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Smoothing of the higher-order Stokes phenomenon

Chris J. Howls, John R. King, Gergő Nemes, Adri B. Olde Daalhuis

TL;DR

The paper addresses the smoothness of the higher-order Stokes phenomenon in asymptotic expansions for functions with multiple Borel-plane singularities. It develops a rigorous framework in which the second hyperterminant $F^{(2)}$ induces a universal smoothing through a new special function $ ext{erfc}(x;y;oldsymbol{\lambda})$, a Gaussian-convolution of an error function, and provides uniform asymptotics that cover all relevant Stokes configurations. The authors verify the theory with multiple applications, including the gamma function and its reciprocal, a second-order nonlinear ODE, and the telegraph equation, revealing ghost-like smooth contributions that appear near Stokes lines but decay away. They also offer rigorous proofs, a detailed notation setup, and a pathway toward higher-order hyperterminant smoothing, suggesting a hierarchical structure of smoothing functions with potential physical interpretations in wave propagation and related fields.

Abstract

For nearly a century and a half the Stokes phenomenon had been perceived as a discontinuous change in the asymptotic representation of a function. In 1989 Berry demonstrated how it is possible to smooth out this discontinuity in broad classes of problems with the prefactor for the exponentially small contribution that is being switched on/off taking the universal form of an error function. Following pioneering work of Berk {\it et al.} \cite{BNR82} and the Japanese school of formally exact asymptotics \cite{Aokietal1994,AKT01}, the concept of the higher-order Stokes phenomenon was introduced in \cite{HLO04} and \cite{CM05}, whereby the ability for the exponentially small terms to cause a Stokes phenomenon may change, depending on the values of parameters in the problem, corresponding to the associated singularities in the Borel plane transitioning between different Riemann sheets. Until now, the higher-order Stokes phenomenon has also been treated as a discontinuous event. In this paper we show how the higher-order Stokes phenomenon is, in fact, also smooth and occurs universally with a prefactor that takes the form of a new special function, based on a Gaussian convolution of an error function that gives rise to a rich structure. We provide a rigorous derivation of the result, with examples spanning the gamma function, a second-order nonlinear ODE and the telegraph equation, giving rise to a ghost-like smooth contribution that is present in the vicinity of a Stokes line, but which rapidly tends to zero on either side. We also include a rigorous derivation and example of the effect of the smoothed higher-order Stokes phenomenon on the individual terms in the asymptotic series, where the additional contributions appear prefactored by an error function.

Smoothing of the higher-order Stokes phenomenon

TL;DR

The paper addresses the smoothness of the higher-order Stokes phenomenon in asymptotic expansions for functions with multiple Borel-plane singularities. It develops a rigorous framework in which the second hyperterminant induces a universal smoothing through a new special function , a Gaussian-convolution of an error function, and provides uniform asymptotics that cover all relevant Stokes configurations. The authors verify the theory with multiple applications, including the gamma function and its reciprocal, a second-order nonlinear ODE, and the telegraph equation, revealing ghost-like smooth contributions that appear near Stokes lines but decay away. They also offer rigorous proofs, a detailed notation setup, and a pathway toward higher-order hyperterminant smoothing, suggesting a hierarchical structure of smoothing functions with potential physical interpretations in wave propagation and related fields.

Abstract

For nearly a century and a half the Stokes phenomenon had been perceived as a discontinuous change in the asymptotic representation of a function. In 1989 Berry demonstrated how it is possible to smooth out this discontinuity in broad classes of problems with the prefactor for the exponentially small contribution that is being switched on/off taking the universal form of an error function. Following pioneering work of Berk {\it et al.} \cite{BNR82} and the Japanese school of formally exact asymptotics \cite{Aokietal1994,AKT01}, the concept of the higher-order Stokes phenomenon was introduced in \cite{HLO04} and \cite{CM05}, whereby the ability for the exponentially small terms to cause a Stokes phenomenon may change, depending on the values of parameters in the problem, corresponding to the associated singularities in the Borel plane transitioning between different Riemann sheets. Until now, the higher-order Stokes phenomenon has also been treated as a discontinuous event. In this paper we show how the higher-order Stokes phenomenon is, in fact, also smooth and occurs universally with a prefactor that takes the form of a new special function, based on a Gaussian convolution of an error function that gives rise to a rich structure. We provide a rigorous derivation of the result, with examples spanning the gamma function, a second-order nonlinear ODE and the telegraph equation, giving rise to a ghost-like smooth contribution that is present in the vicinity of a Stokes line, but which rapidly tends to zero on either side. We also include a rigorous derivation and example of the effect of the smoothed higher-order Stokes phenomenon on the individual terms in the asymptotic series, where the additional contributions appear prefactored by an error function.
Paper Structure (23 sections, 10 theorems, 215 equations, 15 figures)

This paper contains 23 sections, 10 theorems, 215 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Borel plane background
  3. The role of hyperterminants in encoding the Stokes and higher-order Stokes phenomena
  4. Summary of main results
  5. The smoothing when $\arg(\sigma_0 z)=\pi$
  6. The smoothing across $\arg\sigma_0=\arg\sigma_1$ when $\arg(\sigma_0 z)\not=\pi$
  7. Uniform approximation covering all three types of cases
  8. The smoothing across $\arg\sigma_0=\arg\sigma_1$ when $\arg(\sigma_0 z)=\pi$
  9. List of applications of the main results
  10. Application: The gamma function and its reciprocal
  11. Application: A second-order nonlinear ODE
  12. Application: The telegraph equation
  13. Application: Late-late coefficient smoothing
  14. Rigorous proofs of the higher-order Stokes smoothings
  15. Notation
  16. ...and 8 more sections

Key Result

Theorem 9.1

Assuming that $\sigma_0$ and its reciprocal are bounded, we have as $\operatorname{Re}(N_0)\to +\infty$, uniformly with respect to $\left|\arg\left({\rm e}^{-\pi{\rm i}}\sigma_0zN_0^{-1} \right)\right|\le 2\pi-\delta$ ($<2\pi$).

Figures (15)

  • Figure 1: Sketch of the Borel $t$-plane with singularities $\lambda_k, \lambda_j, j\ne k$ and the associated integration contour $\gamma_k({\bf a})$.
  • Figure 2: The smoothing of the ordinary Stokes phenomenon according to \ref{['localStokes1']}. The black curve represents $\left|\frac{{\rm e}^{-\sigma_0 z}}{2\pi{\rm i} z^{N_0}} F^{(1)}\left(z;\genfrac{}{}{0pt}{}{N_0+1}{\sigma_0}\right)\right|$, while the red curve shows $\left|\frac{{\rm e}^{-\sigma_0 z}}{2\pi{\rm i} z^{N_0}}F^{(1)}\left(z;\genfrac{}{}{0pt}{}{N_0+1}{\sigma_0}\right)- \frac{1}{2}\operatorname{erfc}\left(\alpha_0(z)\sqrt{\tfrac{1}{2}N_0}\right)\right|$, a measure of the error of the approximation, for $\sigma_0={\rm i}\sqrt2$, $N_0=30.3$ and $z=20{\rm e}^{{\rm i}\theta}$.
  • Figure 3: The three cases giving rise to higher-order Stokes phenomena in \ref{['F2']}. On the left diagram, the case $\arg(\sigma_0 z)=\pi$ generates a pole in \ref{['F2']} that leads to a (higher-order) Stokes phenomenon with connection formula \ref{['connect2']}. In the middle diagram, where $\arg\sigma_0=\arg\sigma_1$, a pole occurs in a $F^{(2)}$ hyperterminant and leads to a (higher-order) Stokes phenomenon with connection formula \ref{['connect3']}. In the right-hand diagram, where $\arg(\sigma_0 z)=\arg(\sigma_1 z)=\pi$ two poles simultaneously occur in a $F^{(2)}$ hyperterminant, leading to a combined (higher-order) Stokes phenomenon with the uniform approximation \ref{['simpledouble']}.
  • Figure 4: An example of the smoothing when $\arg(\sigma_0 z)=\pi$, based on \ref{['Stokes2a']}. The black curve depicts $\left|\frac{{\rm e}^{-\sigma_0 z}}{2\pi{\rm i} z^{N_0}} \frac{F^{(2)}\left(z;\genfrac{}{}{0pt}{}{N_0+1,}{\sigma_0,}\genfrac{}{}{0pt}{}{N_1+1}{\sigma_1}\right)}{F^{(1)}\left(z;\genfrac{}{}{0pt}{}{N_1+1}{\sigma_1}\right)}\right|$, and the red curve shows $\left|\frac{{\rm e}^{-\sigma_0 z}}{2\pi{\rm i} z^{N_0}} \frac{F^{(2)}\left(z;\genfrac{}{}{0pt}{}{N_0+1,}{\sigma_0,}\genfrac{}{}{0pt}{}{N_1+1}{\sigma_1}\right)}{F^{(1)}\left(z;\genfrac{}{}{0pt}{}{N_1+1}{\sigma_1}\right)}- \frac{1}{2}\operatorname{erfc}\left(\alpha_0(z)\sqrt{\tfrac{1}{2}N_0}\right)\right|$, a measure of the error of the approximation, for indicative values $\sigma_0={\rm i}\sqrt2$, $N_0=30.3$ , $\sigma_1={\rm i} -1$, $N_1=29$ and $z=20{\rm e}^{{\rm i}\theta}$.
  • Figure 5: An example of the smoothing across $\arg(\sigma_0)=\arg(\sigma_1)$, $\arg(\sigma_0 z)\not=\pi$, based on \ref{['simplesigma1int3']}. The black curve represents $\left|\frac{F^{(2)}\left(z;\genfrac{}{}{0pt}{}{N_0+1,}{\sigma_0,}\genfrac{}{}{0pt}{}{N_1+1}{\sigma_1}\right)}{2\pi{\rm i} F^{(1)}\left(z;\genfrac{}{}{0pt}{}{N_0+N_1+1}{\sigma_0+\sigma_1}\right)}\right|$, while the red curve shows $\left|\frac{F^{(2)}\left(z;\genfrac{}{}{0pt}{}{N_0+1,}{\sigma_0,}\genfrac{}{}{0pt}{}{N_1+1}{\sigma_1}\right)}{2\pi{\rm i} F^{(1)}\left(z;\genfrac{}{}{0pt}{}{N_0+N_1+1}{\sigma_0+\sigma_1}\right)}+ \frac{1}{2}\operatorname{erfc}\left(\gamma(\tfrac{\sigma_1}{\sigma_0})\sqrt{\tfrac{1}{2}N_1}\right)\right|$, a measure of the error of the approximation, for indicative values $\sigma_0=1.5{\rm e}^{{\rm i}\theta}$, $N_0=30.3$, $\sigma_1={\rm i}\sqrt2$, $N_1=29$ and $z=20$.
  • ...and 10 more figures

Theorems & Definitions (18)

  • Theorem 9.1
  • proof
  • Theorem 9.2
  • proof
  • Theorem 9.3
  • Theorem 9.4
  • Corollary 9.1
  • Lemma 9.1
  • Lemma 9.2
  • proof : Proof of Lemma \ref{['2F1Lemma']}
  • ...and 8 more