nonlinear harmonic oscillator

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1: 14.30 Spherical and Spheroidal Harmonics

§14.30 Spherical and Spheroidal Harmonics

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§14.30(i) Definitions

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§14.30(iii) Sums

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2: 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. …

3: 32.2 Differential Equations

… ►be a nonlinear second-order differential equation in which $F$ is a rational function of $w$ and $dw/dz$, and is locally analytic in $z$, that is, analytic except for isolated singularities in $ℂ$. … ►When $\beta =0$ this is a nonlinear harmonic oscillator. …

4: 22.19 Physical Applications

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§22.19(ii) Classical Dynamics: The Quartic Oscillator

… ►Such oscillations, of period $2K\left(k\right)/\sqrt{\eta }$, with modulus $k=1/\sqrt{2-{\eta }^{-1}}$ are given by: … ►

§22.19(iii) Nonlinear ODEs and PDEs

►Many nonlinear ordinary and partial differential equations have solutions that may be expressed in terms of Jacobian elliptic functions. These include the time dependent, and time independent, nonlinear Schrödinger equations (NLSE) (Drazin and Johnson (1993, Chapter 2), Ablowitz and Clarkson (1991, pp. 42, 99)), the Korteweg–de Vries (KdV) equation (Kruskal (1974), Li and Olver (2000)), the sine-Gordon equation, and others; see Drazin and Johnson (1993, Chapter 2) for an overview. …

5: Bibliography M

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Spherical Harmonics. An Elementary Treatise on Harmonic Functions with Applications. 3rd edition, International Series of Monographs in Pure and Applied Mathematics, Vol. 98, Pergamon Press, Oxford.

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External Links:

MathReview (A. Erdélyi), ZentralBlatt

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Ch.14.

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New ladder operators for a rational extension of the harmonic oscillator and superintegrability of some two-dimensional systems. J. Math. Phys. 54 (10), pp. Paper 102102, 12 pp..

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MathReview Entry

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§18.36(vi).

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Connection between quantum systems involving the fourth Painlevé transcendent and $k$-step rational extensions of the harmonic oscillator related to Hermite exceptional orthogonal polynomial. J. Math. Phys. 57 (5), pp. Paper 052101, 15 pp..

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ISSN 0022-2488, Document, Link, MathReview Entry

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§18.38(iii).

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Hierarchies and logarithmic oscillations in the temporal relaxation patterns of proteins and other complex systems. Proc. Nat. Acad. Sci. U .S. A. 96 (20), pp. 11085–11089.

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FullText

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§8.24(i).

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The Second Painlevé Transcendent: A Nonlinear Airy Function. In Mechanics Today, Vol. 5, pp. 297–313.

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External Links:

MathReview (H. E. Fettis), ZentralBlatt

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§32.11(ii), §32.17, §32.3(ii).

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6: Peter A. Clarkson

… ►His well-known book Solitons, Nonlinear Evolution Equations and Inverse Scattering (with M. …He is also coauthor of the book From Nonlinearity to Coherence: Universal Features of Nonlinear Behaviour in Many-Body Physics (with J. …

7: Bibliography D

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The constrained quantum mechanical harmonic oscillator. Proc. Cambridge Philos. Soc. 62, pp. 277–286.

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Document, MathReview (T. Erber)

Referenced by:

§12.11(i), §12.11(i), §12.17.

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The KP equation with quasiperiodic initial data. Phys. D 123 (1-4), pp. 123–152.

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Nonlinear waves and solitons in physical systems (Los Alamos, NM, 1997)

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ISSN 0167-2789, Document, MathReview (Pavel Krejčí), ZentralBlatt

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§21.9.

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Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations. CWI Monographs, Vol. 2, North-Holland Publishing Co., Amsterdam.

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ISBN 0-444-87634-0, MathReview (A. de Castro Brzezicki), ZentralBlatt

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§3.7(v).

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From Nonlinearity to Coherence: Universal Features of Nonlinear Behaviour in Many-Body Physics. Oxford University Press, Oxford.

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Profile Peter A. Clarkson.

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8: Mark J. Ablowitz

… ►Ablowitz is an applied mathematician who is interested in solutions of nonlinear wave equations. Certain nonlinear equations are special; e. …Some of the relationships between IST and Painlevé equations are discussed in two books: Solitons and the Inverse Scattering Transform and Solitons, Nonlinear Evolution Equations and Inverse Scattering. …

9: Bibliography B

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Avoided crossings of the quartic oscillator. J. Phys. A 30 (9), pp. 3057–3067.

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ISSN 0305-4470, Document, MathReview (Gabriel Alvarez), ZentralBlatt

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§31.17(ii).

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Algebro-geometric Approach to Nonlinear Integrable Problems. Springer Series in Nonlinear Dynamics, Springer-Verlag, Berlin.

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ISBN 3-540-50265-3, ZentralBlatt

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§21.1, 1st item, §21.6(ii), §21.7(i), §21.7(i), §21.9, Ch.21, Profile Alexander I. Bobenko, Profile Alexander A. Its, Notations T.

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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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Document

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§22.19(ii).

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Stokes surfaces of diffraction catastrophes with codimension three. Nonlinearity 3 (2), pp. 281–291.

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ISSN 0951-7715, Document, MathReview (Maria Carmen Romero-Fuster), ZentralBlatt

Referenced by:

§36.2(i), §36.5(ii), §36.5(iii), §36.5(iv).

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Associated Legendre polynomials, ordinary and modified spherical harmonics. Comput. Phys. Comm. 5 (5), pp. 390–394.

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Single-precision Fortran, maximum accuracy 16S

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Document, Code

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2nd item.

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10: Bernard Deconinck

… ►Deconinck is interested in nonlinear waves. He has worked on integrable systems, algorithms for computations with Riemann surfaces, Bose-Einstein condensates, and methods to investigate the stability of solutions of nonlinear wave equations. …