Modulational Instability of Stokes Waves

• The paper demonstrates that modulational instability in Stokes waves is rigorously characterized through a detailed spectral and Floquet-Bloch analysis.
• It employs high-order asymptotic expansions to quantify growth rates and critical thresholds, revealing a distinctive figure-eight eigenvalue structure.
• The analysis contrasts low-frequency Benjamin–Feir modes with isolated high-frequency instability bands, offering insights into wave breaking and frequency downshift mechanisms.

A modulational (Benjamin–Feir) instability of Stokes waves refers to the spectral and nonlinear instability of small-amplitude, periodic traveling gravity waves with constant velocity—classically described as Stokes waves—under long-wavelength sideband perturbations. This phenomenon is central to the transition from stable periodic wavetrains to complex, modulated, and potentially extreme events in water waves, with mathematically rigorous characterizations now available in both finite and infinite depth regimes.

1. Governing Equations, Stokes Wave Branch, and Spectral Setup

Stokes waves are exact, spatially periodic traveling-wave solutions of the 1D irrotational, inviscid Euler equations with free surface and gravity. In nondimensional variables, the Eulerian formulation for the free-surface elevation $\eta(x,t)$ and surface potential $q(x,t)$ (in a frame moving at speed $c$) is: $$\int_{-\pi}^{\pi} e^{-imx}\left[ (\eta_t - c\eta_x)\cosh(m(\eta+\alpha)) + i q_x \sinh(m(\eta+\alpha))\right] dx = 0,$$

$$q_t - c q_x + \tfrac{1}{2} q_x^2 + \eta - \tfrac{1}{2} \frac{(\eta_t - c\eta_x + \eta_x q_x)^2}{1+\eta_x^2} = 0,$$

for $m \ne 0$, with $\alpha = \kappa h$ representing nondimensional depth.

A small-amplitude Stokes-wave solution $(\eta_S(x;\epsilon),q_S(x;\epsilon))$ is constructed as an analytic branch in the parameter $\epsilon \ll 1$ (wave steepness), with the characteristic expansion: $$\eta_S(x) = \epsilon \cos x + \epsilon^2(\cdots) + \epsilon^3(\cdots) + \cdots,$$ and similarly for $q_S(x;\epsilon)$ (Berti et al., 2023).

To probe stability, one linearizes about the Stokes wave, seeking perturbations of the form: $$\eta(x,t) = \eta_S(x) + \rho \eta_1(x,t) + O(\rho^2),\quad q(x,t) = q_S(x) + \rho q_1(x,t) + O(\rho^2),$$ and applies a Floquet-Bloch ansatz for sideband modes: $\begin{pmatrix} \eta_1 \ q_1 \end{pmatrix}(x,t) = e^{\lambda t + i \mu x} \begin{pmatrix} N(x) \ Q(x) \end{pmatrix} + \text{c.c.},\qquad N, Q \text{ %%%%8%%%%-periodic in %%%%9%%%%},$ where $\mu \in (-\tfrac{1}{2}, \tfrac{1}{2}]$ is the Floquet exponent (sideband detuning) (Creedon et al., 2022, Berti et al., 2023).

The linearized spectral problem is of the form: $$L_{\mu,\epsilon}[N, Q] = \lambda R_{\mu,\epsilon}[N, Q],$$ and the solution spectrum $\{\lambda(\mu, \epsilon)\}$ encodes growth rates of perturbations.

2. High-Order Asymptotics and Structure of the Unstable Spectrum

For $\epsilon \ll 1$ and $\mu = O(\epsilon)$, the unstable eigenvalues near the origin admit an asymptotic expansion (Creedon et al., 2022): $$\lambda(\mu, \epsilon) = \epsilon^2 \lambda_2(\mu_1) + \epsilon^3 \lambda_3(\mu_1) + \epsilon^4 \lambda_4(\mu_1) + O(\epsilon^5),$$ with $$\mu = \epsilon \mu_1(1+ r_1 \epsilon + r_2 \epsilon^2 + ...)$$, $\mu_1 \in (0, M)$, and coefficients explicitly computable to any formal order.

At leading order, for finite depth $\alpha > \alpha_{BW} = 1.3627827...$, one finds: $$\operatorname{Re} \lambda = \epsilon^2 \left(\frac{\mu_1}{8}\right) \Delta_{BW}(\mu_1, \alpha) + \epsilon^4 \Gamma_4(\mu_1, \alpha) + O(\epsilon^6),$$ where

$$\Delta_{BW}(\mu_1, \alpha) = \sqrt{e_2(\alpha)[8e_{BW}(\alpha) - e_2(\alpha)\mu_1^2]}$$

and $e_{BW}>0$ precisely for $\alpha > \alpha_{BW}$ (Creedon et al., 2022).

In infinite depth ($\alpha \to \infty$), this simplifies to: $$\operatorname{Re} \lambda = \epsilon^2 \frac{\mu_1}{8} \sqrt{8 - \mu_1^2} + O(\epsilon^3)$$ for $\mu_1 \in (0, 2\sqrt{2})$.

Elimination of $\mu_1$ at leading order in $$(\operatorname{Re} \lambda, \operatorname{Im} \lambda)$$ yields the classical lemniscate ("figure-eight") in the complex spectral plane, parametrized by the Floquet exponent: $$64 c_g^4 (\operatorname{Re} \lambda)^2 = e_2 (\operatorname{Im} \lambda)^2 [8 e_{BW} \epsilon^2 c_g^2 - e_2 (\operatorname{Im} \lambda)^2],$$ with $c_g$ the group velocity at $k=1$. This figure-eight structure is a hallmark of the modulational instability band (Creedon et al., 2022).

3. Transition Thresholds and Critical Depth: Benjamin-Feir Boundary

Modulational instability is tied to the sign of a depth-dependent Benjamin–Feir (BF) discriminant. Introducing a detailed block-diagonalization and high-order expansion, one finds that for finite depth, the transition occurs at a critical nondimensional depth $\alpha_{BW}$ (Bridges–Mielke threshold) (Berti et al., 2023):

• For $h > h_{WB}=1.363...$, the Stokes wave is modulationally unstable.
• For $h < h_{WB}$, all four small Floquet eigenvalues remain on the imaginary axis (spectral stability).

Near critical depth, a higher-order degeneracy emerges: the BF discriminant is of order $\epsilon^8 - e_{22}\mu^2/8 + ...$, so the width of the unstable region in $\mu$ shrinks as $O(\epsilon^2)$ at $h=h_{WB}$. This requires resolving terms up to fourth order in the Stokes wave expansion.

The instability condition thus forms a stability–instability boundary at: $$h > h_{WB} - c\epsilon^2 + O(\epsilon^3), \quad c > 0$$ with $c$ explicitly computable (Berti et al., 2023).

4. Comparison with High-Frequency Instabilities and Infinite Isolas

Beyond the low-frequency (sideband) instability, the linear operator admits high-frequency (HF) collisions—isolated double eigenvalues at fixed spectral points (isolas). For each integer $p \geq 2$, there is a family of high-frequency instability bands with size $O(\epsilon^p)$ (Berti et al., 26 Jan 2024, Berti et al., 9 May 2024):

• Each isola arises from a collision of branches of opposite Krein signature at a double eigenvalue $\lambda_*^{(p)}(h)$ with Floquet exponent $\mu = (p,h)$ (unique per depth),
• The unstable spectrum contains infinitely many such isolas in finite depth, with the real part scaling as $|\beta_1^{(p)}(h)|\epsilon^p + O(\epsilon^{p+1})$ for $p$th isola (with analytic, depth-dependent coefficient $\beta_1^{(p)}(h)$) (Berti et al., 9 May 2024).
• In infinite depth, the first isola (centered at $\operatorname{Im} \lambda = 3/4$) is much narrower than in finite depth ($O(\epsilon^4)$ in real part, $O(\epsilon^2)$ in imaginary part) (Berti et al., 26 Jan 2024).

For depths just above the BF threshold, the Benjamin–Feir mode dominates ($\sim \epsilon^2$), but as depth increases, there exist parameter regimes where HF growth can compete with or even exceed the BF growth rate (Creedon et al., 2022).

5. Phase Dynamics, Whitham Modulation, and Frequency Downshift Mechanisms

The modulational instability is classically predicted at the PDE level by a cubic nonlinear Schrödinger (NLS) envelope equation for the wave envelope $A(X,T)$,

$$iA_T + \frac12 \omega''(k)A_{XX} + \gamma |A|^2A = 0,$$

with the sign of $\omega''(k)\gamma$ governing stability (Deconinck et al., 2022). The instability criterion is typically $\omega''(k)\gamma < 0$. The Lighthill–Whitham criteria generalize to arbitrary depth or extended models via:

• Hyperbolicity of the Whitham system (real characteristics: marginally stable)
• Ellipticity (complex characteristics: modulational instability), with the transition at the physical BF threshold (Johnson et al., 27 May 2025).

In the vicinity of the BF transition, quadratic nonlinearities in the phase dynamics vanish, necessitating a higher-order, Boussinesq-type normal form with leading cubic nonlinearity (Ratliff et al., 22 Oct 2024). This equation supports explicit heteroclinic fronts connecting stable and unstable wavetrain families, resulting in traveling phase kinks and permanent frequency/wavenumber downshift, even in the absence of dissipation—a mechanism confirmed by both asymptotic and numerical approaches (Ratliff et al., 22 Oct 2024).

6. Nonlinear and Fully Nonlinear Effects

Rigorous analyses have demonstrated that the linear spectral instability mechanism (Benjamin–Feir) extends to nonlinear instability at the level of the two-dimensional water-wave equations:

• For sufficiently small $\epsilon$, all (finite/infinite depth) Stokes waves are nonlinearly unstable to sideband modulations, leading to $O(1)$ deviation over timescales $t \sim \epsilon^{-2} \log(\mu/\delta)$ (Chen et al., 2020).
• For large steepness (near-extreme Stokes waves), new, rapidly growing, localized (crest-trapped) instabilities dominate over the classical BF mechanism, ultimately causing wave breaking (Deconinck et al., 2022).
• The fully nonlinear Stokes expansion (Wilton representation) supports modulational instability in every harmonic $n$, with instability bands $0 < \delta_n < \sqrt{2} k n^n s_n/(n-1)!$ shrinking with $n$, typically dominated by $n=1$ (Sajjadi et al., 2017).

7. Generalizations, Models, and Physical Interpretation

The fundamental modulational instability mechanism persists in a variety of model settings:

• Whitham-type, full-dispersion, and Camassa–Holm equations reproduce the BF threshold and instability growth rates more accurately than long-wave models (KdV, BBM), though with slightly shifted critical wavenumbers due to model-dependent dispersion (Hur et al., 2016, Hur et al., 2013, Hur et al., 2017).
• Nonlinear Schrödinger envelope reductions capture higher-order and higher-sideband cascades, with experiments and numerics confirming the triangular sideband cascade and recurrence phenomena in water waves (Kimmoun et al., 2017).
• In conservative settings, the frequency downshift seen near the BF transition is intrinsic to the phase-geometry of the modulated wavetrain and not reliant on viscous or wind-driven mechanisms (Ratliff et al., 22 Oct 2024).
• In shallow water or rotating fluids (Ostrovsky equation), the criterion for the onset of BF instability is determined via the sign of $\omega_0''(k) \omega_2(k)$, with a critical wavenumber depending on rotation and shallowness (Johnson et al., 27 May 2025, Marin et al., 4 Mar 2025).

Summary Table: Key Regimes and Instability Behaviors

Depth Regime	Instability Mechanism	Instability Thresholds	Max Growth Rate Scaling
Infinite Depth	BF (modulational, $p=1$)	None: always unstable	$O(\epsilon^2)$
Finite Depth	BF (modulational, $p=1$)	$\alpha > \alpha_{BW} \approx 1.3627$	$O(\epsilon^2)$
Finite/Infinite	High-frequency “isolas”, $p\ge2$	At each $(p,h)$ (discrete band centers)	$O(\epsilon^p)$
Near BF threshold	Phase front, energetic downshift	$h \gtrsim \alpha_{BW}$	Downshift front, geometric
High Steepness	Localized crest instability	Steepness $s > s_2$	$>O(\epsilon^2)$

In all regimes, the modulational (Benjamin–Feir) instability is robustly observed for Stokes waves of small amplitude in infinite and sufficiently deep finite depth, with precise quantitative criteria and spectrum now available. The spectral bands generically take closed (figure-eight/ellipse) shapes in the complex plane, with the low-frequency band dominating at small amplitude and depth, and infinite further high-frequency isolas present throughout the spectrum in both finite and infinite depth (Creedon et al., 2022, Berti et al., 9 May 2024, Berti et al., 2023).