An algebraic study of parametric Stokes phenomena

Abstract

We investigate geometric aspects of co-equational parametric resurgence, by studying physical problems whose formal asymptotic solutions give rise to Borel transforms lying on an algebraic curve. This perspective allows us to elucidate concepts unique to parametric resurgence such as singularity structures, (virtual) turning points and the higher-order Stokes phenomenon. We construct examples as solutions to Borel plane partial differential equations using an algebraic curve ansatz before turning to the general analytic structure of co-equational resurgence problems, where we provide a systematic description of analytic continuation and Stokes constants through a Borel plane inner-outer matching procedure.

Figures (7)

• Figure 1: A visualisation of $\Sigma_z$ with parametric dependence on $z$. In the top-left panel we show the sheets at a generic $z$. Bottom-left are the sheets at $z=1$ where we have a $(0,1)$ turning point. Bottom-right are the sheets at $z=0$ where we have a $(1,2)$ turning point. Top-right is the configuration at the virtual turning point where $\pi \chi_0(z) = \pi \chi_2(z)$ but $\chi_0(z) \neq \chi_2(z)$.
• Figure 2: Singularity graphs. The graph on the left is associated to our quadratic running example \ref{['eq:curve-main-example']}, while the graph of the right is associated to the second example \ref{['eq:eg2curve']}.
• Figure 3: An illustration of $\Sigma_z$ for our running example. $\chi_0 = 0$ is also highlighted here since later we will consider this curve as a solution to an inhomogeneous problem where this point is distinguished.
• Figure 4: The exponential conformal map showing relation between Stokes lines in original and conformal planes. The dashed lines and the different integrations contours. Stokes lines emanating from $\log \chi$ now appear.
• Figure 5: This is the Stokes graph in $\mathbb{C}_z$ for our running example \ref{['eq:curve-main-example']}. In the terminology of section \ref{['sec:algcurves']}, $z=0$ and $z=1$ are genuine $(0,1)$ and $(1,2)$ turning points respectively whereas $\hat{z}=1/2$ is a virtual turning point. The Stokes line $\ell_{01}$ is shown in light blue, while $\ell_{12}$ is shown in green. The naïve Stokes line $\hat{\ell}_{02}$ (red) is truncated along the higher-order stokes line (dashed brown) $h_{012}$. Finally two illustrative paths $\gamma_1$ and $\gamma_2$ of analytic continuation are shown, the latter undergoing higher-order Stokes phenomena.

Theorems & Definitions (15)

• Remark
• Example 2.1
• Example 2.2
• Remark
• Example 2.3
• Example 2.4
• Example 2.5
• Example 2.6
• Remark
• Example 3.1