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On the spectral stability of periodic capillary-gravity waves

Changzhen Sun, Erik Wahlén

TL;DR

This work analyzes the spectral stability of small-amplitude periodic capillary-gravity waves in a finite-depth two-dimensional water-wave system. By performing a Bloch–Floquet reduction, the authors derive a finite-dimensional index $C(\alpha,\beta)$ that dictates stability: $C>0$ yields spectral stability for waves in region I∪III, while $C<0$ indicates potential instability. A key insight is the alignment of modulational and global stability analyses through a unified index, with surface tension (nonzero $\beta$) stabilizing the waves by ensuring identical Krein signatures away from the origin. The proof combines Kato perturbation theory, a careful construction of bases for the perturbed kernel, and a two-stage diagonalization of a 4×4 representation matrix to rigorously rule out instabilities away from the origin and to characterize modulational stability near origin. Overall, the results establish a rigorous link between a derived index and the stabilizing effect of surface tension on periodic capillary-gravity waves, providing a comprehensive spectral stability picture.

Abstract

In this paper, we investigate the spectral stability of periodic traveling waves in the two dimensional gravity-capillary water wave problem. We derive a stability criterion based on an index function, whose sign determines the spectral stability of the waves. This result aligns with earlier formal analyses by Djordjević \& Redekopp [15] and Ablowitz \& Segur [1], which employed the nonlinear Schrödinger approximation in the modulational regime. In particular, we show that instability is excluded near spectral crossings away from the origin when the surface tension is positive and the inverse square of the Froude number $α\in(0,1),$ which results from the fact that the corresponding Krein signatures are identical. It is also shown that there exists $α_1 = (23 - 3\sqrt{41})/8$ and a curve $β: (α_1, 1]\rightarrow \mathbb{R}_{+},$ such that for any $α\in (α_1, 1]$, small amplitude periodic waves are spectrally stable when $β> β(α)$. These findings highlight the stabilizing effect of surface tension on periodic capillary-gravity waves.

On the spectral stability of periodic capillary-gravity waves

TL;DR

This work analyzes the spectral stability of small-amplitude periodic capillary-gravity waves in a finite-depth two-dimensional water-wave system. By performing a Bloch–Floquet reduction, the authors derive a finite-dimensional index that dictates stability: yields spectral stability for waves in region I∪III, while indicates potential instability. A key insight is the alignment of modulational and global stability analyses through a unified index, with surface tension (nonzero ) stabilizing the waves by ensuring identical Krein signatures away from the origin. The proof combines Kato perturbation theory, a careful construction of bases for the perturbed kernel, and a two-stage diagonalization of a 4×4 representation matrix to rigorously rule out instabilities away from the origin and to characterize modulational stability near origin. Overall, the results establish a rigorous link between a derived index and the stabilizing effect of surface tension on periodic capillary-gravity waves, providing a comprehensive spectral stability picture.

Abstract

In this paper, we investigate the spectral stability of periodic traveling waves in the two dimensional gravity-capillary water wave problem. We derive a stability criterion based on an index function, whose sign determines the spectral stability of the waves. This result aligns with earlier formal analyses by Djordjević \& Redekopp [15] and Ablowitz \& Segur [1], which employed the nonlinear Schrödinger approximation in the modulational regime. In particular, we show that instability is excluded near spectral crossings away from the origin when the surface tension is positive and the inverse square of the Froude number which results from the fact that the corresponding Krein signatures are identical. It is also shown that there exists and a curve such that for any , small amplitude periodic waves are spectrally stable when . These findings highlight the stabilizing effect of surface tension on periodic capillary-gravity waves.

Paper Structure

This paper contains 24 sections, 26 theorems, 290 equations, 3 figures.

Table of Contents

  1. Introduction
  2. On the condition $C(\alpha, \beta)>0$ when $(\alpha,\beta)\in \rm I \cup III$
  3. Background on the stability of Stokes waves
  4. Strategy of the proof
  5. Spectral preliminaries
  6. Bloch transform
  7. Study of the constant operator $L_{\xi}^0$
  8. Exclusion of unstable spectrum with large imaginary part
  9. Stability near the crossings away from the origin
  10. Modulational stability
  11. Construction of the basis and dual basis
  12. Analysis of the representation matrix
  13. Computation of $\rm D_{\xi}^{0}$
  14. Some useful properties of the entries
  15. Transformation of the representation matrix
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $(\alpha,\beta)$ belong to the region $\rm I \cup \rm III$ defined in region-I-III, and $\kappa=\kappa(\alpha,\beta)$ be uniquely determined by the dispersion relation Let There exists a number $C=C(\alpha, \beta)$, given by such that sufficiently small-amplitude periodic waves $(\zeta_{\varepsilon}, \varphi_{\varepsilon})$ are spectrally stable if $C > 0$ and unstable if $C < 0$.

Figures (3)

  • Figure 1: The stability diagram in the $(\tilde{T},\kappa)$ plane with $\tilde{T}=\frac{\beta}{\alpha}\kappa^2$. The black curves are such that $C=0$. The cyan curve is where $w_1"(0)=0$. The purple curve is where $\sigma^2-\tilde{T}(3-\sigma^2)=0$ whereas the blue curve represents $e_{*}=\sqrt{\alpha}$. The green curve corresponds to $\alpha=1$. The region between the green curve and the right black curve indicates where the small-amplitude periodic waves are (globally) stable.
  • Figure 2: The curve $\beta=\beta_0(\alpha)$ (in blue) and the curve $\beta=\beta_1(\alpha)$ (in red) are such that $C(\beta,\alpha(\beta))=0$. The red curve diverges to $+\infty$ as $\alpha$ approaches $\alpha_1$. The region enclosed by these two curves and lying above $\beta_0(\alpha)$ corresponds to the domain where $C(\alpha,\beta)>0,$ which is defined as $I_1$.
  • Figure 3: The picture of curves $\sigma_{+}(k)$ (in blue) and $\sigma_{-}(k)$ (in red) when $\alpha=0.9,\beta=0.2$.

Theorems & Definitions (49)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1: Crossing of $L_{\xi}^0$
  • proof
  • Theorem 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 4.1
  • ...and 39 more