# quartic oscillator

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## 1—10 of 28 matching pages

##### 1: 22.19 Physical Applications
###### §22.19(ii) Classical Dynamics: The QuarticOscillator
For an initial displacement with $\sqrt{1/\beta}\leq|a|<\sqrt{2/\beta}$, 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
The oscillating intensity of the interference fringes across the caustic is described by the Airy function.
##### 3: 17.17 Physical Applications
See Kassel (1995). … It involves $q$-generalizations of exponentials and Laguerre polynomials, and has been applied to the problems of the harmonic oscillator and Coulomb potentials. …
##### 4: 6.17 Physical Applications
Lebedev (1965) gives an application to electromagnetic theory (radiation of a linear half-wave oscillator), in which sine and cosine integrals are used.
##### 5: 10.73 Physical Applications
###### §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 $\nabla^{4}W+\lambda^{2}\frac{{\partial}^{2}W}{{\partial t}^{2}}=0.$
##### 6: Bibliography B
• K. Bay, W. Lay, and A. Akopyan (1997) Avoided crossings of the quartic oscillator. J. Phys. A 30 (9), pp. 3057–3067.
• C. M. Bender and T. T. Wu (1973) Anharmonic oscillator. II. A study of perturbation theory in large order. Phys. Rev. D 7, pp. 1620–1636.
• ##### 7: 8.24 Physical Applications
The function $\gamma\left(a,x\right)$ 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)). …
##### 8: 18.39 Physical Applications
For a harmonic oscillator, the potential energy is given by …
##### 9: 7.21 Physical Applications
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
$\mbox{P}_{\mbox{\scriptsize VI}}$ also has quadratic and quartic transformations. …The quartic transformation …