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hyperbolic umbilic catastrophe

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1: 36.2 Catastrophes and Canonical Integrals
Normal Forms for Umbilic Catastrophes with Codimension K = 3
36.2.3 Φ ( H ) ( s , t ; x ) = s 3 + t 3 + z s t + y t + x s , x = { x , y , z } ,
36.2.5 Ψ ( U ) ( x ) = - - exp ( i Φ ( U ) ( s , t ; x ) ) d s d t , U = E , H .
36.2.11 Ψ ( U ) ( x ; k ) = k - - exp ( i k Φ ( U ) ( s , t ; x ) ) d s d t , U = E , H ; k > 0 .
2: 36.1 Special Notation
The main functions covered in this chapter are cuspoid catastrophes Φ K ( t ; x ) ; umbilic catastrophes with codimension three Φ ( E ) ( s , t ; x ) , Φ ( H ) ( s , t ; x ) ; canonical integrals Ψ K ( x ) , Ψ ( E ) ( x ) , Ψ ( H ) ( x ) ; diffraction catastrophes Ψ K ( x ; k ) , Ψ ( E ) ( x ; k ) , Ψ ( H ) ( x ; k ) generated by the catastrophes. …
3: 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 ) .
4: 36.5 Stokes Sets
See accompanying text
Figure 36.5.5: Elliptic umbilic catastrophe with z = constant . … Magnify
See accompanying text
Figure 36.5.6: Hyperbolic umbilic catastrophe with z = constant . Magnify
See accompanying text
Figure 36.5.9: Sheets of the Stokes surface for the hyperbolic umbilic catastrophe (colored and with mesh) and the bifurcation set (gray). Magnify
5: Bibliography U
  • T. Uzer, J. T. Muckerman, and M. S. Child (1983) Collisions and umbilic catastrophes. The hyperbolic umbilic canonical diffraction integral. Molecular Phys. 50 (6), pp. 1215–1230.
  • 6: 36.4 Bifurcation Sets
    See accompanying text
    Figure 36.4.4: Bifurcation set of hyperbolic umbilic catastrophe. Magnify
    7: Bibliography N
  • J. F. Nye (2006) Dislocation lines in the hyperbolic umbilic diffraction catastrophe. Proc. Roy. Soc. Lond. Ser. A 462, pp. 2299–2313.
  • 8: 36.6 Scaling Relations
    §36.6 Scaling Relations
    9: 36.10 Differential Equations
    In terms of the normal forms (36.2.2) and (36.2.3), the Ψ ( U ) ( x ) satisfy the following operator equations …
    Φ s ( U ) ( s , t ; x ) = s Φ ( U ) ( s , t ; x ) ,
    Φ t ( U ) ( s , t ; x ) = t Φ ( U ) ( s , t ; x ) .
    36.10.15 3 2 Ψ ( H ) x 2 + i z Ψ ( H ) y - x Ψ ( H ) = 0 ,
    36.10.16 3 2 Ψ ( H ) y 2 + i z Ψ ( H ) x - y Ψ ( H ) = 0 .
    10: 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 ) ) .
    36.11.7 Ψ ( E ) ( 0 , 0 , z ) = π z ( i + 3 exp ( 4 27 i z 3 ) + o ( 1 ) ) , z ± ,
    36.11.8 Ψ ( H ) ( 0 , 0 , z ) = 2 π z ( 1 - i 3 exp ( 1 27 i z 3 ) + o ( 1 ) ) , z ± .