# hyperbolic umbilic catastrophe

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##### 1: 36.2 Catastrophes and Canonical Integrals
###### Normal Forms for UmbilicCatastrophes with Codimension $K=3$
36.2.3 $\Phi^{(\mathrm{H})}\left(s,t;\mathbf{x}\right)=s^{3}+t^{3}+zst+yt+xs,$ $\mathbf{x}=\{x,y,z\}$,
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: 36.7 Zeros
The zeros of these functions are curves in $\mathbf{x}=(x,y,z)$ space; see Nye (2007) for $\Phi_{3}$ and Nye (2006) for $\Phi^{(\mathrm{H})}$.
##### 4: 36.5 Stokes Sets Figure 36.5.5: Elliptic umbilic catastrophe with z = constant . … Magnify Figure 36.5.6: Hyperbolic umbilic catastrophe with z = constant . Magnify Figure 36.5.9: Sheets of the Stokes surface for the hyperbolic umbilic catastrophe (colored and with mesh) and the bifurcation set (gray). Magnify
##### 5: Bibliography U
• T. Uzer, J. T. Muckerman, and M. S. Child (1983) Collisions and umbilic catastrophes. The hyperbolic umbilic canonical diffraction integral. Molecular Phys. 50 (6), pp. 1215–1230.
##### 7: Bibliography N
• J. F. Nye (2006) Dislocation lines in the hyperbolic umbilic diffraction catastrophe. Proc. Roy. Soc. Lond. Ser. A 462, pp. 2299–2313.
##### 9: 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 …
${\Phi_{s}}^{(\mathrm{U})}\left(s,t;\mathbf{x}\right)=\frac{\partial}{\partial s% }\Phi^{(\mathrm{U})}\left(s,t;\mathbf{x}\right),$
${\Phi_{t}}^{(\mathrm{U})}\left(s,t;\mathbf{x}\right)=\frac{\partial}{\partial t% }\Phi^{(\mathrm{U})}\left(s,t;\mathbf{x}\right).$
36.10.15 $3\frac{{\partial}^{2}\Psi^{(\mathrm{H})}}{{\partial x}^{2}}+iz\frac{\partial% \Psi^{(\mathrm{H})}}{\partial y}-x\Psi^{(\mathrm{H})}=0,$
36.10.16 $3\frac{{\partial}^{2}\Psi^{(\mathrm{H})}}{{\partial y}^{2}}+iz\frac{\partial% \Psi^{(\mathrm{H})}}{\partial x}-y\Psi^{(\mathrm{H})}=0.$
36.11.2 $\Psi_{K}\left(\mathbf{x}\right)=\sqrt{2\pi}\sum\limits_{j=1}^{j_{\max}(\mathbf% {x})}\exp\left(i\left(\Phi_{K}\left(t_{j}(\mathbf{x});\mathbf{x}\right)+\tfrac% {1}{4}\pi(-1)^{j+K+1}\right)\right)\left|\frac{{\partial}^{2}\Phi_{K}\left(t_{% j}(\mathbf{x});\mathbf{x}\right)}{{\partial t}^{2}}\right|^{-1/2}(1+o\left(1% \right)).$
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 $\Psi^{(\mathrm{H})}\left(0,0,z\right)=\frac{2\pi}{z}\left(1-\frac{i}{\sqrt{3}}% \exp\left(\frac{1}{27}iz^{3}\right)+o\left(1\right)\right),$ $z\to\pm\infty$.