# elliptic umbilic catastrophe

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## 10 matching pages

##### 1: 36.2 Catastrophes and Canonical Integrals
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###### Normal Forms for UmbilicCatastrophes with Codimension $K=3$
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36.2.2 $\Phi^{(\mathrm{E})}\left(s,t;\mathbf{x}\right)=s^{3}-3st^{2}+z(s^{2}+t^{2})+yt% +xs,$ $\mathbf{x}=\{x,y,z\}$,
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36.2.5 $\Psi^{(\mathrm{U})}\left(\mathbf{x}\right)=\int_{-\infty}^{\infty}\int_{-% \infty}^{\infty}\exp\left(i\Phi^{(\mathrm{U})}\left(s,t;\mathbf{x}\right)% \right)\,\mathrm{d}s\,\mathrm{d}t,$ $\mathrm{U}=\mathrm{E},\mathrm{H}$.
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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
βΊThe 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. …
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##### 4: 36.4 Bifurcation Sets
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36.4.4 $\frac{{\partial}^{2}}{{\partial s}^{2}}\Phi^{(\mathrm{U})}\left(s,t;\mathbf{x}% \right)\frac{{\partial}^{2}}{{\partial t}^{2}}\Phi^{(\mathrm{U})}\left(s,t;% \mathbf{x}\right)-\left(\frac{\,{\partial}^{2}}{\,\partial s\,\partial t}\Phi^% {(\mathrm{U})}\left(s,t;\mathbf{x}\right)\right)^{2}=0.$
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##### 5: 36.10 Differential Equations
βΊIn 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 … βΊ
${\Phi_{s}}^{(\mathrm{U})}\left(s,t;\mathbf{x}\right)=\frac{\partial}{\partial s% }\Phi^{(\mathrm{U})}\left(s,t;\mathbf{x}\right),$
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${\Phi_{t}}^{(\mathrm{U})}\left(s,t;\mathbf{x}\right)=\frac{\partial}{\partial t% }\Phi^{(\mathrm{U})}\left(s,t;\mathbf{x}\right).$
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36.10.13 $6\frac{\,{\partial}^{2}\Psi^{(\mathrm{E})}}{\,\partial x\,\partial y}-2iz\frac% {\partial\Psi^{(\mathrm{E})}}{\partial y}+y\Psi^{(\mathrm{E})}=0,$
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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
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• 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
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36.11.2 $\Psi_{K}\left(\mathbf{x}\right)=\sqrt{2\pi}\sum\limits_{j=1}^{j_{\max}(\mathbf% {x})}\exp\left(i\left(\Phi_{K}\left(t_{j}(\mathbf{x});\mathbf{x}\right)+\tfrac% {1}{4}\pi(-1)^{j+K+1}\right)\right)\left|\frac{{\partial}^{2}\Phi_{K}\left(t_{% j}(\mathbf{x});\mathbf{x}\right)}{{\partial t}^{2}}\right|^{-1/2}(1+o\left(1% \right)).$
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36.11.7 $\Psi^{(\mathrm{E})}\left(0,0,z\right)=\frac{\pi}{z}\left(i+\sqrt{3}\exp\left(% \frac{4}{27}iz^{3}\right)+o\left(1\right)\right),$ $z\to\pm\infty$,
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36.11.8 $\Psi^{(\mathrm{H})}\left(0,0,z\right)=\frac{2\pi}{z}\left(1-\frac{i}{\sqrt{3}}% \exp\left(\frac{1}{27}iz^{3}\right)+o\left(1\right)\right),$ $z\to\pm\infty$.
##### 9: 36.7 Zeros
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###### §36.7(iii) EllipticUmbilic Canonical Integral
βΊThe 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 … βΊ
###### §36.7(iv) Swallowtail and Hyperbolic Umbilic Canonical Integrals
βΊThe zeros of these functions are curves in $\mathbf{x}=(x,y,z)$ space; see Nye (2007) for $\Phi_{3}$ and Nye (2006) for $\Phi^{(\mathrm{H})}$.
##### 10: 36.12 Uniform Approximation of Integrals
βΊDefine 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 $\Phi_{K}$. … βΊThis 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. … βΊFor 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).