Appearance of the higher-order Stokes phenomenon in a discrete Airy equation

TL;DR

This paper analyzes a discrete, advance–delay variant of the Airy equation to show that discretization introduces a richer Stokes phenomenon than in the continuous case. Using steepest descent and factorial-over-power approaches, the authors derive a transseries representation built from an infinite family of saddle points and identify a virtual turning point and two primary turning points. The resulting Stokes structure features higher-order Stokes switching and an accumulation of Stokes/anti-Stokes curves, a nonlinear-like phenomenon arising in a linear discrete system. Numerical experiments corroborate the asymptotic predictions and highlight how discretization reshapes local Airy envelopes and global analytic structure. The work emphasizes that discrete linear systems can exhibit sophisticated asymptotic behavior traditionally associated with nonlinear or higher-order continuous problems.

Abstract

We study a discrete variant of the Airy equation, formulated as an advance-delay equation, to reveal that discretization induces the higher-order Stokes phenomenon, which is not present in the continuous Airy function and is typically only encountered in solutions to third-order or higher linear homogeneous, or nonlinear, differential equations. Using steepest descent and direct series methods, we derive asymptotic solutions and the Stokes structure. Our analysis shows that discretization produces a more intricate Stokes structure, containing higher-order Stokes phenomena and infinite accumulations of Stokes and anti-Stokes curves. The latter feature is a strictly nonlinear effect in continuous differential equations. We show that this unusual behavior can be generated in a discrete equation from a linear discretization. Numerical simulations confirm the predictions, and a direct comparison with the continuous Airy equation explains how the discretization alters the Stokes structure.

Figures (12)

• Figure 1: Numerical solutions $y_m$ of the discrete Airy equation \ref{['eqn:latAi']}, which decay as $x_m \to \pm \infty$, for several values of $\epsilon$ and $h$. Each solution is computed with $y_0 = 1$ at $x_0 = -2$ and rescaled so that the maximum value is $1$ for comparison. The solutions are oscillatory with slowly varying amplitude between the turning points at $x = 0$ and $x = 4$, and decay exponentially outside this region.
• Figure 2: Stokes structure of the $\mathop{\mathrm{Ai}}\nolimits$ solution to the Airy equation \ref{['eqn:spAi']}. Stokes (black) and anti-Stokes (red) curves emerge from the turning point at $x = 0$. In region $\mathcal{D}_1$, the solution has a single decaying exponential as $x \to \infty$ on the real axis. Crossing the anti-Stokes curve causes this contribution to grow instead of decay. Crossing the Stokes curve into $\mathcal{D}_2$ causes a second exponential contribution to appear, resulting in oscillatory behavior on the negative real axis.
• Figure 3: Stokes structure of the advance-delay Airy equation \ref{['eqn:soadvAi']} for $\sigma=1$, with turning points at $x = 0$ and $x = -4$ (gray circles). In region $\mathcal{D}_1$, contributions from upper saddle points $z_s^+$ are exponentially decaying on the positive real axis but become growing after crossing anti-Stokes curves (red). An infinite number of anti-Stokes curves accumulate near the real axis, each for a different $z_s^+$ (only first three shown). Similar behaviour occurs for lower saddle points $z_s^-$ along the negative real axis. Crossing Stokes curves (black) causes contributions to switch on and off. In $\mathcal{D}_2$, contributions from $z_s^+$ vanish and $z_s^-$ appear. In $\mathcal{D}_3$, all contributions appear. Higher-order Stokes curves (dashed blue) truncate the active Stokes curves at the Stokes crossing points (white circles).
• Figure 4: Steepest descent analysis schematics for the advance-delay Airy equation \ref{['eqn:soadvAi']} for $\sigma=1$, showing the integration path (white), saddle points $z_s^\pm$ (circles), and constant phase contours (dashed). Figures (a), (b), and (c) show typical schematics in regions $\mathcal{D}_1$, $\mathcal{D}_2$, and $\mathcal{D}_3$, respectively.
• Figure 5: Schematics of Stokes switching for the Stokes curves $\mathcal{S}_{s,s}^{-,+}$ with $\sigma = 1$. (a) Stokes structure schematic. Showing Stokes curves $\mathcal{S}_{s,s}^{-,+}$ (yellow), each emanate from the turning point $x = 0$. The legend otherwise matches that of Figure \ref{['fig:dAistokstruc']} with additional inactive Stokes curves (dotted yellow and black). Points ①--⑥ correspond to the steepest descent schematics shown in (b)--(g). (b)--(g) Steepest descent schematics, showing the integration path (white) and saddle points $z_s^\pm$ (circles).