# §36.10 Differential Equations

## §36.10(i) Equations for $\Psi_{K}\left(\mathbf{x}\right)$

In terms of the normal form (36.2.1) the $\Psi_{K}\left(\mathbf{x}\right)$ satisfy the operator equation

 36.10.1 $\Phi_{K}'\left(-i\frac{\partial}{\partial x_{1}};\mathbf{x}\right)\Psi_{K}% \left(\mathbf{x}\right)=0,$

or explicitly,

 36.10.2 $\frac{{\partial}^{K+1}\Psi_{K}\left(\mathbf{x}\right)}{{\partial x_{1}}^{K+1}}% +\sum_{m=1}^{K}(-i)^{m-K-2}\left(\frac{mx_{m}}{K+2}\right)\frac{{\partial}^{m-% 1}\Psi_{K}\left(\mathbf{x}\right)}{{\partial x_{1}}^{m-1}}=0.$

### Special Cases

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

 36.10.3 $\frac{{\partial}^{2}\Psi_{1}}{{\partial x}^{2}}-\frac{x}{3}\Psi_{1}=0.$

$K=2$, cusp:

 36.10.4 $\frac{{\partial}^{3}\Psi_{2}}{{\partial x}^{3}}-\frac{1}{2}y\frac{\partial\Psi% _{2}}{\partial x}-\frac{i}{4}x\Psi_{2}=0.$

$K=3$, swallowtail:

 36.10.5 $\frac{{\partial}^{4}\Psi_{3}}{{\partial x}^{4}}-\frac{3}{5}z\frac{{\partial}^{% 2}\Psi_{3}}{{\partial x}^{2}}-\frac{2i}{5}y\frac{\partial\Psi_{3}}{\partial x}% +\frac{1}{5}x\Psi_{3}=0.$

## §36.10(ii) Partial Derivatives with Respect to the $x_{n}$

 36.10.6 $\frac{{\partial}^{ln}\Psi_{K}}{{\partial x_{m}}^{ln}}=i^{n(l-m)}\frac{{% \partial}^{mn}\Psi_{K}}{{\partial x_{l}}^{mn}},$ $1\leq m\leq K$, $1\leq l\leq K$.

### Special Cases

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

$K=2$, cusp:

 36.10.7 $\frac{{\partial}^{2n}\Psi_{2}}{{\partial x}^{2n}}=i^{n}\frac{{\partial}^{n}% \Psi_{2}}{{\partial y}^{n}}.$

$K=3$, swallowtail:

 36.10.8 $\displaystyle\frac{{\partial}^{2n}\Psi_{3}}{{\partial x}^{2n}}$ $\displaystyle=i^{n}\frac{{\partial}^{n}\Psi_{3}}{{\partial y}^{n}},$ 36.10.9 $\displaystyle\frac{{\partial}^{3n}\Psi_{3}}{{\partial x}^{3n}}$ $\displaystyle=(-1)^{n}\frac{{\partial}^{n}\Psi_{3}}{{\partial z}^{n}},$ 36.10.10 $\displaystyle\frac{{\partial}^{3n}\Psi_{3}}{{\partial y}^{3n}}$ $\displaystyle=i^{n}\frac{{\partial}^{2n}\Psi_{3}}{{\partial z}^{2n}}.$

## §36.10(iii) Operator Equations

In terms of the normal forms (36.2.2) and (36.2.3), the $\Psi^{(\mathrm{U})}\left(\mathbf{x}\right)$ satisfy the following operator equations

 36.10.11 $\displaystyle{\Phi_{s}^{(\mathrm{U})}}\left(-i\frac{\partial}{\partial x},-i% \frac{\partial}{\partial y};\mathbf{x}\right)\Psi^{(\mathrm{U})}\left(\mathbf{% x}\right)$ $\displaystyle=0,$ $\displaystyle{\Phi_{t}^{(\mathrm{U})}}\left(-i\frac{\partial}{\partial x},-i% \frac{\partial}{\partial y};\mathbf{x}\right)\Psi^{(\mathrm{U})}\left(\mathbf{% x}\right)$ $\displaystyle=0,$

where

 36.10.12 $\displaystyle{\Phi_{s}^{(\mathrm{U})}}\left(s,t;\mathbf{x}\right)$ $\displaystyle=\frac{\partial}{\partial s}\Phi^{(\mathrm{U})}\left(s,t;\mathbf{% x}\right),$ $\displaystyle{\Phi_{t}^{(\mathrm{U})}}\left(s,t;\mathbf{x}\right)$ $\displaystyle=\frac{\partial}{\partial t}\Phi^{(\mathrm{U})}\left(s,t;\mathbf{% x}\right).$

Explicitly,

 36.10.13 $6\frac{{\partial}^{2}\Psi^{(\mathrm{E})}}{\partial x\partial y}-2iz\frac{% \partial\Psi^{(\mathrm{E})}}{\partial y}+y\Psi^{(\mathrm{E})}=0,$
36.10.14 $3\left(\frac{{\partial}^{2}\Psi^{(\mathrm{E})}}{{\partial x}^{2}}-\frac{{% \partial}^{2}\Psi^{(\mathrm{E})}}{{\partial y}^{2}}\right)+2iz\frac{\partial% \Psi^{(\mathrm{E})}}{\partial x}-x\Psi^{(\mathrm{E})}=0.$
 36.10.15 $3\frac{{\partial}^{2}\Psi^{(\mathrm{H})}}{{\partial x}^{2}}+iz\frac{\partial% \Psi^{(\mathrm{H})}}{\partial y}-x\Psi^{(\mathrm{H})}=0,$
 36.10.16 $3\frac{{\partial}^{2}\Psi^{(\mathrm{H})}}{{\partial y}^{2}}+iz\frac{\partial% \Psi^{(\mathrm{H})}}{\partial x}-y\Psi^{(\mathrm{H})}=0.$

## §36.10(iv) Partial $z$-Derivatives

 36.10.17 $i\frac{\partial\Psi^{(\mathrm{E})}}{\partial z}=\frac{{\partial}^{2}\Psi^{(% \mathrm{E})}}{{\partial x}^{2}}+\frac{{\partial}^{2}\Psi^{(\mathrm{E})}}{{% \partial y}^{2}},$
 36.10.18 $i\frac{\partial\Psi^{(\mathrm{H})}}{\partial z}=\frac{{\partial}^{2}\Psi^{(% \mathrm{H})}}{\partial x\partial y}.$

Equation (36.10.17) is the paraxial wave equation.