# quartic oscillator

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## 1—10 of 32 matching pages

##### 1: 22.19 Physical Applications

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###### §22.19(ii) Classical Dynamics: The Quartic Oscillator

… ►For an initial displacement with $$, bounded oscillations take place near one of the two points of stable equilibrium $x=\pm \sqrt{1/\beta}$. Such oscillations, of period $2K\left(k\right)/\sqrt{\eta}$, with modulus $k=1/\sqrt{2-{\eta}^{-1}}$ are given by: …##### 2: Sidebar 9.SB2: Interference Patterns in Caustics

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►The oscillating intensity of the interference fringes across the caustic is described by the Airy function.

##### 3: 18.39 Applications in the Physical Sciences

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► a) The Harmonic Oscillator
…Then $\omega =2\pi \nu =\sqrt{k/m}$ is the

*circular frequency*of oscillation (with $\nu $ the ordinary frequency), independent of the amplitude of the oscillations. … ► b) The Morse Oscillator …The finite system of functions ${\psi}_{n}$ is orthonormal in ${L}^{2}(\mathbb{R},dx)$, see (18.34.7_3). …The corresponding eigenfunction transform is a generalization of the Kontorovich–Lebedev transform §10.43(v), see Faraut (1982, §IV). …##### 4: 17.17 Physical Applications

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►See Kassel (1995).
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►It involves $q$-generalizations of exponentials and Laguerre polynomials, and has been applied to the problems of the harmonic oscillator and Coulomb potentials.
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##### 5: 6.17 Physical Applications

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►Lebedev (1965) gives an application to electromagnetic theory (radiation of a linear half-wave oscillator), in which sine and cosine integrals are used.

##### 6: 10.73 Physical Applications

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###### §10.73(i) Bessel and Modified Bessel Functions

►Bessel functions first appear in the investigation of a physical problem in Daniel Bernoulli’s analysis of the small oscillations of a uniform heavy flexible chain. … ►In the theory of plates and shells, the oscillations of a circular plate are determined by the differential equation ►
10.73.3
$${\nabla}^{4}W+{\lambda}^{2}\frac{{\partial}^{2}W}{{\partial t}^{2}}=0.$$

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##### 7: Bibliography B

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Avoided crossings of the quartic oscillator.
J. Phys. A 30 (9), pp. 3057–3067.
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Anharmonic oscillator. II. A study of perturbation theory in large order.
Phys. Rev. D 7, pp. 1620–1636.
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##### 8: 8.24 Physical Applications

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►The function $\gamma (a,x)$ appears in: discussions of power-law relaxation times in complex physical systems (Sornette (1998)); logarithmic oscillations in relaxation times for proteins (Metzler et al. (1999)); Gaussian orbitals and exponential (Slater) orbitals in quantum chemistry (Shavitt (1963), Shavitt and Karplus (1965)); population biology and ecological systems (Camacho et al. (2002)).
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##### 9: 7.21 Physical Applications

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►Fried and Conte (1961) mentions the role of $w\left(z\right)$ in the theory of linearized waves or oscillations in a hot plasma; $w\left(z\right)$ is called the

*plasma dispersion function*or*Faddeeva (or Faddeyeva) function*; see Faddeeva and Terent’ev (1954). …##### 10: 32.7 Bäcklund Transformations

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${\text{P}}_{\text{VI}}$ also has quadratic and quartic transformations.
…The quartic transformation
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