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<p>We study the spectral stability of small-amplitude Stokes waves in a family of weakly nonlinear, unidirectional models of the form <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(u_t + L u + (u^2)_x = 0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>u</mi> <mi>t</mi> </msub> <mo>+</mo> <mi>L</mi> <mi>u</mi> <mo>+</mo> <msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>u</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mi>x</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. 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.</p>

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Sheets of Spectral Data of Stokes Waves in Weakly Nonlinear Models

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年4月20日

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  • Benjamin Akers,
  • Ryan Creedon

摘要

We study the spectral stability of small-amplitude Stokes waves in a family of weakly nonlinear, unidirectional models of the form \(u_t + L u + (u^2)_x = 0\) 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.