elliptic umbilic catastrophe

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1: 36.2 Catastrophes and Canonical Integrals

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Normal Forms for Umbilic Catastrophes with Codimension $K=3$

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36.2.2 $${\mathrm{\Phi }}^{(\mathrm{E})}(s,t;\mathbf{x})=s^3-3s{t}^2+z(s^2+t^2)+yt+xs,$$ $\mathbf{x}=\{x,y,z\}$,

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Defines:

${\mathrm{\Phi }}^{(\mathrm{E})}(s,t;\mathbf{x})$: elliptic umbilic catastrophe

Symbols:

$y$: real parameter, $z$: real parameter, $t$: variable, $s$: variable and $x$: real parameter

Referenced by:

§36.10(iii), §36.2(i), §36.7(iii), §36.8

Permalink:

http://dlmf.nist.gov/36.2.E2

Encodings:

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See also:

Annotations for §36.2(i), §36.2(i), §36.2 and Ch.36

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36.2.5 $${\mathrm{\Psi }}^{(\mathrm{U})}\left(\mathbf{x}\right)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\exp \left(\mathrm{i}{\mathrm{\Phi }}^{(\mathrm{U})}(s,t;\mathbf{x})\right)dsdt,$$ $\mathrm{U}=\mathrm{E},\mathrm{H}$.

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Defines:

${\mathrm{\Psi }}^{(\mathrm{E})}\left(\mathbf{x}\right)$: elliptic umbilic canonical integral function, ${\mathrm{\Psi }}^{(\mathrm{H})}\left(\mathbf{x}\right)$: hyperbolic umbilic canonical integral function and ${\mathrm{\Psi }}^{(\mathrm{U})}\left(\mathbf{x}\right)$: umbilic canonical integral function

Symbols:

$dx$: differential, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\int$: integral, ${\mathrm{\Phi }}^{(\mathrm{U})}(s,t;\mathbf{x})$: elliptic umbilic catastrophe for $\mathrm{U}=\mathrm{E}\text{ or }\mathrm{K}$ , $t$: variable and $s$: variable

Referenced by:

§36.10(iii), §36.10(iv), §36.2(i), §36.7(iii), §36.8

Permalink:

http://dlmf.nist.gov/36.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §36.2(i), §36.2(i), §36.2 and Ch.36

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36.2.11 $${\mathrm{\Psi }}^{(\mathrm{U})}(\mathbf{x};k)=k{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\exp \left(\mathrm{i}k{\mathrm{\Phi }}^{(\mathrm{U})}(s,t;\mathbf{x})\right)dsdt,$$ $\mathrm{U}=\mathrm{E},\mathrm{H}$; $k>0$.

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Defines:

${\mathrm{\Psi }}^{(\mathrm{E})}(\mathbf{x};k)$: elliptic umbilic canonical integral function, ${\mathrm{\Psi }}^{(\mathrm{H})}(\mathbf{x};k)$: hyperbolic umbilic canonical integral function and ${\mathrm{\Psi }}^{(\mathrm{U})}(\mathbf{x};k)$: umbilic canonical integral function

Symbols:

$dx$: differential, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\int$: integral, ${\mathrm{\Phi }}^{(\mathrm{U})}(s,t;\mathbf{x})$: elliptic umbilic catastrophe for $\mathrm{U}=\mathrm{E}\text{ or }\mathrm{K}$ , $k$: variable, $t$: variable and $s$: variable

Referenced by:

§36.2(i)

Permalink:

http://dlmf.nist.gov/36.2.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §36.2(i), §36.2(i), §36.2 and Ch.36

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2: 36.1 Special Notation

… ►The main functions covered in this chapter are cuspoid catastrophes ${\mathrm{\Phi }}_K(t;\mathbf{x})$; umbilic catastrophes with codimension three ${\mathrm{\Phi }}^{(\mathrm{E})}(s,t;\mathbf{x})$, ${\mathrm{\Phi }}^{(\mathrm{H})}(s,t;\mathbf{x})$; canonical integrals ${\mathrm{\Psi }}_K\left(\mathbf{x}\right)$, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(\mathbf{x}\right)$, ${\mathrm{\Psi }}^{(\mathrm{H})}\left(\mathbf{x}\right)$; diffraction catastrophes ${\mathrm{\Psi }}_K(\mathbf{x};k)$, ${\mathrm{\Psi }}^{(\mathrm{E})}(\mathbf{x};k)$, ${\mathrm{\Psi }}^{(\mathrm{H})}(\mathbf{x};k)$ generated by the catastrophes. …

3: 36.5 Stokes Sets

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Figure 36.5.5: Elliptic umbilic catastrophe with $z=\mathrm{constant}$. Magnify

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Figure 36.5.8: Sheets of the Stokes surface for the elliptic umbilic catastrophe (colored and with mesh) and the bifurcation set (gray). Magnify

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4: 36.4 Bifurcation Sets

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36.4.4 $$\frac{{\partial }^2}{{\partial s}^2}{\mathrm{\Phi }}^{(\mathrm{U})}(s,t;\mathbf{x})\frac{{\partial }^2}{{\partial t}^2}{\mathrm{\Phi }}^{(\mathrm{U})}(s,t;\mathbf{x})-{\left(\frac{{\partial }^2}{\partial s\partial t}{\mathrm{\Phi }}^{(\mathrm{U})}(s,t;\mathbf{x})\right)}^2=0.$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential, ${\mathrm{\Phi }}^{(\mathrm{U})}(s,t;\mathbf{x})$: elliptic umbilic catastrophe for $\mathrm{U}=\mathrm{E}\text{ or }\mathrm{K}$ , $t$: variable and $s$: variable

Permalink:

http://dlmf.nist.gov/36.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §36.4(i), §36.4(i), §36.4 and Ch.36

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Figure 36.4.3: Bifurcation set of elliptic umbilic catastrophe. Magnify

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5: 36.10 Differential Equations

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

${\mathrm{\Phi }}_s^{(\mathrm{U})}(s,t;\mathbf{x})={\displaystyle \frac{\partial }{\partial s}}{\mathrm{\Phi }}^{(\mathrm{U})}(s,t;\mathbf{x}),$

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${\mathrm{\Phi }}_t^{(\mathrm{U})}(s,t;\mathbf{x})={\displaystyle \frac{\partial }{\partial t}}{\mathrm{\Phi }}^{(\mathrm{U})}(s,t;\mathbf{x}).$

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36.10.13 $$6\frac{{\partial }^2{\mathrm{\Psi }}^{(\mathrm{E})}}{\partial x\partial y}-2\mathrm{i}z\frac{\partial {\mathrm{\Psi }}^{(\mathrm{E})}}{\partial y}+y{\mathrm{\Psi }}^{(\mathrm{E})}=0,$$

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Symbols:

${\mathrm{\Psi }}^{(\mathrm{E})}\left(\mathbf{x}\right)$: elliptic umbilic canonical integral function, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential, $y$: real parameter, $z$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.10.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §36.10(iii), §36.10 and Ch.36

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

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Symbols:

${\mathrm{\Psi }}^{(\mathrm{E})}\left(\mathbf{x}\right)$: elliptic umbilic canonical integral function, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential, $y$: real parameter, $z$: real parameter and $x$: real parameter

Referenced by:

§36.10(iv)

Permalink:

http://dlmf.nist.gov/36.10.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §36.10(iv), §36.10 and Ch.36

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6: Bibliography B

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The elliptic umbilic diffraction catastrophe. Phil. Trans. Roy. Soc. Ser. A 291 (1382), pp. 453–484.

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External Links:

FullText

Referenced by:

§36.15(iv), §36.2(i), §36.7(ii), §36.7(iii).

Encodings:

BibTeX

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7: 36.6 Scaling Relations

§36.6 Scaling Relations

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8: 36.11 Leading-Order Asymptotics

§36.11 Leading-Order Asymptotics

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36.11.2 $${\mathrm{\Psi }}_K\left(\mathbf{x}\right)=\sqrt{2\pi }\sum _{j=1}^{j_{\max }(\mathbf{x})}\exp \left(\mathrm{i}\left({\mathrm{\Phi }}_K(t_j(\mathbf{x});\mathbf{x})+\frac{1}{4}\pi {(-1)}^{j+K+1}\right)\right){\left|\frac{{\partial }^2{\mathrm{\Phi }}_K(t_j(\mathbf{x});\mathbf{x})}{{\partial t}^2}\right|}^{-1/2}(1+o\left(1\right)).$$

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Symbols:

${\mathrm{\Psi }}_K\left(\mathbf{x}\right)$: canonical integral function, $\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{\Phi }}_K(t;\mathbf{x})$: cuspoid catastrophe of codimension $K$ , $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $o\left(x\right)$: order less than, $\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential, $t$: variable, $K$: codimension and $t_j(\mathbf{x})$: solutions

Permalink:

http://dlmf.nist.gov/36.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §36.11 and Ch.36

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36.11.7 $${\mathrm{\Psi }}^{(\mathrm{E})}\left(0,0,z\right)=\frac{\pi }{z}\left(\mathrm{i}+\sqrt{3}\exp \left(\frac{4}{27}\mathrm{i}{z}^3\right)+o\left(1\right)\right),$$ $z\to \pm \mathrm{\infty }$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(\mathbf{x}\right)$: elliptic umbilic canonical integral function, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $o\left(x\right)$: order less than and $z$: real parameter

Permalink:

http://dlmf.nist.gov/36.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §36.11, §36.11 and Ch.36

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36.11.8 $${\mathrm{\Psi }}^{(\mathrm{H})}\left(0,0,z\right)=\frac{2\pi }{z}\left(1-\frac{\mathrm{i}}{\sqrt{3}}\exp \left(\frac{1}{27}\mathrm{i}{z}^3\right)+o\left(1\right)\right),$$ $z\to \pm \mathrm{\infty }$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, ${\mathrm{\Psi }}^{(\mathrm{H})}\left(\mathbf{x}\right)$: hyperbolic umbilic canonical integral function, $\mathrm{i}$: imaginary unit, $o\left(x\right)$: order less than and $z$: real parameter

Permalink:

http://dlmf.nist.gov/36.11.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §36.11, §36.11 and Ch.36

9: 36.7 Zeros

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§36.7(iii) Elliptic Umbilic Canonical Integral

… ►The zeros are lines in $\mathbf{x}=(x,y,z)$ space where $\mathrm{ph}{\mathrm{\Psi }}^{(\mathrm{E})}\left(\mathbf{x}\right)$ is undetermined. …Near $z=z_n$, and for small $x$ and $y$, the modulus $|{\mathrm{\Psi }}^{(\mathrm{E})}\left(\mathbf{x}\right)|$ 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 … ►

§36.7(iv) Swallowtail and Hyperbolic Umbilic Canonical Integrals

►The zeros of these functions are curves in $\mathbf{x}=(x,y,z)$ space; see Nye (2007) for ${\mathrm{\Phi }}_3$ and Nye (2006) for ${\mathrm{\Phi }}^{(\mathrm{H})}$.

10: 36.12 Uniform Approximation of Integrals

… ►Define a mapping $u(t;\mathbf{y})$ by relating $f(u;\mathbf{y})$ to the normal form (36.2.1) of ${\mathrm{\Phi }}_K(t;\mathbf{x})$ 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 ${\mathrm{\Phi }}_K$. … ►This technique can be applied to generate a hierarchy of approximations for the diffraction catastrophes ${\mathrm{\Psi }}_K(\mathbf{x};k)$ in (36.2.10) away from $\mathbf{x}=\mathbf{0}$, in terms of canonical integrals ${\mathrm{\Psi }}_J\left(\xi (\mathbf{x};k)\right)$ for . For example, the diffraction catastrophe ${\mathrm{\Psi }}_2(x,y;k)$ defined by (36.2.10), and corresponding to the Pearcey integral (36.2.14), can be approximated by the Airy function ${\mathrm{\Psi }}_1\left(\xi (x,y;k)\right)$ 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).