Appearance of the higher-order Stokes phenomenon in a discrete Airy equation
Appearance of the higher-order Stokes phenomenon in a discrete Airy equation
We study a discrete variant of the Airy equation and show that discretization produces a more intricate Stokes structure than in the continuous case, inducing the higher-order Stokes phenomenon and infinite accumulations of Stokes and anti-Stokes curves. These features are absent in the continuous Airy equation and are typically seen only in solutions of at least third-order linear homogeneous, second-order or higher linear inhomogeneous, or nonlinear differential equations. Remarkably, this behavior is seen here to arise in a second-order homogeneous linear difference equation. Using exponential asymptotic methods, we derive the asymptotic solutions and the corresponding Stokes structure, with numerical simulations confirming our predictions. We conjecture that the higher order Stokes phenomenon is able to be present in other second order linear difference equations.
Higher-order Stokes Phenomenon, Exponential Asymptotics, Discrete Equations
Moston-Duggan, Aaron J.
9fb314cd-ee7c-4e31-a5c1-fd27ecb5811d
Howls, Christopher J.
66d3f0f0-376c-4f7a-a206-093935e6c560
Lustri, Christopher J.
c647ba7f-24e3-4ee3-b83f-08e9b32cfb58
23 January 2026
Moston-Duggan, Aaron J.
9fb314cd-ee7c-4e31-a5c1-fd27ecb5811d
Howls, Christopher J.
66d3f0f0-376c-4f7a-a206-093935e6c560
Lustri, Christopher J.
c647ba7f-24e3-4ee3-b83f-08e9b32cfb58
Moston-Duggan, Aaron J., Howls, Christopher J. and Lustri, Christopher J.
(2026)
Appearance of the higher-order Stokes phenomenon in a discrete Airy equation.
Journal of Physics A: Mathematical and Theoretical.
(doi:10.1088/1751-8121/ae31c3).
Abstract
We study a discrete variant of the Airy equation and show that discretization produces a more intricate Stokes structure than in the continuous case, inducing the higher-order Stokes phenomenon and infinite accumulations of Stokes and anti-Stokes curves. These features are absent in the continuous Airy equation and are typically seen only in solutions of at least third-order linear homogeneous, second-order or higher linear inhomogeneous, or nonlinear differential equations. Remarkably, this behavior is seen here to arise in a second-order homogeneous linear difference equation. Using exponential asymptotic methods, we derive the asymptotic solutions and the corresponding Stokes structure, with numerical simulations confirming our predictions. We conjecture that the higher order Stokes phenomenon is able to be present in other second order linear difference equations.
Text
Moston-Duggan+et+al_2025_J._Phys._A%3A_Math._Theor._10.1088_1751-8121_ae31c3
- Accepted Manuscript
Text
Moston-Duggan_2026_J._Phys._A__Math._Theor._59_045202
- Version of Record
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Accepted/In Press date: 29 December 2025
Published date: 23 January 2026
Keywords:
Higher-order Stokes Phenomenon, Exponential Asymptotics, Discrete Equations
Identifiers
Local EPrints ID: 508024
URI: http://eprints.soton.ac.uk/id/eprint/508024
ISSN: 1751-8113
PURE UUID: ebd2046d-cb7d-4f67-bea6-331983dad75c
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Date deposited: 09 Jan 2026 17:57
Last modified: 03 Feb 2026 02:37
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Contributors
Author:
Aaron J. Moston-Duggan
Author:
Christopher J. Lustri
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