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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.14090 |
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| _version_ | 1866915863199744000 |
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| author | Akers, Benjamin Creedon, Ryan P. |
| author_facet | Akers, Benjamin Creedon, Ryan P. |
| contents | We study the spectral stability of small-amplitude Stokes waves in a family of weakly nonlinear, unidirectional models of the form $u_t + L u + (u^2)_x = 0$. We introduce a perturbation method to expand the spectral data in wave amplitude near flat-state eigenvalue collisions, with the ratio of the colliding modes as a free parameter. This yields sheets of spectral data whose slices at fixed amplitude give isolas of instability. The same perturbation framework treats both high-frequency and Benjamin--Feir instabilities, extends to discontinuous dispersion relations (including the Akers--Milewski equation), and, for the first time, provides an analytic approximation of the Benjamin--Feir spectrum for this model and a direct comparison of high-frequency and Benjamin--Feir growth rates across the full family of models. Asymptotic predictions are validated against numerical spectra computed by Floquet--Fourier--Hill and quasi-Newton methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_14090 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sheets of Spectral Data of Stokes Waves in Weakly Nonlinear Models Akers, Benjamin Creedon, Ryan P. Analysis of PDEs We study the spectral stability of small-amplitude Stokes waves in a family of weakly nonlinear, unidirectional models of the form $u_t + L u + (u^2)_x = 0$. We introduce a perturbation method to expand the spectral data in wave amplitude near flat-state eigenvalue collisions, with the ratio of the colliding modes as a free parameter. This yields sheets of spectral data whose slices at fixed amplitude give isolas of instability. The same perturbation framework treats both high-frequency and Benjamin--Feir instabilities, extends to discontinuous dispersion relations (including the Akers--Milewski equation), and, for the first time, provides an analytic approximation of the Benjamin--Feir spectrum for this model and a direct comparison of high-frequency and Benjamin--Feir growth rates across the full family of models. Asymptotic predictions are validated against numerical spectra computed by Floquet--Fourier--Hill and quasi-Newton methods. |
| title | Sheets of Spectral Data of Stokes Waves in Weakly Nonlinear Models |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2603.14090 |