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On higher order isolas of unstable Stokes waves

This paper computes the asymptotic expansion of the analytic function $\beta_1^{(\mathtt{p})}(\mathtt{h})$ governing high-frequency instability isolas of Stokes waves in the deep-water limit for $\mathtt{p}=2,3,4$ , demonstrating that the function vanishes exponentially fast as depth approaches infinity.

Original authors: Massimiliano Berti, Livia Corsi, Alberto Maspero, Paolo Ventura

Published 2026-07-28

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CC BY 4.0

Original authors: Massimiliano Berti, Livia Corsi, Alberto Maspero, Paolo Ventura

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the ocean not as a chaotic, churning mess, but as a giant, rhythmic drum. When you strike it, waves travel across the surface. Some of these waves are perfect, repeating patterns known as "Stokes waves." They are the idealized, mathematical version of a swell that rolls endlessly without changing shape. For over a century, scientists have been obsessed with one question: Are these perfect waves stable? If you poke them slightly, do they bounce back to their perfect shape, or do they unravel into chaos?

The answer is surprisingly complex. In the 1960s, researchers discovered that if the water is deep enough, these waves can become unstable when poked by long, gentle ripples. This is called the "Benjamin-Feir instability," and it's like a perfectly balanced spinning top that suddenly starts wobbling and falling over. But in the 1980s, a new mystery emerged. Scientists found that even in water of any depth, these waves can also become unstable when poked by very specific, high-frequency ripples. These instabilities don't look like a single wobbling top; instead, they appear as tiny, isolated islands of chaos floating in a sea of stability. In the language of mathematics, these are called "isolas." They are like secret, microscopic whirlpools that only appear under very precise conditions.

This paper, written by Massimiliano Berti, Livia Corsi, Alberto Maspero, and Paolo Ventura, dives deep into the mathematics of these "islands of instability." Specifically, they investigate what happens to these islands when the water becomes infinitely deep—the "deep-water limit." Previous work had shown that these islands exist and that they are incredibly small, shrinking rapidly as the wave gets more complex. However, a crucial piece of the puzzle was missing: exactly how do these islands behave as the water gets deeper and deeper? Do they vanish completely? Do they wiggle around before disappearing? The authors set out to calculate the precise mathematical formula that describes this behavior for the first few types of these islands.

The team focused on the first three "higher-order" islands (labeled $p=2, 3,$ and $4$). Think of these as the first, second, and third most complex types of these instability islands. Using a mix of rigorous hand-calculated math and powerful computer algebra software (Mathematica), they derived a new formula for a key number, $\beta_1^{(p)}(h)$ , which acts like a "thermometer" for the size of these islands. Their main finding is a bit of a shocker: as the depth of the water ( $h$ ) goes to infinity, these islands don't just shrink; they vanish exponentially fast.

The authors proved that for the first three types of islands, the size of the instability shrinks at a rate proportional to $e^{-h/2}$ , $e^{-2h}$ , and $e^{-2h}$ respectively. To put this in perspective, if you double the depth of the ocean, the size of these instability islands doesn't just get half as big; it gets astronomically smaller, shrinking by a factor of millions or billions depending on the depth. This confirms that in truly deep water, these specific high-frequency instabilities become almost non-existent, vanishing so quickly they are practically invisible.

Crucially, the paper also addresses a potential worry. Because these islands are so small, one might wonder if they wiggle up and down, crossing zero size and reappearing infinitely many times as the water gets deeper. The authors' calculations show that for the first three types of islands ( $p=2, 3, 4$ ), this does not happen. The function describing their size approaches zero smoothly and steadily, without oscillating. This suggests that there are only a finite number of depths where these islands might completely disappear (where the function hits exactly zero) before vanishing forever in the deep. While the paper proves this non-oscillatory behavior for the first three cases, the authors admit that for even more complex islands ( $p > 4$ ), the math gets so tangled that the answer remains an open question for now, though they conjecture the same smooth behavior holds true.

In short, this paper provides a precise map of how these hidden islands of chaos behave in the deepest parts of the ocean. It tells us that while the ocean is full of complex dynamics, the specific high-frequency instabilities that plague shallower waters essentially disappear into the abyss as the depth increases, shrinking away with mathematical elegance and speed.

Technical Summary: On Higher Order Isolas of Unstable Stokes Waves

Problem Statement
The paper addresses the spectral stability of Stokes waves, which are periodic traveling wave solutions to the gravity water waves equations. While the instability of these waves under long-wave perturbations (Benjamin-Feir or modulational instability) is well-understood, the focus here is on high-frequency instabilities. These manifest as "isolated" spectral bands (isolated from the continuous spectrum and the origin) in the linearized operator governing the water waves.

Previous work by Deconinck and Oliveras numerically identified a sequence of these unstable isolas, parameterized by an integer $p \ge 2$ , which appear as small loops in the complex plane centered on the imaginary axis. Their size scales as $O(\epsilon^p)$ , where $\epsilon$ is the wave amplitude. A rigorous mathematical proof of the existence of these isolas for all $p \ge 2$ was recently established in [3] (Berti, Maspero, Ventura). However, a critical component of this existence proof relies on a coefficient function, $\beta^{(p)}_1(h)$ , which determines the width of the unstable band. While [3] proved that $\beta^{(p)}_1(h)$ is not identically zero by analyzing the shallow-water limit ( $h \to 0^+$ ), the behavior of this function in the deep-water limit ( $h \to +\infty$ ) remained an open question. Specifically, it was known that $\lim_{h \to +\infty} \beta^{(p)}_1(h) = 0$ , but the rate of decay and the precise asymptotic behavior were unknown.

Methodology
The authors employ a rigorous asymptotic analysis of the coefficient $\beta^{(p)}_1(h)$ as the water depth $h$ tends to infinity. The methodology involves:

Linearization and Spectral Analysis: Utilizing the Hamiltonian formulation of the water waves equations and the Bloch-Floquet decomposition to analyze the spectrum of the linearized operator $L_{\mu, \epsilon}(h)$ .
Perturbation Theory: Expanding the eigenvalues of the operator near the double eigenvalues of the unperturbed system ( $\epsilon = 0$ ). The existence and shape of the $p$ -th isola depend on the discriminant of a characteristic equation, where $\beta^{(p)}_1(h)$ appears as a leading-order coefficient.
Asymptotic Expansion: The core of the work involves deriving the asymptotic expansion of $\beta^{(p)}_1(h)$ $β_{1}^{(p)} (h)$ for $h \to +\infty$ $h \to + \infty$ . This requires:
- Expanding the wavenumber $\phi(p, h)$ where the unperturbed spectral bands intersect.
- Expanding the Taylor-Fourier coefficients of the Stokes wave profile ( $a^{[\ell]}_\ell, p^{[\ell]}_\ell$ ) in powers of $e^{-2h}$ .
- Systematically expanding the complex algebraic expressions defining $\beta^{(p)}_1(h)$ , which are sums of many terms involving hyperbolic functions and square roots.
Computational Assistance: For $p=2$ , the expansions are derived analytically. For $p=3$ and $p=4$ , the algebraic complexity increases significantly (involving 9 and 27 terms, respectively). The authors utilize the computer algebra system Mathematica to perform these high-order expansions and verify the cancellations required to obtain the leading-order terms.

Key Contributions and Results
The primary contribution of this manuscript is the computation of the sharp asymptotic expansion of the coefficient $\beta^{(p)}_1(h)$ for $p=2, 3, 4$ in the deep-water limit. The main result, Theorem 2.1, establishes that:

Exponential Decay: The coefficient $\beta^{(p)}_1(h)$ vanishes exponentially fast as $h \to +\infty$ .
Explicit Leading Terms:
- For $p=2$ : $\beta^{(2)}_1(h) = \frac{3\sqrt{3}}{64} e^{-h/2} + O(e^{-3h/4})$ .
- For $p=3$ : $\beta^{(3)}_1(h) = \frac{2\sqrt{2}}{3} e^{-2h} + O(e^{-3h})$ .
- For $p=4$ : $\beta^{(4)}_1(h) = -\frac{5\sqrt{5}}{8\sqrt{3}} e^{-2h} + O(e^{-4h})$ .

These results confirm that $\beta^{(p)}_1(h)$ is not identically zero for these values of $p$ , thereby providing an alternative proof of the existence of the corresponding instability isolas in deep water. Furthermore, the explicit form of the expansion rules out the possibility of infinitely many zeros of $\beta^{(p)}_1(h)$ accumulating at infinity.

Significance and Claims
The paper positions itself as a complement to the recent existence proof in [3]. Its significance lies in:

Completing the Deep-Water Picture: It resolves the asymptotic behavior of the instability coefficient in the deep-water limit, showing that while the isolas shrink exponentially fast (making them harder to detect numerically as $h$ increases), they theoretically persist for $p=2, 3, 4$ .
Structural Insight: The derivation highlights "astonishing structural properties" of the Taylor-Fourier coefficients of Stokes waves, where subtle cancellations occur in the deep-water limit to produce the specific exponential decay rates.
Conjecture: Based on the results for $p=2, 3, 4$ , the authors formulate Conjecture 2.2, stating that for any integer $p \ge 2$ , the function $h \mapsto \beta^{(p)}_1(h)$ has a finite number of zeros in $(0, +\infty)$ . This implies that the instability isolas exist for all but a finite set of depth values.

The authors explicitly note that for $p > 4$ , the problem remains analytically open due to the dramatic increase in computational complexity, though the methodology remains applicable. The paper does not propose new physical applications or experimental setups but strictly advances the rigorous mathematical understanding of the spectral structure of Stokes waves.

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