diffraction catastrophes
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1: 36.2 Catastrophes and Canonical Integrals
2: 36.1 Special Notation
§36.1 Special Notation
… ►The main functions covered in this chapter are cuspoid catastrophes ; umbilic catastrophes with codimension three , ; canonical integrals , , ; diffraction catastrophes , , generated by the catastrophes. …3: 36.6 Scaling Relations
4: 36.5 Stokes Sets
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►Stokes sets are surfaces (codimension one) in space, across which or acquires an exponentially-small asymptotic contribution (in ), associated with a complex critical point of or .
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5: 36.14 Other Physical Applications
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►Diffraction catastrophes describe the (linear) wave amplitudes that smooth the geometrical caustic singularities and decorate them with interference patterns.
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►Diffraction catastrophes describe the connection between ray optics and wave optics.
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►Diffraction catastrophes describe the “semiclassical” connections between classical orbits and quantum wavefunctions, for integrable (non-chaotic) systems.
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6: 36.12 Uniform Approximation of Integrals
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36.12.1
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►This technique can be applied to generate a hierarchy of approximations for the diffraction catastrophes
in (36.2.10) away from , in terms of canonical integrals for .
For example, the diffraction catastrophe
defined by (36.2.10), and corresponding to the Pearcey integral (36.2.14), can be approximated by the Airy function when is large, provided that and are not small.
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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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Dislocation lines in the swallowtail diffraction catastrophe.
Proc. Roy. Soc. Lond. Ser. A 463, pp. 343–355.
8: 36.7 Zeros
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9: Bibliography B
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Stokes surfaces of diffraction catastrophes with codimension three.
Nonlinearity 3 (2), pp. 281–291.
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Axial and focal-plane diffraction catastrophe integrals.
J. Phys. A 43 (37), pp. 375206, 13.
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The elliptic umbilic diffraction catastrophe.
Phil. Trans. Roy. Soc. Ser. A 291 (1382), pp. 453–484.
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Phase-space projection identities for diffraction catastrophes.
J. Phys. A 13 (1), pp. 149–160.
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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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10: Bibliography T
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On the analysis of diffraction catastrophes.
J. Phys. A 10, pp. L11–L16.
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