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36 Integrals with Coalescing SaddlesProperties

§36.10 Differential Equations

Contents
  1. §36.10(i) Equations for ΨK(x)
  2. §36.10(ii) Partial Derivatives with Respect to the xn
  3. §36.10(iii) Operator Equations
  4. §36.10(iv) Partial z-Derivatives

§36.10(i) Equations for ΨK(x)

In terms of the normal form (36.2.1) the ΨK(x) satisfy the operator equation

36.10.1 ΦK(ix1;x)ΨK(x)=0,

or explicitly,

36.10.2 K+1ΨK(x)x1K+1+m=1K(i)mK2(mxmK+2)m1ΨK(x)x1m1=0.

Special Cases

K=1, fold: (36.10.1) becomes Airy’s equation (§9.2(i))

36.10.3 2Ψ1x2x3Ψ1=0.

K=2, cusp:

36.10.4 3Ψ2x312yΨ2xi4xΨ2=0.

K=3, swallowtail:

36.10.5 4Ψ3x435z2Ψ3x22i5yΨ3x+15xΨ3=0.

§36.10(ii) Partial Derivatives with Respect to the xn

36.10.6 lnΨKxmln=in(lm)mnΨKxlmn,
1mK, 1lK.

Special Cases

K=1, fold: (36.10.6) is an identity.

K=2, cusp:

36.10.7 2nΨ2x2n=innΨ2yn.

K=3, swallowtail:

36.10.8 2nΨ3x2n =innΨ3yn,
36.10.9 3nΨ3x3n =(1)nnΨ3zn,
36.10.10 3nΨ3y3n =in2nΨ3z2n.

§36.10(iii) Operator Equations

In terms of the normal forms (36.2.2) and (36.2.3), the Ψ(U)(x) satisfy the following operator equations

36.10.11 Φs(U)(ix,iy;x)Ψ(U)(x) =0,
Φt(U)(ix,iy;x)Ψ(U)(x) =0,

where

36.10.12 Φs(U)(s,t;x) =sΦ(U)(s,t;x),
Φt(U)(s,t;x) =tΦ(U)(s,t;x).

Explicitly,

36.10.13 62Ψ(E)xy2izΨ(E)y+yΨ(E)=0,
36.10.14 3(2Ψ(E)x22Ψ(E)y2)+2izΨ(E)xxΨ(E)=0.
36.10.15 32Ψ(H)x2+izΨ(H)yxΨ(H)=0,
36.10.16 32Ψ(H)y2+izΨ(H)xyΨ(H)=0.

§36.10(iv) Partial z-Derivatives

36.10.17 iΨ(E)z=2Ψ(E)x2+2Ψ(E)y2,
36.10.18 iΨ(H)z=2Ψ(H)xy.

Equation (36.10.17) is the paraxial wave equation.