Sheets of Spectral Data of Stokes Waves in Weakly Nonlinear Models

Abstract

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.

Figures (7)

• Figure 1: Left: An isola of a triad instability in the deep water ($h\rightarrow\infty$) gravity-capillary Whitham equation near $(\lambda_0,p_0)\approx(-0.0608i,0.2681)$ with $\sigma=2.5$ at $\varepsilon=10^{-3}$. The circles are computed with a quasi-Newton iteration; the curves are the asymptotic prediction of \ref{['TriadLam1']}. Right: An isola of a triad instability in the Akers--Milewski equation near $(\lambda_0,p_0)=(0.3536i,0.1464)$ with $\sigma=2$ at $\varepsilon=10^{-3}$. The circles are computed with a quasi-Newton iteration; the curves are the asymptotic prediction of \ref{['TriadLam1']}.
• Figure 2: The surface of spectral data predicted by \ref{['TriadLam1']} in the deep water ($h\rightarrow\infty$) gravity-capillary Whitham equation is visualized. The spectrum bifurcated from $(\lambda_0,p_0)\approx(-0.0608i,0.2681)$ with $\sigma=2.5$. This is the same configuration which is compared to numerical predictions in Figure \ref{['TriadIsolaFig']}.
• Figure 3: Left:The surface of spectral data predicted by \ref{['QuartetLam2']} in the Kawahara equation is visualized. The spectrum bifurcates from $(\lambda_0,p_0)\approx(0.2277i,0.3675)$ with $(a,b)=(1,-0.25)$. This is the same configuration which is compared to numerical predictions in Figure \ref{['KawaharaQuartet']}. Right The surface of spectral data predicted by \ref{['QuartetLam2']} in the deep water gravity-capillary Whitham equation is visualized. The spectrum bifurcates from $(\lambda_0,p_0)\approx(0.2177i,0.1363)$ with $\sigma=0.25$. This is the same configuration which is compared to numerical predictions in Figure \ref{['WhithamQuartet']}.
• Figure 4: Left: An isola of a quartet instability in the Kawahara equation near $(\lambda_0,p_0)=(0.2277i,0.3675)$ with $(a,b)=(1,-0.25)$ at $\varepsilon=10^{-3}$. The circles are computed with a quasi-Newton iteration; the curves are the second order asymptotic prediction of \ref{['QuartetLam2']}. Center: The imaginary part of the most unstable eigenvalue of \ref{['QuartetLam2']} (solid line) against numerical computations from a quasi-Newton iteration (circles). Right: The error between the numerically computed and asymptotics \ref{['QuartetLam2']} (starred line); the solid line marks $y=6\varepsilon^3$.
• Figure 5: Left: An isola of a quartet instability in the deep-water gravity-capillary Whitham equation near $(\lambda_0,p_0)=(0.2177i,0.1363)$ with $\sigma=0.25$ at $\varepsilon=10^{-4}$. The circles are computed with a quasi-Newton iteration; the curves are the second order asymptotic prediction of \ref{['QuartetLam2']}. Center: The imaginary part of the most unstable eigenvalue of \ref{['QuartetLam2']} (solid line) against numerical computations from a quasi-Newton iteration (circles). Right: The error between the numerically computed and asymptotics \ref{['QuartetLam2']} (starred line); the solid line marks $y=10^3\varepsilon^3$.