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1: 36.2 Catastrophes and Canonical Integrals
Normal Forms for Umbilic Catastrophes with Codimension K = 3
(elliptic umbilic). …(hyperbolic umbilic).
Canonical Integrals
Addendum: For further special cases see §36.2(iv)
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: 36.3 Visualizations of Canonical Integrals
Figure 36.3.6: Modulus of elliptic umbilic canonical integral function | Ψ ( E ) ( x , y , 0 ) | .
Figure 36.3.7: Modulus of elliptic umbilic canonical integral function | Ψ ( E ) ( x , y , 2 ) | .
Figure 36.3.8: Modulus of elliptic umbilic canonical integral function | Ψ ( E ) ( x , y , 4 ) | .
Figure 36.3.9: Modulus of hyperbolic umbilic canonical integral function | Ψ ( H ) ( x , y , 0 ) | .
Figure 36.3.10: Modulus of hyperbolic umbilic canonical integral function | Ψ ( H ) ( x , y , 1 ) | .
4: 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 …
36.10.13 6 2 Ψ ( E ) x y - 2 i z Ψ ( E ) y + y Ψ ( E ) = 0 ,
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 .
36.10.17 i Ψ ( E ) z = 2 Ψ ( E ) x 2 + 2 Ψ ( E ) y 2 ,
5: 36.6 Scaling Relations
§36.6 Scaling Relations
Ψ ( U ) ( x ; k ) = k β ( U ) Ψ ( U ) ( y ( U ) ( k ) ) ,
Indices for k -Scaling of Magnitude of Ψ K or Ψ ( U ) (Singularity Index)
umbilics β ( U ) = 1 3 .
umbilics γ x ( U ) = 2 3 ,
6: 36.4 Bifurcation Sets
Critical Points for Umbilics
Bifurcation (Catastrophe) Set for Umbilics
Elliptic umbilic cusp lines (ribs): … Hyperbolic umbilic bifurcation set (codimension three): … Hyperbolic umbilic cusp line (rib): …
7: 36.5 Stokes Sets
Stokes sets are surfaces (codimension one) in x space, across which Ψ K ( x ; k ) or Ψ ( U ) ( x ; k ) acquires an exponentially-small asymptotic contribution (in k ), associated with a complex critical point of Φ K or Φ ( U ) . …
§36.5(iii) Umbilics
Elliptic Umbilic Stokes Set (Codimension three)
Hyperbolic Umbilic Stokes Set (Codimension three)
See accompanying text
Figure 36.5.5: Elliptic umbilic catastrophe with z = constant . … Magnify
8: 36.8 Convergent Series Expansions
§36.8 Convergent Series Expansions
36.8.3 3 2 / 3 4 π 2 Ψ ( H ) ( 3 1 / 3 x ) = Ai ( x ) Ai ( y ) n = 0 ( - 3 - 1 / 3 i z ) n c n ( x ) c n ( y ) n ! + Ai ( x ) Ai ( y ) n = 2 ( - 3 - 1 / 3 i z ) n c n ( x ) d n ( y ) n ! + Ai ( x ) Ai ( y ) n = 2 ( - 3 - 1 / 3 i z ) n d n ( x ) c n ( y ) n ! + Ai ( x ) Ai ( y ) n = 1 ( - 3 - 1 / 3 i z ) n d n ( x ) d n ( y ) n ! ,
36.8.4 Ψ ( E ) ( x ) = 2 π 2 ( 2 3 ) 2 / 3 n = 0 ( - i ( 2 / 3 ) 2 / 3 z ) n n ! ( f n ( x + i y 12 1 / 3 , x - i y 12 1 / 3 ) ) ,
9: 36.9 Integral Identities
§36.9 Integral Identities
36.9.8 | Ψ ( H ) ( x , y , z ) | 2 = 8 π 2 ( 2 9 ) 1 / 3 - - Ai ( ( 4 3 ) 1 / 3 ( x + z v + 3 u 2 ) ) Ai ( ( 4 3 ) 1 / 3 ( y + z u + 3 v 2 ) ) d u d v .
36.9.9 | Ψ ( E ) ( x , y , z ) | 2 = 8 π 2 3 2 / 3 0 0 2 π ( Ai ( 1 3 1 / 3 ( x + i y + 2 z u exp ( i θ ) + 3 u 2 exp ( - 2 i θ ) ) ) Bi ( 1 3 1 / 3 ( x - i y + 2 z u exp ( - i θ ) + 3 u 2 exp ( 2 i θ ) ) ) ) u d u d θ .
10: 36.11 Leading-Order Asymptotics
§36.11 Leading-Order Asymptotics
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 ± .