36 Integrals with Coalescing Saddles Properties 36.4 Bifurcation Sets 36.6 Scaling Relations

§36.5 Stokes Sets

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Permalink:: http://dlmf.nist.gov/36.5
See also:: Annotations for Ch.36

§36.5(i) Definitions

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Keywords:: Stokes sets, definitions
Referenced by:: §36.5(iv)
Permalink:: http://dlmf.nist.gov/36.5.i
See also:: Annotations for §36.5 and Ch.36

Stokes sets are surfaces (codimension one) in $\mathbf{x}$ space, across which $\Psi_{K}(\mathbf{x};k)$ or $\Psi^{(\mathrm{U})}(\mathbf{x};k)$ acquires an exponentially-small asymptotic contribution (in $k$ ), associated with a complex critical point of $\Phi_{K}$ or $\Phi^{(\mathrm{U})}$ . The Stokes sets are defined by the exponential dominance condition:

36.5.1	$\displaystyle\Re\left(\Phi_{K}\left(t_{j}(\mathbf{x});\mathbf{x}\right)-\Phi_{% K}\left(t_{\mu}(\mathbf{x});\mathbf{x}\right)\right)$	$\displaystyle=0,$
36.5.1	$\displaystyle\Re\left(\Phi^{(\mathrm{U})}\left(s_{j}(\mathbf{x}),t_{j}(\mathbf% {x});\mathbf{x}\right)-\Phi^{(\mathrm{U})}\left(s_{\mu}(\mathbf{x}),t_{\mu}(% \mathbf{x});\mathbf{x}\right)\right)$	$\displaystyle=0,$
ⓘ Symbols: $\Phi_{\NVar{K}}\left(\NVar{t};\NVar{\mathbf{x}}\right)$ : cuspoid catastrophe of codimension $K$ , $\Re$ : real part, $\Phi^{(\mathrm{U})}\left(\NVar{s},\NVar{t};\NVar{\mathbf{x}}\right)$ : elliptic umbilic catastrophe for $\mathrm{U}=\mathrm{E}\mbox{ or }\mathrm{K}$ , $s$ : variable, $K$ : codimension, $t_{j}(\mathbf{x})$ : solutions and $j$ : real critical point Referenced by: §36.5(ii) Permalink: http://dlmf.nist.gov/36.5.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §36.5(i), §36.5 and Ch.36

where $j$ denotes a real critical point (36.4.1) or (36.4.2), and $\mu$ denotes a critical point with complex $t$ or $s, t$ , connected with $j$ by a steepest-descent path (that is, a path where $\Re\Phi=\mathrm{constant}$ ) in complex $t$ or $(s,t)$ space.

In the following subsections, only Stokes sets involving at least one real saddle are included unless stated otherwise.

§36.5(ii) Cuspoids

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Keywords:: Pearcey integral, Stokes sets, cuspoids, formula for Stokes set, formulas for Stokes set, swallowtail canonical integral
Notes:: See Wright (1980), Berry and Howls (1990). The common strategy employed in deriving the formulas in this subsection involves using the critical-point condition (36.4.1) to reduce the order of the catastrophe polynomials in (36.2.1), then solving (36.5.1) for the imaginary part of the complex critical point in terms of the value of the real critical point, which is itself determined by (36.4.1) and then used to generate the Stokes sets parametrically.
Referenced by:: §36.5(iii)
Permalink:: http://dlmf.nist.gov/36.5.ii
See also:: Annotations for §36.5 and Ch.36

$K=1$ . Airy Function

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See also:: Annotations for §36.5(ii), §36.5 and Ch.36

The Stokes set consists of the rays $\operatorname{ph}x=\pm 2\pi/3$ in the complex $x$ -plane.

$K=2$ . Cusp

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See also:: Annotations for §36.5(ii), §36.5 and Ch.36

The Stokes set is itself a cusped curve, connected to the cusp of the bifurcation set:

36.5.2		$y^{3}=\tfrac{27}{4}\left(\sqrt{27}-5\right)x^{2}=1.32403x^{2}.$
ⓘ Symbols: $y$ : real parameter and $x$ : real parameter Permalink: http://dlmf.nist.gov/36.5.E2 Encodings: TeX, pMML, png See also: Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

$K=3$ . Swallowtail

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See also:: Annotations for §36.5(ii), §36.5 and Ch.36

The Stokes set takes different forms for $z=0$ , $z<0$ , and $z>0$ .

For $z=0$ , the set consists of the two curves

36.5.3	$\displaystyle x$	$\displaystyle=B_{\pm}\|y\|^{4/3},$
36.5.3	$\displaystyle B_{\pm}$	$\displaystyle=10^{-1/3}\left(2x_{\pm}^{4/3}-\tfrac{1}{2}x_{\pm}^{-2/3}\right),$
ⓘ Symbols: $y$ : real parameter, $x_{i}$ : real parameter and $x$ : real parameter Permalink: http://dlmf.nist.gov/36.5.E3 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

where $x_{\pm}$ are the two smallest positive roots of the equation

36.5.4		$80x^{5}-40x^{4}-55x^{3}+5x^{2}+20x-1=0,$
ⓘ Symbols: $x$ : real parameter Permalink: http://dlmf.nist.gov/36.5.E4 Encodings: TeX, pMML, png See also: Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

and

36.5.5	$\displaystyle B_{-}$	$\displaystyle=-1.69916,$
36.5.5	$\displaystyle B_{+}$	$\displaystyle=0.33912.$
ⓘ Permalink: http://dlmf.nist.gov/36.5.E5 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

For $z\neq 0$ , the Stokes set is expressed in terms of scaled coordinates

36.5.6	$\displaystyle X$	$\displaystyle=x/z^{2},$
36.5.6	$\displaystyle Y$	$\displaystyle=y/\|z\|^{3/2},$
ⓘ Defines: $X$ : scaled coordinate (locally) and $Y$ : scaled coordinate (locally) Symbols: $y$ : real parameter, $z$ : real parameter and $x$ : real parameter Referenced by: §36.5(ii) Permalink: http://dlmf.nist.gov/36.5.E6 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

36.5.7		$X=\dfrac{9}{20}+20u^{4}-\frac{Y^{2}}{20u^{2}}+6u^{2}\operatorname{sign}\left(z% \right),$
ⓘ Symbols: $\operatorname{sign}\NVar{x}$ : sign of, $z$ : real parameter, $X$ : scaled coordinate and $Y$ : scaled coordinate Permalink: http://dlmf.nist.gov/36.5.E7 Encodings: TeX, pMML, png See also: Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

where $u$ satisfies the equation

36.5.8		$16u^{5}-\frac{Y^{2}}{10u}+4u^{3}\operatorname{sign}\left(z\right)-\frac{3}{10}% \|Y\|\operatorname{sign}\left(z\right)+4t^{5}+2t^{3}\operatorname{sign}\left(z% \right)+\|Y\|t^{2}=0,$
ⓘ Symbols: $\operatorname{sign}\NVar{x}$ : sign of, $z$ : real parameter, $t$ : variable and $Y$ : scaled coordinate Permalink: http://dlmf.nist.gov/36.5.E8 Encodings: TeX, pMML, png See also: Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

in which

36.5.9		$t=-u+\left(\dfrac{\|Y\|}{10u}-u^{2}-\dfrac{3}{10}\operatorname{sign}\left(z% \right)\right)^{1/2}.$
ⓘ Symbols: $\operatorname{sign}\NVar{x}$ : sign of, $z$ : real parameter, $t$ : variable and $Y$ : scaled coordinate Referenced by: §36.5(ii) Permalink: http://dlmf.nist.gov/36.5.E9 Encodings: TeX, pMML, png See also: Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

For $z<0$ , there are two solutions $u$ , provided that $|Y|>(\frac{2}{5})^{1/2}$ . They generate a pair of cusp-edged sheets connected to the cusped sheets of the swallowtail bifurcation set (§36.4).

For $z>0$ the Stokes set has two sheets. The first sheet corresponds to $x<0$ and is generated as a solution of Equations (36.5.6)–(36.5.9). The second sheet corresponds to $x>0$ and it intersects the bifurcation set (§36.4) smoothly along the line generated by $X=X_{1}=6.95643$ , $\left|Y\right|=\left|Y_{1}\right|=6.81337$ . For $\left|Y\right|>Y_{1}$ the second sheet is generated by a second solution of (36.5.6)–(36.5.9), and for $\left|Y\right|<Y_{1}$ it is generated by the roots of the polynomial equation

36.5.10		$160u^{6}+40u^{4}=Y^{2}.$
ⓘ Symbols: $Y$ : scaled coordinate Permalink: http://dlmf.nist.gov/36.5.E10 Encodings: TeX, pMML, png See also: Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

§36.5(iii) Umbilics

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Keywords:: Stokes sets, elliptic umbilic canonical integral, formulas for Stokes set, hyperbolic umbilic canonical integral, umbilics
Notes:: See Berry and Howls (1990). The strategy used to derive the formulas in this subsection is the same as in §36.5(ii), but using the exponents in the representations (36.2.6) and (36.2.8).
Permalink:: http://dlmf.nist.gov/36.5.iii
See also:: Annotations for §36.5 and Ch.36

Elliptic Umbilic Stokes Set (Codimension three)

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See also:: Annotations for §36.5(iii), §36.5 and Ch.36

This consists of three separate cusp-edged sheets connected to the cusp-edged sheets of the bifurcation set, and related by rotation about the $z$ -axis by $2\pi/3$ . One of the sheets is symmetrical under reflection in the plane $y=0$ , and is given by

36.5.11		$\frac{x}{z^{2}}=-1-12u^{2}+8u-\left\|\frac{y}{z^{2}}\right\|\dfrac{\frac{1}{3}-u% }{\left(u\left(\frac{2}{3}-u\right)\right)^{1/2}}.$
ⓘ Symbols: $y$ : real parameter, $z$ : real parameter and $x$ : real parameter Referenced by: §36.5(iv) Permalink: http://dlmf.nist.gov/36.5.E11 Encodings: TeX, pMML, png See also: Annotations for §36.5(iii), §36.5(iii), §36.5 and Ch.36

Here $u$ is the root of the equation

36.5.12		$8u^{3}-4u^{2}-\left\|\frac{y}{3z^{2}}\right\|\left(\frac{u}{\tfrac{2}{3}-u}% \right)^{1/2}=\frac{y^{2}}{6wz^{4}}-2w^{3}-2w^{2},$
ⓘ Symbols: $y$ : real parameter, $z$ : real parameter and $w$ : quantity Permalink: http://dlmf.nist.gov/36.5.E12 Encodings: TeX, pMML, png See also: Annotations for §36.5(iii), §36.5(iii), §36.5 and Ch.36

with

36.5.13		$w=u-\tfrac{2}{3}+\left(\left(\tfrac{2}{3}-u\right)^{2}+\left\|\frac{y}{6z^{2}}% \right\|\left(\frac{\frac{2}{3}-u}{u}\right)^{1/2}\right)^{1/2},$
ⓘ Defines: $w$ : quantity (locally) Symbols: $y$ : real parameter and $z$ : real parameter Permalink: http://dlmf.nist.gov/36.5.E13 Encodings: TeX, pMML, png See also: Annotations for §36.5(iii), §36.5(iii), §36.5 and Ch.36

and such that

36.5.14		$0<u<\tfrac{1}{6}.$
ⓘ Permalink: http://dlmf.nist.gov/36.5.E14 Encodings: TeX, pMML, png See also: Annotations for §36.5(iii), §36.5(iii), §36.5 and Ch.36

Hyperbolic Umbilic Stokes Set (Codimension three)

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See also:: Annotations for §36.5(iii), §36.5 and Ch.36

This consists of a cusp-edged sheet connected to the cusp-edged sheet of the bifurcation set and intersecting the smooth sheet of the bifurcation set. With coordinates

36.5.15	$\displaystyle X$	$\displaystyle=(x-y)/z^{2},$
36.5.15	$\displaystyle Y$	$\displaystyle=\tfrac{1}{2}+\left((x+y)/z^{2}\right),$
ⓘ Defines: $X$ : scaled coordinate (locally) Symbols: $y$ : real parameter, $z$ : real parameter and $x$ : real parameter Permalink: http://dlmf.nist.gov/36.5.E15 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §36.5(iii), §36.5(iii), §36.5 and Ch.36

the intersection lines with the bifurcation set are generated by $|X|=X_{2}=0.45148$ , $Y=Y_{2}=0.59693$ . Define

36.5.16	$\displaystyle Y(u,X)$	$\displaystyle=8u-24u^{2}+X\dfrac{u-\tfrac{1}{6}}{\left(u\left(u-\tfrac{1}{3}% \right)\right)^{1/2}},$
36.5.16	$\displaystyle f(u,X)$	$\displaystyle=16u^{3}-4u^{2}-\tfrac{1}{6}\|X\|\left(\dfrac{u}{u-\tfrac{1}{3}}% \right)^{1/2}.$
ⓘ Symbols: $X$ : scaled coordinate Permalink: http://dlmf.nist.gov/36.5.E16 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §36.5(iii), §36.5(iii), §36.5 and Ch.36

When $|X|>X_{2}$ the Stokes set $Y_{\mathrm{S}}(X)$ is given by

36.5.17		$Y_{\mathrm{S}}(X)=Y(u,\|X\|),$
ⓘ Symbols: $X$ : scaled coordinate Permalink: http://dlmf.nist.gov/36.5.E17 Encodings: TeX, pMML, png See also: Annotations for §36.5(iii), §36.5(iii), §36.5 and Ch.36

where $u$ is the root of the equation

36.5.18		$f(u,X)=f(-u+\tfrac{1}{3},X),$
ⓘ Symbols: $X$ : scaled coordinate Permalink: http://dlmf.nist.gov/36.5.E18 Encodings: TeX, pMML, png See also: Annotations for §36.5(iii), §36.5(iii), §36.5 and Ch.36

such that $u>\tfrac{1}{3}$ . This part of the Stokes set connects two complex saddles.

Alternatively, when $|X|<X_{2}$

36.5.19		$Y_{\mathrm{S}}(X)=Y(-u,-\|X\|),$
ⓘ Symbols: $X$ : scaled coordinate Permalink: http://dlmf.nist.gov/36.5.E19 Encodings: TeX, pMML, png See also: Annotations for §36.5(iii), §36.5(iii), §36.5 and Ch.36

where $u$ is the positive root of the equation

36.5.20		$f(-u,X)=\dfrac{X^{2}}{12w}+4w^{3}-2w^{2},$
ⓘ Symbols: $w$ : quantity and $X$ : scaled coordinate Permalink: http://dlmf.nist.gov/36.5.E20 Encodings: TeX, pMML, png See also: Annotations for §36.5(iii), §36.5(iii), §36.5 and Ch.36

in which

36.5.21		$w=(\tfrac{1}{3}+u)\left(1-\left(1-\dfrac{\|X\|}{12u^{1/2}(\tfrac{1}{3}+u)^{3/2}}% \right)^{1/2}\right).$
ⓘ Symbols: $w$ : quantity and $X$ : scaled coordinate Referenced by: §36.5(iv) Permalink: http://dlmf.nist.gov/36.5.E21 Encodings: TeX, pMML, png See also: Annotations for §36.5(iii), §36.5(iii), §36.5 and Ch.36

§36.5(iv) Visualizations

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Keywords:: Pearcey integral, Stokes sets, hyperbolic umbilic catastrophe, picture of Stokes set, swallowtail canonical integral, visualizations
Notes:: The graphics were generated by the authors. For Figures 36.5.2–36.5.9, Eqs. (36.5.11)–(36.5.21) were used in parametric form $x=x(y)$ , and checked against the numerical computations in Berry and Howls (1990) (which were based directly on the definitions given in §36.5(i)).
Permalink:: http://dlmf.nist.gov/36.5.iv
See also:: Annotations for §36.5 and Ch.36

In Figures 36.5.1–36.5.6 the plane is divided into regions by the dashed curves (Stokes sets) and the continuous curves (bifurcation sets). Red and blue numbers in each region correspond, respectively, to the numbers of real and complex critical points that contribute to the asymptotics of the canonical integral away from the bifurcation sets. In Figure 36.5.4 the part of the Stokes surface inside the bifurcation set connects two complex saddles. The distribution of real and complex critical points in Figures 36.5.5 and 36.5.6 follows from consistency with Figure 36.5.1 and the fact that there are four real saddles in the inner regions.