# elliptic umbilic catastrophe

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##### 1: 36.2 Catastrophes and Canonical Integrals
###### Normal Forms for UmbilicCatastrophes with Codimension $K=3$
36.2.2 $\Phi^{(\mathrm{E})}\left(s,t;\mathbf{x}\right)=s^{3}-3st^{2}+z(s^{2}+t^{2})+yt% +xs,$ $\mathbf{x}=\{x,y,z\}$,
36.2.5 $\Psi^{(\mathrm{U})}\left(\mathbf{x}\right)=\int_{-\infty}^{\infty}\int_{-% \infty}^{\infty}\exp\left(i\Phi^{(\mathrm{U})}\left(s,t;\mathbf{x}\right)% \right)\mathrm{d}s\mathrm{d}t,$ $\mathrm{U}=\mathrm{E},\mathrm{H}$.
36.2.11 $\Psi^{(\mathrm{U})}(\mathbf{x};k)=k\int_{-\infty}^{\infty}\int_{-\infty}^{% \infty}\exp\left(ik\Phi^{(\mathrm{U})}\left(s,t;\mathbf{x}\right)\right)% \mathrm{d}s\mathrm{d}t,$ $\mathrm{U=E,H}$; $k>0$.
##### 2: 36.1 Special Notation
The main functions covered in this chapter are cuspoid catastrophes $\Phi_{K}\left(t;\mathbf{x}\right)$; umbilic catastrophes with codimension three $\Phi^{(\mathrm{E})}\left(s,t;\mathbf{x}\right)$, $\Phi^{(\mathrm{H})}\left(s,t;\mathbf{x}\right)$; canonical integrals $\Psi_{K}\left(\mathbf{x}\right)$, $\Psi^{(\mathrm{E})}\left(\mathbf{x}\right)$, $\Psi^{(\mathrm{H})}\left(\mathbf{x}\right)$; diffraction catastrophes $\Psi_{K}(\mathbf{x};k)$, $\Psi^{(\mathrm{E})}(\mathbf{x};k)$, $\Psi^{(\mathrm{H})}(\mathbf{x};k)$ generated by the catastrophes. …
##### 3: 36.5 Stokes Sets Figure 36.5.5: Elliptic umbilic catastrophe with z = constant . Magnify Figure 36.5.8: Sheets of the Stokes surface for the elliptic umbilic catastrophe (colored and with mesh) and the bifurcation set (gray). Magnify
##### 4: 36.4 Bifurcation Sets
36.4.4 $\frac{{\partial}^{2}}{{\partial s}^{2}}\Phi^{(\mathrm{U})}\left(s,t;\mathbf{x}% \right)\frac{{\partial}^{2}}{{\partial t}^{2}}\Phi^{(\mathrm{U})}\left(s,t;% \mathbf{x}\right)-\left(\frac{{\partial}^{2}}{\partial s\partial t}\Phi^{(% \mathrm{U})}\left(s,t;\mathbf{x}\right)\right)^{2}=0.$
##### 5: 36.10 Differential 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 …
${\Phi_{s}}^{(\mathrm{U})}\left(s,t;\mathbf{x}\right)=\frac{\partial}{\partial s% }\Phi^{(\mathrm{U})}\left(s,t;\mathbf{x}\right),$
${\Phi_{t}}^{(\mathrm{U})}\left(s,t;\mathbf{x}\right)=\frac{\partial}{\partial t% }\Phi^{(\mathrm{U})}\left(s,t;\mathbf{x}\right).$
36.10.13 $6\frac{{\partial}^{2}\Psi^{(\mathrm{E})}}{\partial x\partial y}-2iz\frac{% \partial\Psi^{(\mathrm{E})}}{\partial y}+y\Psi^{(\mathrm{E})}=0,$
36.10.17 $i\frac{\partial\Psi^{(\mathrm{E})}}{\partial z}=\frac{{\partial}^{2}\Psi^{(% \mathrm{E})}}{{\partial x}^{2}}+\frac{{\partial}^{2}\Psi^{(\mathrm{E})}}{{% \partial y}^{2}},$
##### 6: Bibliography B
• M. V. Berry, J. F. Nye, and F. J. Wright (1979) The elliptic umbilic diffraction catastrophe. Phil. Trans. Roy. Soc. Ser. A 291 (1382), pp. 453–484.
• ##### 7: 36.6 Scaling Relations
###### §36.6 Scaling Relations
36.11.2 $\Psi_{K}\left(\mathbf{x}\right)=\sqrt{2\pi}\sum\limits_{j=1}^{j_{\max}(\mathbf% {x})}\exp\left(i\left(\Phi_{K}\left(t_{j}(\mathbf{x});\mathbf{x}\right)+\tfrac% {1}{4}\pi(-1)^{j+K+1}\right)\right)\left|\frac{{\partial}^{2}\Phi_{K}\left(t_{% j}(\mathbf{x});\mathbf{x}\right)}{{\partial t}^{2}}\right|^{-1/2}(1+o\left(1% \right)).$
36.11.7 $\Psi^{(\mathrm{E})}\left(0,0,z\right)=\frac{\pi}{z}\left(i+\sqrt{3}\exp\left(% \frac{4}{27}iz^{3}\right)+o\left(1\right)\right),$ $z\to\pm\infty$,
36.11.8 $\Psi^{(\mathrm{H})}\left(0,0,z\right)=\frac{2\pi}{z}\left(1-\frac{i}{\sqrt{3}}% \exp\left(\frac{1}{27}iz^{3}\right)+o\left(1\right)\right),$ $z\to\pm\infty$.
The zeros are lines in $\mathbf{x}=(x,y,z)$ space where $\operatorname{ph}\Psi^{(\mathrm{E})}\left(\mathbf{x}\right)$ is undetermined. …Near $z=z_{n}$, and for small $x$ and $y$, the modulus $|\Psi^{(\mathrm{E})}\left(\mathbf{x}\right)|$ has the symmetry of a lattice with a rhombohedral unit cell that has a mirror plane and an inverse threefold axis whose $z$ and $x$ repeat distances are given by …
The zeros of these functions are curves in $\mathbf{x}=(x,y,z)$ space; see Nye (2007) for $\Phi_{3}$ and Nye (2006) for $\Phi^{(\mathrm{H})}$.
Define a mapping $u(t;\mathbf{y})$ by relating $f(u;\mathbf{y})$ to the normal form (36.2.1) of $\Phi_{K}\left(t;\mathbf{x}\right)$ in the following way: …with the $K+1$ functions $A(\mathbf{y})$ and $\mathbf{x}(\mathbf{y})$ determined by correspondence of the $K+1$ critical points of $f$ and $\Phi_{K}$. … This technique can be applied to generate a hierarchy of approximations for the diffraction catastrophes $\Psi_{K}(\mathbf{x};k)$ in (36.2.10) away from $\mathbf{x}=\boldsymbol{{0}}$, in terms of canonical integrals $\Psi_{J}\left(\xi(\mathbf{x};k)\right)$ for $J. For example, the diffraction catastrophe $\Psi_{2}(x,y;k)$ defined by (36.2.10), and corresponding to the Pearcey integral (36.2.14), can be approximated by the Airy function $\Psi_{1}\left(\xi(x,y;k)\right)$ when $k$ is large, provided that $x$ and $y$ are not small. … For further information concerning integrals with several coalescing saddle points see Arnol’d et al. (1988), Berry and Howls (1993, 1994), Bleistein (1967), Duistermaat (1974), Ludwig (1966), Olde Daalhuis (2000), and Ursell (1972, 1980).