# §36.10 Differential Equations

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

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

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

or explicitly,

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

### Special Cases

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

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

$K=2$, cusp:

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

$K=3$, swallowtail:

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

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

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

### Special Cases

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

$K=2$, cusp:

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

$K=3$, swallowtail:

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

## §36.10(iii) Operator Equations

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

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

where

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

Explicitly,

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

## §36.10(iv) Partial $z$-Derivatives

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

Equation (36.10.17) is the paraxial wave equation.