§36.5 Stokes Sets

§36.5(i) Definitions

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 $\displaystyle\Re\left(\Phi_{K}\left(t_{j}(\mathbf{x});\mathbf{x}\right)-\Phi_{% K}\left(t_{\mu}(\mathbf{x});\mathbf{x}\right)\right)$ $\displaystyle=0,$ $\displaystyle\Re\left(\Phi^{(\mathrm{U})}\left(s_{j}(\mathbf{x}),t_{j}(\mathbf% {x});\mathbf{x}\right)-\Phi^{(\mathrm{U})}\left(s_{\mu}(\mathbf{x}),t_{\mu}(% \mathbf{x});\mathbf{x}\right)\right)$ $\displaystyle=0,$

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

$K=1$. Airy Function

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

$K=2$. Cusp

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 36

$K=3$. Swallowtail

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 $\displaystyle x$ $\displaystyle=B_{\pm}|y|^{4/3},$ $\displaystyle B_{\pm}$ $\displaystyle=10^{-1/3}\left(2x_{\pm}^{4/3}-\tfrac{1}{2}x_{\pm}^{-2/3}\right),$ ⓘ 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 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 36

and

 36.5.5 $\displaystyle B_{-}$ $\displaystyle=-1.69916,$ $\displaystyle B_{+}$ $\displaystyle=0.33912.$ ⓘ 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 36

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

 36.5.6 $\displaystyle X$ $\displaystyle=x/z^{2},$ $\displaystyle Y$ $\displaystyle=y/|z|^{3/2},$ ⓘ 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 36

by

 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 $x$, $z$: 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 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 $x$, $z$: 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 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 $x$, $z$: 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 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| 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 36

§36.5(iii) Umbilics

Elliptic Umbilic Stokes Set (Codimension three)

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 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 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 36

and such that

 36.5.14 $0 ⓘ 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 36

Hyperbolic Umbilic Stokes Set (Codimension three)

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 $\displaystyle X$ $\displaystyle=(x-y)/z^{2},$ $\displaystyle Y$ $\displaystyle=\tfrac{1}{2}+\left((x+y)/z^{2}\right),$ ⓘ 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 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 $\displaystyle Y(u,X)$ $\displaystyle=8u-24u^{2}+X\dfrac{u-\tfrac{1}{6}}{\left(u\left(u-\tfrac{1}{3}% \right)\right)^{1/2}},$ $\displaystyle f(u,X)$ $\displaystyle=16u^{3}-4u^{2}-\tfrac{1}{6}|X|\left(\dfrac{u}{u-\tfrac{1}{3}}% \right)^{1/2}.$ ⓘ 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 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 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 36

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

Alternatively, when $|X|

 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 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 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 36

§36.5(iv) Visualizations

In Figures 36.5.136.5.6 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.