diffraction catastrophes

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1: 36.2 Catastrophes and Canonical Integrals

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Diffraction Catastrophes

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36.2.10 $${\mathrm{\Psi }}_K(𝐱;k)=\sqrt{k}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\exp \left(\mathrm{i}k{\mathrm{\Phi }}_K(t;𝐱)\right)dt,$$ $k>0$.

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Defines:

${\mathrm{\Psi }}_K(𝐱;k)$: diffraction catastrophe

Symbols:

${\mathrm{\Phi }}_K(t;𝐱)$: cuspoid catastrophe of codimension $K$ , $dx$: differential of $x$ , $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\int$: integral, $k$: variable, $t$: variable and $K$: codimension

Referenced by:

§36.12(i)

Permalink:

http://dlmf.nist.gov/36.2.E10

Encodings:

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See also:

Annotations for §36.2(i), §36.2(i), §36.2 and Ch.36

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36.2.11 $\mathrm{U}=\mathrm{E},\mathrm{H}$; $k>0$.

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Defines:

${\mathrm{\Psi }}^{(\mathrm{E})}(𝐱;k)$: elliptic umbilic canonical integral function, ${\mathrm{\Psi }}^{(\mathrm{H})}(𝐱;k)$: hyperbolic umbilic canonical integral function and ${\mathrm{\Psi }}^{(\mathrm{U})}(𝐱;k)$: umbilic canonical integral function

Symbols:

$dx$: differential of $x$ , $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\int$: integral, : elliptic umbilic catastrophe for $\mathrm{U}=\mathrm{E}\text{ or }\mathrm{K}$ , $k$: variable, $t$: variable and $s$: variable

Referenced by:

§36.2(i)

Permalink:

http://dlmf.nist.gov/36.2.E11

Encodings:

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See also:

Annotations for §36.2(i), §36.2(i), §36.2 and Ch.36

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2: 36.1 Special Notation

§36.1 Special Notation

… ►The main functions covered in this chapter are cuspoid catastrophes ${\mathrm{\Phi }}_K(t;𝐱)$; umbilic catastrophes with codimension three ${\mathrm{\Phi }}^{(\mathrm{E})}(s,t;𝐱)$, ${\mathrm{\Phi }}^{(\mathrm{H})}(s,t;𝐱)$; canonical integrals ${\mathrm{\Psi }}_K\left(𝐱\right)$, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$, ${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$; diffraction catastrophes ${\mathrm{\Psi }}_K(𝐱;k)$, ${\mathrm{\Psi }}^{(\mathrm{E})}(𝐱;k)$, ${\mathrm{\Psi }}^{(\mathrm{H})}(𝐱;k)$ generated by the catastrophes. …

3: 36.6 Scaling Relations

§36.6 Scaling Relations

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Diffraction Catastrophe Scaling

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${\mathrm{\Psi }}_K(𝐱;k)=k^{{\beta }_K}{\mathrm{\Psi }}_K\left(𝐲(k)\right),$

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${\mathrm{\Psi }}^{(\mathrm{U})}(𝐱;k)=k^{{\beta }^{(\mathrm{U})}}{\mathrm{\Psi }}^{(\mathrm{U})}\left({𝐲}^{(\mathrm{U})}(k)\right),$

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4: 36.14 Other Physical Applications

… ►Diffraction catastrophes describe the (linear) wave amplitudes that smooth the geometrical caustic singularities and decorate them with interference patterns. … ►Diffraction catastrophes describe the connection between ray optics and wave optics. … ►Diffraction catastrophes describe the “semiclassical” connections between classical orbits and quantum wavefunctions, for integrable (non-chaotic) systems. …

5: 36.5 Stokes Sets

… ►Stokes sets are surfaces (codimension one) in $𝐱$ space, across which ${\mathrm{\Psi }}_K(𝐱;k)$ or ${\mathrm{\Psi }}^{(\mathrm{U})}(𝐱;k)$ acquires an exponentially-small asymptotic contribution (in $k$), associated with a complex critical point of ${\mathrm{\Phi }}_K$ or . …

6: 36.12 Uniform Approximation of Integrals

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36.12.1 $$I(𝐲,k)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\exp \left(\mathrm{i}kf(u;𝐲)\right)g(u,𝐲)du,$$

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Defines:

$I(𝐲,k)$: oscillatory integral (locally)

Symbols:

$dx$: differential of $x$ , $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\int$: integral, $k$: variable, $g(u,𝐲)$: function, $f(u,𝐲)$: function and $u(t;𝐲)$: mapping

Referenced by:

§36.13

Permalink:

http://dlmf.nist.gov/36.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §36.12(i), §36.12 and Ch.36

… ►This technique can be applied to generate a hierarchy of approximations for the diffraction catastrophes ${\mathrm{\Psi }}_K(𝐱;k)$ in (36.2.10) away from $𝐱=\mathrm{𝟎}$, in terms of canonical integrals ${\mathrm{\Psi }}_J\left(\xi (𝐱;k)\right)$ for . For example, the diffraction catastrophe ${\mathrm{\Psi }}_2(x,y;k)$ defined by (36.2.10), and corresponding to the Pearcey integral (36.2.14), can be approximated by the Airy function ${\mathrm{\Psi }}_1\left(\xi (x,y;k)\right)$ when $k$ is large, provided that $x$ and $y$ are not small. …

7: Bibliography N

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Dislocation lines in the hyperbolic umbilic diffraction catastrophe. Proc. Roy. Soc. Lond. Ser. A 462, pp. 2299–2313.

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External Links:

ISSN 1364-5021, Document, MathReview, ZentralBlatt

Referenced by:

§36.7(iv).

Encodings:

BibTeX

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Dislocation lines in the swallowtail diffraction catastrophe. Proc. Roy. Soc. Lond. Ser. A 463, pp. 343–355.

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External Links:

ISSN 1364-5021, Document, ZentralBlatt

Referenced by:

§36.7(iv).

Encodings:

BibTeX

8: Bibliography B

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

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External Links:

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

Referenced by:

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

Encodings:

BibTeX

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Axial and focal-plane diffraction catastrophe integrals. J. Phys. A 43 (37), pp. 375206, 13.

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External Links:

ISSN 1751-8113, Document, FullText, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

(36.2.28), (36.2.29), §36.2(iv).

Encodings:

BibTeX

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The elliptic umbilic diffraction catastrophe. Phil. Trans. Roy. Soc. Ser. A 291 (1382), pp. 453–484.

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External Links:

FullText

Referenced by:

§36.15(iv), §36.2(i), §36.7(ii), §36.7(iii).

Encodings:

BibTeX

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Phase-space projection identities for diffraction catastrophes. J. Phys. A 13 (1), pp. 149–160.

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External Links:

ISSN 0305-4470, Document, MathReview (J. S. Joel), ZentralBlatt

Referenced by:

§36.9.

Encodings:

BibTeX

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Some Geometric Aspects of Wave Motion: Wavefront Dislocations, Diffraction Catastrophes, Diffractals. In Geometry of the Laplace Operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), Vol. 36, pp. 13–28.

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External Links:

MathReview (I. Stewart), ZentralBlatt

Referenced by:

§36.14(i).

Encodings:

BibTeX

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9: 36.7 Zeros

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Table 36.7.1: Zeros of cusp diffraction catastrophe to 5D. …

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Zeros $\left\{\genfrac{}{}{0.0pt}{}{x}{y}\right\}$ inside, and zeros $\left[\genfrac{}{}{0.0pt}{}{x}{y}\right]$ outside, the cusp $x^2=\frac{8}{27}{|y|}^3$.
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10: Bibliography T

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On the analysis of diffraction catastrophes. J. Phys. A 10, pp. L11–L16.

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External Links:

Document

Referenced by:

§36.14(iii), §36.2(ii).

Encodings:

BibTeX

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