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On higher order isolas of unstable Stokes waves

Massimiliano Berti, Livia Corsi, Alberto Maspero, Paolo Ventura

TL;DR

The paper analyzes high-frequency instability (isolas in the spectrum) of $2\pi$-periodic Stokes waves under longitudinal perturbations using a Bloch-Floquet decomposition. Instabilities away from the origin occur along isolas indexed by $p\ge 2$ and governed by the analytic function $\beta_1^{(p)}(h)$; prior work showed $\beta_1^{(p)}(h)$ is not identically zero via $\lim_{h\to 0^+} \beta_1^{(p)}(h) = -\infty$. Here, sharp deep-water asymptotics are derived for $p=2,3,4$, proving exponential decay of $\beta_1^{(p)}(h)$ to zero and providing explicit leading terms, which confirms nonvanishing behavior for large depth and suggests a finite number of zeros for these cases. These results sharpen the understanding of the high-frequency instability spectrum in deep water and support the existing framework of infinitely many isolas parameterized by $p\ge 2$, while leaving the general $p>4$ case analytically open.

Abstract

We overview the recent result [3, Theorem 1.1] about the high-frequency instability of Stokes waves subject to longitudinal perturbations. The spectral bands of unstable eigenvalues away from the origin form a sequence of {\it isolas} parameterized by an integer $ \mathtt{p} \geq 2 $ for any value of the depth $ \mathtt{h} > 0 $ such that an explicit analytic function $β_1^{(\mathtt{p})}(\mathtt{h}) $ is not zero. In [3] it is proved that the map $ \mathtt{h} \mapsto β_1^{(\mathtt{p})}(\mathtt{h}) $ is not identically zero for any $ \mathtt{p} \geq 2 $ by showing that $ \lim_{\mathtt{h} \to 0^+}β_1^{(\mathtt{p})}(\mathtt{h}) = - \infty $. In this manuscript we compute the asymptotic expansion of $β_1^{(\mathtt{p})}(\mathtt{h}) $ in the deep-water limit $ \mathtt{h} \to + \infty $ -- it vanishes exponentially fast to zero -- for $\mathtt{p}=2$, $3$, $4$.

On higher order isolas of unstable Stokes waves

TL;DR

The paper analyzes high-frequency instability (isolas in the spectrum) of -periodic Stokes waves under longitudinal perturbations using a Bloch-Floquet decomposition. Instabilities away from the origin occur along isolas indexed by and governed by the analytic function ; prior work showed is not identically zero via . Here, sharp deep-water asymptotics are derived for , proving exponential decay of to zero and providing explicit leading terms, which confirms nonvanishing behavior for large depth and suggests a finite number of zeros for these cases. These results sharpen the understanding of the high-frequency instability spectrum in deep water and support the existing framework of infinitely many isolas parameterized by , while leaving the general case analytically open.

Abstract

We overview the recent result [3, Theorem 1.1] about the high-frequency instability of Stokes waves subject to longitudinal perturbations. The spectral bands of unstable eigenvalues away from the origin form a sequence of {\it isolas} parameterized by an integer for any value of the depth such that an explicit analytic function is not zero. In [3] it is proved that the map is not identically zero for any by showing that . In this manuscript we compute the asymptotic expansion of in the deep-water limit -- it vanishes exponentially fast to zero -- for , , .
Paper Structure (5 sections, 5 theorems, 78 equations, 2 figures)

This paper contains 5 sections, 5 theorems, 78 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main result
  3. Case $\mathtt{p} =2$
  4. Case $\mathtt{p} =3$
  5. Case $\mathtt{p} =4$

Key Result

Theorem 1.1

(Unstable spectrum BCMV) For any integer $\mathtt{p} \in \mathbb{N}$, $\mathtt{p} \geq 2$, for any $\mathtt{h} >0$ let $\underline{\mu} = \underline{\varphi}(\mathtt{p},\mathtt{h}) > 0$ such that the operator ${\cal L}_{\underline{\mu},0}(\mathtt{h})$ in WW has a double eigenvalue at $\mathrm{i}\, and, for any $0 < \epsilon < \epsilon_1^{(\mathtt{p})}$, (see Figure fig:instareg) where $T_1^{(\m

Figures (2)

  • Figure 1: Spectral bands with non zero real part of the $L^2 (\mathbb{R})$-spectrum of ${\cal L}_{\epsilon}$. On the right, zoom of the $\mathtt{p}$-th isola of modulational instability. Its center $y_0^{({\mathtt p} )} (\epsilon)$ is $O(\epsilon^2)$ distant from $\mathrm{i}\, \omega_*^{({\mathtt p} )}$ and its size is $\propto \epsilon^{{\mathtt p} }$. As shown in Theorem \ref{['thm:main2']}, two eigenvalues $\lambda^{\pm}_\mathtt{p}(\mu,\epsilon)$ of the Floquet operator ${\cal L}_{\mu,\epsilon}$ span the $\mathtt{p}$-th isola for Floquet exponents $\mu \in (\mu_\wedge^{({\mathtt p} )}(\epsilon),\mu_\vee^{({\mathtt p} )}(\epsilon) )$ and recollide on the imaginary axis at the ends of the interval.
  • Figure 2: The instability region around the curve $\mu_0^{(\mathtt{p})}(\epsilon)$ delimited by the curves $\mu_\wedge^{(\mathtt{p})}(\epsilon)$ and $\mu_\vee^{(\mathtt{p})}(\epsilon)$.

Theorems & Definitions (9)

  • Theorem 1.1
  • Theorem 2.1
  • Conjecture 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof