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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.06511 |
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| _version_ | 1866909830912933888 |
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| author | Moston-Duggan, Aaron J. Howls, Christopher J. Lustri, Christopher J. |
| author_facet | Moston-Duggan, Aaron J. Howls, Christopher J. Lustri, Christopher J. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_06511 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Appearance of the higher-order Stokes phenomenon in a discrete Airy equation Moston-Duggan, Aaron J. Howls, Christopher J. Lustri, Christopher J. Mathematical Physics Complex Variables 30E, 30B, 39, 41 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. |
| title | Appearance of the higher-order Stokes phenomenon in a discrete Airy equation |
| topic | Mathematical Physics Complex Variables 30E, 30B, 39, 41 |
| url | https://arxiv.org/abs/2510.06511 |