Automorphisms of Stokes multipliers in higher-order WKBJ theory

Abstract

We consider the Stokes phenomenon and higher-order Stokes phenomenon (HOSP) of formal asymptotic transseries arising in the WKBJ analysis of linear differential equations and integral problems. We introduce a framework of automorphisms that act on the Stokes constants of the divergent expansion, explained via late-late-term expansions and parametric Alien calculus, to capture this phenomenon. Our method is applied to a paradigmatic example: we obtain the full Stokes line structure and automorphisms for the Swallowtail problem from catastrophe theory, which contains four WKBJ components. We demonstrate that, in a system with four or more WKBJ components, the automorphism associated with the HOSP can itself change value across another higher-order Stokes line, which occurs when different higher-order Stokes lines intersect. We then argue that no additional special behaviour emerges for transseries with five or more WKBJ components.

Figures (7)

• Figure 1: The values of $a$ and $b$ are shown that result in coalescence of the three singular points \ref{['eq:singularity']}. Along $b=0$ two singularities coincide at $z=9a^2/20$, and on $a=-(5/2)^{1/3} |b|^{2/3}$ two singularities coincide at $z=-3a^2/20$. At $a=b=0$ all three singularities coincide at $z=0$. In each of these cases, the transseries \ref{['eq:swallowtailexp']} still contains four distinct WKBJ components, but the Stokes geometry is simplified. The case of one singularity with $a=b=0$ contains no intersecting Stokes lines and so is analogous to the Stokes geometry for the Airy integral, and that with two singularities contains no crossing higher-order Stokes lines and so is analogous to the Stokes geometry for the Pearcey integral. The Stokes line structures for these simplified cases are shown later in figure \ref{['fig:otherab']}.
• Figure 2: The paths of steepest descent $C_i(\arg[\epsilon])$, defined as where $(f(t,z)-\chi_i)/\epsilon$ is real and positive emerging from saddle $i$, are shown for the swallowtail integral \ref{['eq:newswallowint']} at $z=3+0.5\mathrm{i}$. $(a)$ shows the steepest descent curves $C_i(0)$ through each saddle for the case where $\arg[\epsilon]=0$. The arrows along each of these define the direction of integration in \ref{['eq:intseriessol']}. In $(b1$--$b2)$, the special steepest descent curves $C_j(\sigma_{ij})$ are shown for $\sigma_{ij}=\arg[\chi_j-\chi_i]$. The direction along each steepest descent curve is obtained by taking the direction in $(a)$ and changing $\arg[\epsilon]$ continuously from $0$ to $\sigma_{ij}$. Values for the Stokes constants $S_{ij}$ are obtained from \ref{['eq:stokesconstantsgamma']} by deforming the anticlockwise contour around saddle $i$ in panel $(bi)$ onto all adjacent curves $C_{j}(\sigma_{ij})$.
• Figure 3: $(a)$ The higher-order Stokes line geometry for the swallowtail problem \ref{['eq:swallowtailrescaled']} is shown for the parameter values $(a,b)=$$(1,3)$. Turning points are shown with black circles, virtual turning points with grey circles, branch cuts with grey lines, and higher-order Stokes lines with black lines. Arrows across $h_{i>k;j}$ show the direction to which the automorphism $\mathcal{T}_{i>k;j}$ is applied. In $(b)$, the region near the turning points is shown. In $(c)$, the region with four intersecting higher-order Stokes lines is shown. Note that both $h_{i>k;j}$ and $h_{k>i;j}$ coincide along the same curve, but that the automorphisms are applied in opposite directions.
• Figure 4: Stokes lines $l_{i>j}$ (bold lines) and higher-order Stokes lines $h_{i>k;j}$ (thin lines) are shown for the swallowtail problem \ref{['eq:swallowtailmain']}. The Stokes lines $l_{i>j}$ satisfy criteria \ref{['eq:level1line']} with the singulant functions \ref{['eq:singsol']}. Note that while every higher-order Stokes line was shown figure \ref{['fig:HOSL']}, only active higher-order Stokes lines are shown here. The higher-order Stokes line $h_{i>k;j}$ is inactive if the automorphism \ref{['eq:HOSA+P2']} is suppressed due to either $S_{ij}=0$ or $S_{jk}=0$.
• Figure 5: The active $l_{i>j}$ Stokes lines are shown for the swallowtail problem \ref{['eq:swallowtailmain']}. These were obtained by taking all the Stokes lines $l_{i>j}$ shown in figure \ref{['fig:HOSLandSL']} and retaining only those with $S_{ij} \neq 0$ along them. Values for the Stokes constants $S_{ij}$ are taken from table \ref{['tab:Sij']}.