oscillator

Biswas

Bhattacharjee

2019

Nonlinear Dyn

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Evolution in time of an arbitrary initial state for a parametrically driven quantum oscillator is an interesting problem since there exist regions in parameter space (defined by the amplitude and frequency of the driving) where the moments of the probability distribution can diverge in time. While the first moment satisfies a Mathieu equation, the higher-order moments follow Mathieu like equations of order greater than two. It is not very often that a physical problem gives rise to higher-order Mathieu equations. Hence, we give a detailed study of the different stability zones associated with the parametric quantum oscillator, using perturbative techniques traditionally associated with the Mathieu equation. We verify our results by numerical analysis, thus demonstrating that for the higher-order Mathieu equations, the traditional perturbation theory methods give a consistent account of the stability zones.

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“…Our goal is to obtain approximate solutions to the differential equations (B.18) and (B.19). We will do so by following the derivation in Ref [52] with some modifications.…”

Section: Appendix E1 Approximate Two-scale Solutionmentioning

confidence: 99%

Time-evolution of nonlinear optomechanical systems: interplay of mechanical squeezing and non-Gaussianity

Qvarfort

Serafini

Xuereb

et al. 2020

J. Phys. A: Math. Theor.

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We analytically solve the time evolution of a nonlinear optomechanical Hamiltonian with arbitrary time-dependent mechanical squeezing and optomechanical coupling. The solution is based on techniques which provide a new set of analytical tools to study the unitary evolution of optomechanical systems in full generality. To demonstrate the applicability of our method, we compute the degree of non-Gaussianity of the time-evolved state of the system by means of a measure based on the relative entropy of the state. We find that the addition of a constant mechanical squeezing term generally decreases the overall non-Gaussian character of the state. For squeezing modulated at resonance, we recover the Mathieu equations, which we solve pertubatively. We find that the non-Gaussianity increases with both time and the squeezing parameter in this specific regime. ‡ Current address: § We note here that non-Gaussianity for the case D 1 = D 2 = 0 has been already studied [9].

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Mathieu's Equation and Its Generalizations: Overview of Stability Charts and Their Features

Cited by 195 publications

References 39 publications

Stability of axion dark matter-photon conversion

Stability of axion dark matter-photon conversion

On the properties of a class of higher-order Mathieu equations originating from a parametric quantum oscillator

Time-evolution of nonlinear optomechanical systems: interplay of mechanical squeezing and non-Gaussianity

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