elliptic umbilic catastrophe
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Normal Forms for Umbilic Catastrophes with Codimension►
36.2.2 ,… ►
36.2.5 .… ►
36.2.11 ; .…
… ►The main functions covered in this chapter are cuspoid catastrophes ; umbilic catastrophes with codimension three , ; canonical integrals , , ; diffraction catastrophes , , generated by the catastrophes. …
… ►In terms of the normal forms (36.2.2) and (36.2.3), the satisfy the following operator equations … ►►… ►
The elliptic umbilic diffraction catastrophe.
Phil. Trans. Roy. Soc. Ser. A 291 (1382), pp. 453–484.
§36.6 Scaling Relations…
9: 36.7 Zeros
§36.7(iii) Elliptic Umbilic Canonical Integral… ►The 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 … ►
§36.7(iv) Swallowtail and Hyperbolic Umbilic Canonical Integrals►The zeros of these functions are curves in space; see Nye (2007) for and Nye (2006) for .
… ►Define 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 . … ►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. … ►For 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).