Stokes' phenomenon in continuous limits of discrete Painlevé I

TL;DR

This work investigates how Stokes' phenomenon manifests in discrete versions of Painlevé I by comparing finite-difference discretisations and the integrable first discrete Painlevé equation through exponential asymptotics. By formulating infinite-order differential equations that interpolate the discrete problems, the authors identify singulants and Stokes lines, quantify exponential switchings, and study inner problems near poles. They demonstrate that finite-K discretisations exhibit increasingly rich Stokes structures that grow with $K$, but the integrable discrete Painlevé I limit eliminates Stokes switching, consistent with the Painlevé property and preventing moveable singularities. A general Delta analysis reveals parameter regimes where Stokes effects vanish (e.g., special $oldsymbol{ riangle}$ values), highlighting a deep link between discrete integrability and continuous markers of integrability, with implications for discretisation choices in preserving qualitative asymptotics.

Abstract

We use exponential asymptotic analysis to identify the relevance of Stokes' phenomenon to integrability in discrete systems. We study Stokes' phenomenon in two discrete problems with the same (leading-order) continuous limit, a finite-difference discretisation of the first continuous Painlevé equation and the first discrete Painlevé equation, as well as a family of differential equation associated with each discrete problem. This analysis reveals two important observations. Firstly, the orderly behaviour that characterises Stokes' phenomenon in discrete equations emerges naturally from corresponding continuous differential equations as the order of the latter increases, although this is not apparent at low orders. Secondly, Stokes' phenomenon vanishes in the continuum limit of the integrable discrete equation, but not the non-integrable discrete equation. This means that subdominant exponentials do not appear in the integrable equation, and therefore do not cause moveable singularities to form in the solution. The results are clarified further by consideration of one-parameter family of difference equations that interpolates between the two considered in detail.

Figures (19)

• Figure 1: Pole locations for a tritronquée solution of the Painlevé I equation \ref{['e:P1']}. Poles are indicated by crosses, and are only found in the sector $\mathrm{arg}(z) \in (4\pi/5,6\pi/5)$. The five sectors of the solution are separated by dashed lines. Pole locations for the other tritronquée solutions may be found by rotating the pole locations by a multiple of $2\pi/5$.
• Figure 2: Approximation of $\Lambda$ obtained by evaluating \ref{['e:K2match']} at different values of $k$, denoted $\Lambda_{\mathrm{app}}$. The value of $\Lambda_{\mathrm{app}}$ obtained as $k$ becomes large is shown as a dashed line.
• Figure 3: The diagrams in (a)--(c) show local Stokes' phenomenon in $y(z)$ for $K=2$ generated by a singularity at $z = z_p$, with the branch cut set (a) horizontally, (b) vertically upwards, and (c) vertically downwards. The legend in each case is shown in (d). Figure (d) shows the combined Stokes structure due to multiple poles. Each pole generates vertical Stokes lines which switch on exponentially small contributions to the left of the Stokes line. There are no active anti-Stokes lines, so the series expansion \ref{['e:dPIseries']} is valid everywhere except on the branch cuts. There are no exponential contributions in the right-half plane (and hence the rightward extending anti-Stokes lines are inactive), as this Stokes structure corresponds to the unique asymptotic solution which is $\mathcal{O}(\sqrt{z})$ as $|z| \to \infty$ for $\mathrm{Re}(z) > 0$.
• Figure 4: The diagrams in (a)--(c) show local Stokes' phenomena in $y(z)$ for $K=3$ generated by a singularity at $z = z_p$ associated with (a) $\chi_1$ and $\chi_2$, (b) $\chi_3$ and $\chi_4$, and (c) the combined contribution of all four contributions. The legend for each is shown in (d). Figure (d) shows the combined Stokes structure due to multiple poles. Each pole generates Stokes lines that switch on exponentially small contributions; these contributions become large when anti-Stokes lines are crossed. The series expansion \ref{['e:dPIseries']} is no longer valid in the red-shaded regions.
• Figure 5: Values of $\chi'$ for different choices of $K$, denoted by circles in the complex $z$-plane. Figures (a)--(d) depict values for $K = 2$ to $5$ respectively. For even values of $K$, there must be at least one imaginary pair of solutions to preserve horizontal symmetry in the solutions. The number of solutions is equal to $2K-2$.