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swallowtail catastrophe

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1: 36.2 Catastrophes and Canonical Integrals
Normal Forms Associated with Canonical Integrals: Cuspoid Catastrophe with Codimension K
Special cases: K = 1 , fold catastrophe; K = 2 , cusp catastrophe; K = 3 , swallowtail catastrophe. …
2: 36.5 Stokes Sets
See accompanying text
Figure 36.5.2: Swallowtail catastrophe with z < 0 . Magnify
See accompanying text
Figure 36.5.3: Swallowtail catastrophe with z = 0 . Magnify
See accompanying text
Figure 36.5.4: Swallowtail catastrophe with z > 0 . … Magnify
3: 36.4 Bifurcation Sets
See accompanying text
Figure 36.4.2: Bifurcation set of swallowtail catastrophe. Magnify
4: Bibliography N
  • J. F. Nye (2007) Dislocation lines in the swallowtail diffraction catastrophe. Proc. Roy. Soc. Lond. Ser. A 463, pp. 343–355.
  • 5: 36.6 Scaling Relations
    §36.6 Scaling Relations
    6: 36.7 Zeros
    The zeros of these functions are curves in x = ( x , y , z ) space; see Nye (2007) for Φ 3 and Nye (2006) for Φ ( H ) .
    7: Bibliography C
  • J. N. L. Connor, P. R. Curtis, and D. Farrelly (1983) A differential equation method for the numerical evaluation of the Airy, Pearcey and swallowtail canonical integrals and their derivatives. Molecular Phys. 48 (6), pp. 1305–1330.
  • J. N. L. Connor and P. R. Curtis (1982) A method for the numerical evaluation of the oscillatory integrals associated with the cuspoid catastrophes: Application to Pearcey’s integral and its derivatives. J. Phys. A 15 (4), pp. 1179–1190.
  • J. N. L. Connor and D. Farrelly (1981) Molecular collisions and cusp catastrophes: Three methods for the calculation of Pearcey’s integral and its derivatives. Chem. Phys. Lett. 81 (2), pp. 306–310.
  • J. N. L. Connor (1976) Catastrophes and molecular collisions. Molecular Phys. 31 (1), pp. 33–55.
  • 8: 36.11 Leading-Order Asymptotics
    §36.11 Leading-Order Asymptotics
    36.11.2 Ψ K ( x ) = 2 π j = 1 j max ( x ) exp ( i ( Φ K ( t j ( x ) ; x ) + 1 4 π ( - 1 ) j + K + 1 ) ) | 2 Φ K ( t j ( x ) ; x ) t 2 | - 1 / 2 ( 1 + o ( 1 ) ) .
    9: 36.10 Differential Equations
    K = 3 , swallowtail: … K = 3 , swallowtail: …
    Φ s ( U ) ( s , t ; x ) = s Φ ( U ) ( s , t ; x ) ,
    Φ t ( U ) ( s , t ; x ) = t Φ ( U ) ( s , t ; x ) .
    10: 36.12 Uniform Approximation of Integrals
    Define a mapping u ( t ; y ) by relating f ( u ; y ) to the normal form (36.2.1) of Φ K ( t ; x ) in the following way: …with the K + 1 functions A ( y ) and x ( y ) determined by correspondence of the K + 1 critical points of f and Φ K . … This technique can be applied to generate a hierarchy of approximations for the diffraction catastrophes Ψ K ( x ; k ) in (36.2.10) away from x = 0 , in terms of canonical integrals Ψ J ( ξ ( x ; k ) ) for J < K . For example, the diffraction catastrophe Ψ 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 Ψ 1 ( ξ ( x , y ; k ) ) when k is large, provided that x and y 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).