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βΊThe main functions covered in this chapter are cuspoid catastrophes
; umbiliccatastrophes with codimension three , ; canonical integrals , , ; diffraction catastrophes
, , generated by the catastrophes.
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βΊβΊβΊFigure 36.5.5: Ellipticumbiliccatastrophe with .
Magnify
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βΊβΊβΊFigure 36.5.8: Sheets of the Stokes surface for the ellipticumbiliccatastrophe (colored and with mesh) and the bifurcation set (gray).
Magnify
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βΊThe zeros are lines in space where is undetermined.
…Near , and for small and , the modulus has the symmetry of a lattice with a rhombohedral unit cell that has a mirror plane and an inverse threefold axis whose and repeat distances are given by
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βΊ
§36.7(iv) Swallowtail and Hyperbolic Umbilic Canonical Integrals
βΊThe zeros of these functions are curves in space; see Nye (2007) for and Nye (2006) for .
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βΊDefine a mapping by relating to the normal form (36.2.1) of in the following way:
…with the functions and determined by correspondence of the critical points of and .
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βΊThis 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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βΊFor further information concerning integrals with several coalescing saddle points see Arnol’d et al. (1988), Berry and Howls (1993, 1994), Bleistein (1967), Duistermaat (1974), Ludwig (1966), Olde Daalhuis (2000), and Ursell (1972, 1980).