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11: 33.2 Definitions and Basic Properties

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§33.2(i) Coulomb Wave Equation

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12: 33.14 Definitions and Basic Properties

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§33.14(i) Coulomb Wave Equation

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13: 9.16 Physical Applications

… ►The frequent appearances of the Airy functions in both classical and quantum physics is associated with wave equations with turning points, for which asymptotic (WKBJ) solutions are exponential on one side and oscillatory on the other. … ►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). …

14: 19.18 Derivatives and Differential Equations

… ►The next four differential equations apply to the complete case of $R_F$ and $R_G$ in the form $R_{-a}(\frac{1}{2},\frac{1}{2};z_1,z_2)$ (see (19.16.20) and (19.16.23)). … ►Also $W=R_{-a}(\frac{1}{2},\frac{1}{2};t+r,t-r)$, with $r=\sqrt{x^2+y^2}$, satisfies a wave equation: …

15: 28.32 Mathematical Applications

… ►The two-dimensional wave equation … ►Kernels $K$ can be found, for example, by separating solutions of the wave equation in other systems of orthogonal coordinates. …

16: 36.10 Differential Equations

… ►Equation (36.10.17) is the paraxial wave equation.

17: Bibliography

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Perturbation solutions of the ellipsoidal wave equation. Quart. J. Math. Oxford Ser. (2) 7, pp. 161–174.

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External Links:

ISSN 0033-5606, Document, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§29.11.

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A new treatment of the ellipsoidal wave equation. Proc. London Math. Soc. (3) 9, pp. 21–50.

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External Links:

ISSN 0024-6115, Document, MathReview (J. Meixner), ZentralBlatt

Referenced by:

§29.11.

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The Whittaker-Hill equation and the wave equation in paraboloidal co-ordinates. Proc. Roy. Soc. Edinburgh Sect. A 67, pp. 265–276.

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External Links:

MathReview (H. Hochstadt), ZentralBlatt

Referenced by:

§28.31(i), §28.31(ii), §28.31(ii), §28.32(ii).

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18: 21.9 Integrable Equations

… ►Particularly important for the use of Riemann theta functions is the Kadomtsev–Petviashvili (KP) equation, which describes the propagation of two-dimensional, long-wave length surface waves in shallow water (Ablowitz and Segur (1981, Chapter 4)): … …

19: Bibliography V

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A Mathieu equation for ships rolling among waves. I, II. Norske Vid. Selsk. Forh., Trondheim 22 (25–26), pp. 113–123.

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External Links:

MathReview (J. V. Wehausen), ZentralBlatt

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2nd item.

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Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation. Constr. Approx. 20 (1), pp. 39–54.

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External Links:

ISSN 0176-4276, Document, MathReview, ZentralBlatt

Referenced by:

§30.16(i), §30.16(ii), §30.16(ii).

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20: Bibliography L

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Solutions to a generalized spheroidal wave equation: Teukolsky’s equations in general relativity, and the two-center problem in molecular quantum mechanics. J. Math. Phys. 27 (5), pp. 1238–1265.

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External Links:

ISSN 0022-2488, Document, MathReview (Basilis C. Xanthopoulos), ZentralBlatt

Referenced by:

§30.12, §31.17(ii).

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Computations of spheroidal harmonics with complex arguments: A review with an algorithm. Phys. Rev. E 58 (5), pp. 6792–6806.

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Notes:

Mathematica package for eigenvalues and solutions of the spheroidal wave equations

External Links:

Code(SpheroidF), Document

Referenced by:

2nd item, 3rd item.

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Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation. J. Differential Equations 162 (1), pp. 27–63.

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External Links:

ISSN 0022-0396, Document, MathReview (John Albert), ZentralBlatt

Referenced by:

§22.19(iii).

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