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

36.5.1 | $\mathrm{\Re}\left({\mathrm{\Phi}}_{K}({t}_{j}(\mathbf{x});\mathbf{x})-{\mathrm{\Phi}}_{K}({t}_{\mu}(\mathbf{x});\mathbf{x})\right)$ | $=0,$ | ||

$\mathrm{\Re}\left({\mathrm{\Phi}}^{(\mathrm{U})}({s}_{j}(\mathbf{x}),{t}_{j}(\mathbf{x});\mathbf{x})-{\mathrm{\Phi}}^{(\mathrm{U})}({s}_{\mu}(\mathbf{x}),{t}_{\mu}(\mathbf{x});\mathbf{x})\right)$ | $=0,$ | |||

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 $\mathrm{\Re}\mathrm{\Phi}=\mathrm{constant}$) 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.

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

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

36.5.2 | $${y}^{3}=\frac{27}{4}\left(\sqrt{27}-5\right){x}^{2}=1.32403{x}^{2}.$$ | ||

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

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

36.5.3 | $x$ | $={B}_{\pm}{|y|}^{4/3},$ | ||

${B}_{\pm}$ | $={10}^{-1/3}\left(2{x}_{\pm}^{4/3}-\frac{1}{2}{x}_{\pm}^{-2/3}\right),$ | |||

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

36.5.4 | $$80{x}^{5}-40{x}^{4}-55{x}^{3}+5{x}^{2}+20x-1=0,$$ | ||

and

36.5.5 | ${B}_{-}$ | $=-1.69916,$ | ||

${B}_{+}$ | $=\mathrm{0.33912.}$ | |||

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

36.5.6 | $X$ | $=x/{z}^{2},$ | ||

$Y$ | $=y/{|z|}^{3/2},$ | |||

by

36.5.7 | $$X=\frac{9}{20}+20{u}^{4}-\frac{{Y}^{2}}{20{u}^{2}}+6{u}^{2}\mathrm{sign}\left(z\right),$$ | ||

where $u$ satisfies the equation

36.5.8 | $$16{u}^{5}-\frac{{Y}^{2}}{10u}+4{u}^{3}\mathrm{sign}\left(z\right)-\frac{3}{10}|Y|\mathrm{sign}\left(z\right)+4{t}^{5}+2{t}^{3}\mathrm{sign}\left(z\right)+|Y|{t}^{2}=0,$$ | ||

in which

36.5.9 | $$t=-u+{\left(\frac{|Y|}{10u}-{u}^{2}-\frac{3}{10}\mathrm{sign}\left(z\right)\right)}^{1/2}.$$ | ||

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

36.5.10 | $$160{u}^{6}+40{u}^{4}={Y}^{2}.$$ | ||

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 | $$\frac{x}{{z}^{2}}=-1-12{u}^{2}+8u-\left|\frac{y}{{z}^{2}}\right|\frac{\frac{1}{3}-u}{{\left(u\left(\frac{2}{3}-u\right)\right)}^{1/2}}.$$ | ||

Here $u$ is the root of the equation

36.5.12 | $$8{u}^{3}-4{u}^{2}-\left|\frac{y}{3{z}^{2}}\right|{\left(\frac{u}{\frac{2}{3}-u}\right)}^{1/2}=\frac{{y}^{2}}{6w{z}^{4}}-2{w}^{3}-2{w}^{2},$$ | ||

with

36.5.13 | $$w=u-\frac{2}{3}+{\left({\left(\frac{2}{3}-u\right)}^{2}+\left|\frac{y}{6{z}^{2}}\right|{\left(\frac{\frac{2}{3}-u}{u}\right)}^{1/2}\right)}^{1/2},$$ | ||

and such that

36.5.14 | $$ | ||

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 | $X$ | $=(x-y)/{z}^{2},$ | ||

$Y$ | $=\frac{1}{2}+\left((x+y)/{z}^{2}\right),$ | |||

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

36.5.16 | $Y(u,X)$ | $=8u-24{u}^{2}+X{\displaystyle \frac{u-\frac{1}{6}}{{\left(u\left(u-\frac{1}{3}\right)\right)}^{1/2}}},$ | ||

$f(u,X)$ | $=16{u}^{3}-4{u}^{2}-\frac{1}{6}|X|{\left({\displaystyle \frac{u}{u-\frac{1}{3}}}\right)}^{1/2}.$ | |||

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

36.5.17 | $${Y}_{\mathrm{S}}(X)=Y(u,|X|),$$ | ||

where $u$ is the root of the equation

36.5.18 | $$f(u,X)=f(-u+\frac{1}{3},X),$$ | ||

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

Alternatively, when $$

36.5.19 | $${Y}_{\mathrm{S}}(X)=Y(-u,-|X|),$$ | ||

where $u$ is the positive root of the equation

36.5.20 | $$f(-u,X)=\frac{{X}^{2}}{12w}+4{w}^{3}-2{w}^{2},$$ | ||

in which

36.5.21 | $$w=(\frac{1}{3}+u)\left(1-{\left(1-\frac{|X|}{12{u}^{1/2}{(\frac{1}{3}+u)}^{3/2}}\right)}^{1/2}\right).$$ | ||

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.