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Long time dynamics of space periodic water waves

Massimiliano Berti

TL;DR

The work surveys long-time dynamics of space-periodic water waves, focusing on three pillars: the emergence of quasi-periodic solutions via KAM for 1d quasi-linear PDEs, long-time well-posedness through paradifferential Birkhoff normal form, and rigorous modulational instability analyses of Stokes waves. It introduces and integrates novel techniques—pseudo-differential KAM, paradifferential BNF, and symplectic Kato perturbation—to handle the quasi-linear, nonlocal Dirichlet–Neumann structure and reversible Hamiltonian nature of the water-wave equations. Key results include the existence of small-amplitude quasi-periodic standing and traveling waves, almost global-in-time stability for small data, and precise descriptions of modulational instabilities near Stokes waves (figure-8 near zero and high-frequency isolas). Collectively, these contributions illuminate how Hamiltonian structure and spectral methods govern the long-time dynamics of space-periodic water waves and point to important open problems in higher dimensions and broader parameter regimes.

Abstract

We review recent advances regarding the long-time dynamics of space-periodic water waves, focusing on 1) bifurcation of quasi-periodic solutions, both standing and traveling; 2) long-time well-posedness results; 3) modulational instability of Stokes waves. These results rely on unconventional approaches to KAM and Birkhoff normal form theories for Hamiltonian quasi-linear PDEs and symplectic Kato perturbation theory for separated eigenvalues of reversible and Hamiltonian operators.

Long time dynamics of space periodic water waves

TL;DR

The work surveys long-time dynamics of space-periodic water waves, focusing on three pillars: the emergence of quasi-periodic solutions via KAM for 1d quasi-linear PDEs, long-time well-posedness through paradifferential Birkhoff normal form, and rigorous modulational instability analyses of Stokes waves. It introduces and integrates novel techniques—pseudo-differential KAM, paradifferential BNF, and symplectic Kato perturbation—to handle the quasi-linear, nonlocal Dirichlet–Neumann structure and reversible Hamiltonian nature of the water-wave equations. Key results include the existence of small-amplitude quasi-periodic standing and traveling waves, almost global-in-time stability for small data, and precise descriptions of modulational instabilities near Stokes waves (figure-8 near zero and high-frequency isolas). Collectively, these contributions illuminate how Hamiltonian structure and spectral methods govern the long-time dynamics of space-periodic water waves and point to important open problems in higher dimensions and broader parameter regimes.

Abstract

We review recent advances regarding the long-time dynamics of space-periodic water waves, focusing on 1) bifurcation of quasi-periodic solutions, both standing and traveling; 2) long-time well-posedness results; 3) modulational instability of Stokes waves. These results rely on unconventional approaches to KAM and Birkhoff normal form theories for Hamiltonian quasi-linear PDEs and symplectic Kato perturbation theory for separated eigenvalues of reversible and Hamiltonian operators.
Paper Structure (16 sections, 5 theorems, 49 equations, 3 figures)

This paper contains 16 sections, 5 theorems, 49 equations, 3 figures.

Table of Contents

  1. The water waves equations.
  2. Linear water waves.
  3. Major questions.
  4. Quasi-periodic water waves.
  5. Quasi-periodic standing waves.
  6. Quasi-periodic Stokes waves.
  7. Ideas of proofs.
  8. Open problems.
  9. Almost global space periodic water waves.
  10. Ideas of proofs.
  11. Open problems.
  12. Modulational instability.
  13. Unstable spectral bands close to zero: the "figure 8".
  14. Ideas of proofs.
  15. Unstable spectral bands away from zero.
  16. ...and 1 more sections

Key Result

Theorem 1.1

(Quasi-periodic standing water waves BBHM) Consider finitely many "tangential" sites $\mathbb{S}^+ \subset \mathbb N$ and denote $\nu := | \mathbb{S}^+|$ its cardinality. Fix a subset $[\mathtt{h}_1, \mathtt{h}_2 ] \subset (0,+\infty)$. Then there exists $\bar{s} > 0$, $\varepsilon_0 \in (0,1)$ such In addition the quasi-periodic solutions QP:soluz0 are linearly stable.

Figures (3)

  • Figure 1: The gravity water waves problem. The velocity field inside the time dependent domain $\mathcal{D}_\eta$ is divergence free and with constant vorticity $\gamma$. Capillary forces may act at the free interface $y = \eta (t,x)$.
  • Figure 2: On the left, standing waves for irrotational fluids ($\gamma =0$); these solutions are even in $x$ at any time. On the right, quasi-periodic traveling Stokes waves with $\nu$ frequencies, see \ref{['def:QPT']}.
  • Figure 3.1: spectral bands with non zero real part of the $L^2 (\mathbb R)$-spectrum of $\mathcal{L}_{\varepsilon}$. The "figure 8" is present for $\mathtt{h} > 1.363...$. For any ${\mathtt p} \geq 2$, for any depth except a closed set, there exists an isola of instability with size $\propto | \beta_1^{(\mathtt{p})}(\mathtt{h})| \varepsilon^{{\mathtt p} }$. Its center $y_0^{({\mathtt p} )} (\varepsilon)$ is $\mathcal{O}(\varepsilon^2)$ distant from ${\rm i} \omega_*^{({\mathtt p} )}$. The notation $a \propto b$ means that $a b^{-1} \to c$ as $\varepsilon \to 0$.

Theorems & Definitions (7)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Remark 3.1
  • Theorem 3.2
  • Theorem 3.3: High frequency instabilities BCMV