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First isola of modulational instability of Stokes waves in deep water

Massimiliano Berti, Alberto Maspero, Paolo Ventura

Abstract

We prove high-frequency modulational instability of small-amplitude Stokes waves in deep water under longitudinal perturbations, providing the first isola of unstable eigenvalues branching off from $\mathtt{i}\frac34$. Unlike the finite depth case this is a degenerate problem and the real part of the unstable eigenvalues has a much smaller size than in finite depth. By a symplectic version of Kato theory we reduce to search the eigenvalues of a $2\times 2$ Hamiltonian and reversible matrix which has eigenvalues with non-zero real part if and only if a certain analytic function is not identically zero. In deep water we prove that the Taylor coefficients up to order three of this function vanish, but not the fourth-order one.

First isola of modulational instability of Stokes waves in deep water

Abstract

We prove high-frequency modulational instability of small-amplitude Stokes waves in deep water under longitudinal perturbations, providing the first isola of unstable eigenvalues branching off from . Unlike the finite depth case this is a degenerate problem and the real part of the unstable eigenvalues has a much smaller size than in finite depth. By a symplectic version of Kato theory we reduce to search the eigenvalues of a Hamiltonian and reversible matrix which has eigenvalues with non-zero real part if and only if a certain analytic function is not identically zero. In deep water we prove that the Taylor coefficients up to order three of this function vanish, but not the fourth-order one.
Paper Structure (8 sections, 18 theorems, 165 equations, 4 figures)

This paper contains 8 sections, 18 theorems, 165 equations, 4 figures.

Table of Contents

  1. Introduction and main result
  2. Perturbation of separated eigenvalues and instability criteria
  3. Abstract instability criteria.
  4. Taylor expansion of ${\cal B}(\mu,\epsilon)$, $P({\mu,\epsilon})$ and $\mathfrak{B}(\mu,\epsilon)$
  5. Entanglement coefficients
  6. Abstract representation formulas
  7. Entanglement coefficients for $p=2$
  8. Taylor expansion of $\mathtt{B}(\mu,\epsilon)$

Key Result

Theorem 1.1

There exist $\epsilon_1,\delta_0>0$ and real analytic functions $\mu_0,\mu_\pm \colon [0, \epsilon_1) \to B_{\delta_0}(\tfrac{1}{4})$ with $\mu_+(\epsilon)\leq \mu_0(\epsilon) \leq \mu_-(\epsilon)$ of the form such that for any $(\mu, \epsilon) \in B_{\delta_0}(\tfrac{1}{4})\times [0,\epsilon_1)$ the following holds true. Defining $\nu_\pm (\epsilon) := \mu_\pm (\epsilon) - \mu_0 (\epsilon)$, the

Figures (4)

  • Figure 1: Part of the spectrum of $\mathcal{L}_\epsilon$ in the deep-water case. The figure "8" was proved in BMV1. The two small symmetric isolas (detailed in Figure \ref{['figure0']}) are the subject of this article.
  • Figure 2: First isola depicted by the two symmetric non-purely imaginary eigenvalues of $\mathcal{L}_{\mu,\epsilon}$ close to $\mathrm{i}\,\tfrac{3}{4}$. The isola does not encircle $\mathrm{i}\, \frac{3}{4}$.
  • Figure 3: The degenerate instability region. We boxed with black dashed lines the validity box of Theorem \ref{['thm:main']}. At any fixed $\epsilon$, for $(\mu,\epsilon)$ in the colored cusp-shape region, one has the formation of the isola of instability in Figure \ref{['figure0']}.
  • Figure 4: The instability region around the curve $\mu_0(\epsilon)$ delimited by the curves $\mu_\vee(\epsilon)$ and $\mu_\wedge(\epsilon)$.

Theorems & Definitions (35)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 1.3: Multiple eigenvalues of $\mathcal{L}_{\mu,0}$ away from $0$
  • Theorem 1.4
  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.5
  • proof
  • ...and 25 more