nonlinear harmonic oscillator

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21: 23.21 Physical Applications

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§23.21(ii) Nonlinear Evolution Equations

►Airault et al. (1977) applies the function $\mathrm{\wp }$ to an integrable classical many-body problem, and relates the solutions to nonlinear partial differential equations. …

22: 18.38 Mathematical Applications

… ►While the Toda equation is an important model of nonlinear systems, the special functions of mathematical physics are usually regarded as solutions to linear equations. … ►

Zonal Spherical Harmonics

►Ultraspherical polynomials are zonal spherical harmonics. …

23: Bibliography C

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On the expansion of a Coulomb potential in spherical harmonics. Proc. Cambridge Philos. Soc. 46, pp. 626–633.

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External Links:

Document, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§34.12.

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Stationary solutions of the one-dimensional nonlinear Schrödinger equation. I. Case of repulsive nonlinearity. Phys. Rev. A 62 (063610), pp. 1–10.

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External Links:

Document

Referenced by:

§22.19(iii).

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The second Painlevé equation, its hierarchy and associated special polynomials. Nonlinearity 16 (3), pp. R1–R26.

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ISSN 0951-7715, Document, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.8(ii).

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Nonclassical Symmetry Reductions and Exact Solutions for Physically Significant Nonlinear Evolution Equations. In Nonlinear and Chaotic Phenomena in Plasmas, Solids and Fluids (Edmonton, AB, 1990), W. Rozmus and J. A. Tuszynski (Eds.), pp. 72–79.

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External Links:

MathReview

Referenced by:

§29.19(ii).

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Painlevé Equations—Nonlinear Special Functions: Computation and Application. In Orthogonal Polynomials and Special Functions, F. Marcellàn and W. van Assche (Eds.), Lecture Notes in Math., Vol. 1883, pp. 331–411.

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External Links:

ISBN 3-540-31062-2, MathReview (Marta Mazzocco), ZentralBlatt

Referenced by:

Ch.32.

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24: 29.19 Physical Applications

… ►Clarkson (1991) solves nonlinear evolution equations. …

25: 9.16 Physical Applications

… ►Within classical physics, they appear prominently in physical optics, electromagnetism, radiative transfer, fluid mechanics, and nonlinear wave propagation. … ►Airy functions play a prominent role in problems defined by nonlinear wave equations. These first appeared in connection with the equation governing the evolution of long shallow water waves of permanent form, generally called solitons, and are predicted by the Korteweg–de Vries (KdV) equation (a third-order nonlinear partial differential equation). …

26: 3.8 Nonlinear Equations

§3.8 Nonlinear Equations

… ►where $z$ is a real or complex variable and the function $f$ is nonlinear. … ► … ►

§3.8(vii) Systems of Nonlinear Equations

►For fixed-point iterations and Newton’s method for solving systems of nonlinear equations, see Gautschi (1997a, Chapter 4, §9) and Ortega and Rheinboldt (1970). …

27: 21.9 Integrable Equations

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21.9.1 $$4u_t=6u{u}_x+u_{xxx},$$

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Symbols:

$u(x,y,t)$: elevation

Permalink:

http://dlmf.nist.gov/21.9.E1

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pMML, png, TeX

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Annotations for §21.9 and Ch.21

►and the nonlinear Schrödinger equations …

28: 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.

29: Bibliography G

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A harmonic mean inequality for the gamma function. SIAM J. Math. Anal. 5 (2), pp. 278–281.

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Document, MathReview (S. Saran), ZentralBlatt

Referenced by:

§5.6(i).

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A code to evaluate prolate and oblate spheroidal harmonics. Comput. Phys. Comm. 108 (2-3), pp. 267–278.

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Notes:

Double-precision Fortran, maximum accuracy 18S

External Links:

Document, Code, ZentralBlatt

Referenced by:

5th item, 1st item.

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Evaluation of toroidal harmonics. Comput. Phys. Comm. 124 (1), pp. 104–122.

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Document, Code, ZentralBlatt

Referenced by:

4th item.

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DTORH3 2.0: A new version of a computer program for the evaluation of toroidal harmonics. Comput. Phys. Comm. 139 (2), pp. 186–191.

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Document, Code, ZentralBlatt

Referenced by:

2nd item.

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Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials. J. Phys. A 47 (1), pp. 015203, 26 pp..

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External Links:

ISSN 1751-8113, Document, Link, MathReview (Brian Simanek)

Referenced by:

§18.36(vi), §18.36(vi).

Encodings:

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30: Donald St. P. Richards

… ►Richards has published numerous papers on special functions of matrix argument, harmonic analysis, multivariate statistical analysis, probability inequalities, and applied probability. He is editor of the book Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications, published by the American Mathematical Society in 1992, and coeditor of Representation Theory and Harmonic Analysis: A Conference in Honor of R. A. Kunze (with T. …