We establish novel existence results of 3d gravity-capillary periodic traveling waves. In particular we prove the bifurcation of multiple, geometrically distinct truly 3d Stokes waves having the same momentum of any non-resonant 2d Stokes wave. This unexpected clustering phenomenon of Stokes waves, observed in physical fluids, is a fundamental consequence of the Hamiltonian nature of the water waves equations, their symmetry groups, and novel topological arguments. We employ a variational Lyapunov-Schmidt reduction combined with equivariant Morse-Conley theory for a functional defined on a joined topological space invariant under a 2-torus action. Although the reduction is a priori singular near the hyperplanes of 2d-waves, we circumvent this difficulty by exhaustive use of the symmetry groups. This approach yields a complete bifurcation picture of 3d gravity-capillary Stokes waves.

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.

The spectral instability of traveling periodic water waves has been investigated for more than sixty years, since the seminal discovery of Benjamin and Feir. Despite an extensive literature, no rigorous theory has been available for arbitrary three-dimensional – longitudinal and transverse – perturbations. We establish the first rigorous description of the 3d unstable spectrum of small-amplitude gravity Stokes waves in deep water in a full neighborhood of the McLean resonant curves. Our results reveal that the Benjamin-Feir instability and the first longitudinal high-frequency isola originate from the same resonant interaction, hidden in the purely longitudinal setting. The dominant instabilities emerge for Fourier-Bloch parameters near the origin, corresponding to the 3d Benjamin-Feir modulational instability. Our approach provides quantitative bounds for the real parts of the unstable eigenvalues and establishes a computable necessary and sufficient criterion for the onset of instability near arbitrary high-frequency McLean curves. These results are enabled by three key innovations: (i) a Kato perturbative analysis allowing Lipschitz-type singularities of the linearized operator with respect to the Fourier-Bloch parameters; (ii) a polar-analytic KAM-type decoupling isolating the unstable eigenvalue pairs near the origin; and (iii) an analytic continuation argument in full neighborhoods of the McLean curves. A primary challenge is to establish fine regularity properties for the Dirichlet-Neumann operator conjugated via the Fourier-Bloch transform.

We prove existence and multiplicity of Cantor families of small amplitude time periodic solutions of completely resonant Klein-Gordon equations on the sphere $\mathbb{S}^3$ with quadratic, cubic and quintic nonlinearity, regarded as toy models in General Relativity. The solutions are obtained by a variational Lyapunov- Schmidt decomposition, which reduces the problem to the search of mountain pass critical points of a restricted Euler-Lagrange action functional. Compactness properties of its gradient are obtained by Strichartz-type estimates for the solutions of the linear Klein-Gordon equation on $\mathbb{S}^3$.

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 \mathrm{i}\frac{3}{4} . 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 nonzero 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.

Rogue waves are extreme ocean events characterized by the sudden formation of anomalously large crests, and remain an important subject of investigation in oceanography and mathematics. A central problem is to quantify the probability of their formation under random Gaussian sea initial data. In this work, we rigorously characterize the tail probability for the formation of rogue waves of the pure gravity water wave equations in deep water, the most accurate quasilinear PDE modeling waves in open ocean. This large deviation result confirms various conjectures from the oceanography literature in the weakly nonlinear regime. Moreover, the result holds up to the optimal timescales allowed by deterministic well-posedness theory. The proof shows that rogue waves most likely arise through "dispersive focusing", where phase quasi-synchronization produces constructive amplification of the water crest. The main difficulty in justifying this mechanism is propagating statistical information over such long timescales, which we overcome by combining normal forms and probabilistic methods. Unlike prior work, this novel approach does not require approximate solutions to be Gaussian. Our general method tracks the tail probability of solutions to Hamiltonian PDEs with an integrable normal form and random Gaussian initial data, even in the absence of (quasi-)invariant measures.

We overview the recent result [3, Theorem 1.1] about the high-frequency instability of pure gravity Stokes waves subject to longitudinal perturbations. The spectral bands of unstable eigenvalues away from the origin form a sequence of isolas parameterized by an integer ≥ 2 for any value of the depth > 0 such that an explicit coefficient β _1^()() is not zero. In [3] it is proved that the map ↦β _1^()() is analytic and it is not identically zero for any ≥ 2 , by showing that lim _→ 0^+β _1^()() = - ∞ . In this manuscript we compute the asymptotic expansion of β _1^()() in the deep-water limit → + ∞ –it vanishes exponentially fast to zero– for =2 , 3, 4.

In this paper we paralinearize the contour dynamics equation for sharp-fronts of $\alpha$-SQG, for any $ \alpha \in (0,1) \cup (1,2) $, close to a circular vortex. This turns out to be a quasi-linear Hamiltonian PDE. After deriving the asymptotic expansion of the linear frequencies of oscillations at the vortex disk and verifying the absence of three wave interactions, we prove that, in the most singular cases $ \alpha \in (1,2) $, any initial vortex patch which is $ \varepsilon $-close to the disk exists for a time interval of size at least $ \sim \varepsilon^{-2} $. This quadratic lifespan result relies on a paradifferential Birkhoff normal form reduction and exploits cancellations arising from the Hamiltonian nature of the equation. This is the first normal form long time existence result of sharp fronts.

We prove the existence of small amplitude time quasi-periodic solutions of the pure gravity water waves equations  with constant vorticity , for a bidimensional fluid over a flat bottom delimited by a space periodic free interface. Using a Nash-Moser implicit function iterative scheme we construct traveling nonlinear waves which pass through each other slightly deforming and retaining forever a quasiperiodic structure. These solutions exist for any fixed value of depth and gravity and restricting the vorticity parameter to a Borel set of asymptotically full Lebesgue measure.

We prove the long-standing conjecture regarding the existence of infinitely many high-frequency modulational instability “isolas" for a Stokes wave in arbitrary depth 𝚑 > 0, subject to longitudinal perturbations. We completely describe the spectral bands with non-zero real part away from the origin of the L^2(ℝ)-spectrum of the water waves system linearized at a Stokes waves of small amplitude ε> 0. The unstable spectrum is the union of isolas of elliptical shape, parameterized by integers 𝚙≥ 2, with semiaxis of size |β_1^(𝚙) (𝚑)| ε^𝚙+ O(ε^𝚙+1 ) where β_1^( 𝚙) (𝚑) is a nonzero analytic function of the depth 𝚑 that depends on the Taylor coefficients of the Stokes waves up to order 𝚙.

We prove that all the solutions of a quasi-periodically forced linear Klein-Gordon equation ψ_tt-ψ_xx+𝚖ψ+Q(ω t)ψ=0 where Q(ω t) := a^(2)(ω t, x) ∂_xx + a^(1)(ω t, x)∂_x + a^(0)(ω t, x) is a differential operator of order 2, parity preserving and reversible, are almost periodic in time and uniformly bounded for all times, provided that the coefficients a^(2) , a^(1) , a^(0) are small enough and the forcing frequency ω∈ℝ^ν belongs to a Borel set of asymptotically full measure. This result is obtained by reducing the Klein-Gordon equation to a diagonal constant coefficient system with purely imaginary eigenvalues. The main difficulty is the presence in the perturbation Q (ω t) of the second order differential operator a^(2)(ω t, x)∂_xx. In suitable coordinates the Klein-Gordon equation is the composition of two backward/forward quasi-periodic in time perturbed transport equations with non-constant coefficients, up to lower order pseudo-differential remainders. A key idea is to straighten this first order pseudo-differential operator with bi-characteristics through a novel quantitative Egorov analysis.

We construct an infinite-dimensional family of smooth integrable magnetic systems on the two-torus which are Zoll, meaning that all the unit-speed magnetic geodesics are periodic. The metric and the magnetic field of such systems are arbitrarily close to the flat metric and to a given constant magnetic field. This extends to the magnetic setting a famous result by Guillemin on the two-sphere. We characterize Zoll magnetic systems as zeros of a suitable action functional $S$, and then look for its zeros by means of a Nash-Moser implicit function theorem. This requires showing the right-invertibility of the linearized operator $\mathrm{d} S$ in a neighborhood of the flat metric and constant magnetic field, and establishing tame estimates for the right inverse. As key step we prove the invertibility of the normal operator $\mathrm{d} S\circ \mathrm{d} S^*$ which, unlike in Guillemin's case, is pseudo-differential only at the highest order. We overcome this difficulty noting that, by the asymptotic properties of Bessel functions, the lower order expansion of $\mathrm{d} S \circ \mathrm{d}S^*$ is a sum of Fourier integral operators. We then use a resolvent identity decomposition which reduces the problem to the invertibility of $\mathrm{d} S \circ \mathrm{d} S^*$ restricted to the subspace of functions corresponding to high Fourier modes. The inversion of such a restricted operator is finally achieved by making the crucial observation that lower order Fourier integral operators satisfy asymmetric tame estimates.

In the last years substantial mathematical progress has been made in KAM theory for quasi-linear/fully nonlinear Hamiltonian partial differential equations, notably for water waves and Euler equations. In this survey we focus on recent advances in quasi-periodic vortex patch solutions of the 2d -Euler equation in ℝ^2 close to uniformly rotating Kirchhoff elliptical vortices, with aspect ratios belonging to a set of asymptotically full Lebesgue measure. The problem is reformulated into a quasi-linear Hamiltonian equation for a radial displacement from the ellipse. A major difficulty of the KAM proof is the presence of a zero normal mode frequency, which is due to the conservation of the angular momentum. The key novelty to overcome this degeneracy is to perform a perturbative symplectic reduction of the angular momentum, introducing it as a symplectic variable in the spirit of the Darboux – Carathéodory theorem of symplectic rectification, valid in finite dimension. This approach is particularly delicate in an infinite-dimensional phase space: our symplectic change of variables is a nonlinear modification of the transport flow generated by the angular momentum itself.

We prove an almost global existence result for space periodic solutions of the 1D gravity-capillary water waves equations with constant vorticity. The result holds for any value of gravity, vorticity and depth, a full measure set of surface tensions, and any small and smooth enough initial datum. The proof demands a novel approach—that we call paradifferential Hamiltonian Birkhoff normal form for quasi-linear PDEs—in presence of resonant wave interactions: the normal form is not integrable but it preserves the Sobolev norms thanks to its Hamiltonian nature. A major difficulty is that paradifferential calculus used to prove local well posedness (as the celebrated Alinhac good unknown) breaks the Hamiltonian structure. A major achievement of this paper is to correct (possibly) unbounded paradifferential transformations to symplectic maps, up to an arbitrary degree of homogeneity. Thanks to a deep cancellation, our symplectic correctors are smoothing perturbations of the identity. Thus we are able to preserve both the paradifferential structure and the Hamiltonian nature of the equations. Such Darboux procedure is written in an abstract functional setting applicable also in other contexts.

The paper fully answers a long standing open question concerning the stability/instability of pure gravity periodic traveling water waves—called Stokes waves—at the critical Whitham–Benjamin depth _WB= 1.363... and nearby values. We prove that Stokes waves of small amplitude 𝒪( ϵ ) are, at the critical depth _WB , linearly unstable under long wave perturbations. The same holds true for slightly smaller values of the depth > _WB- c ϵ ^2 , c > 0 , depending on the amplitude of the wave. This problem was not rigorously solved in previous literature because the expansions degenerate at the critical depth. To solve this degenerate case, and describe in a mathematically exhaustive way how the eigenvalues change their stable-to-unstable nature along this shallow-to-deep water transient, we Taylor-expand the computations of Berti et al. (Arch Ration Mech Anal 247:91, 2023) at a higher degree of accuracy, starting from the fourth order expansion of the Stokes waves. We prove that also in this transient regime a pair of unstable eigenvalues depict a closed figure “8”, of smaller size than for > _WB , as the Floquet exponent varies.

We present the almost global in time existence result in [ 13 ] of small amplitude space periodic solutions of the 1D gravity-capillary water waves equations with constant vorticity and we describe the ideas of proof. This is based on a novel Hamiltonian paradifferential Birkhoff normal form approach for quasi-linear PDEs.

We prove the existence of time quasi-periodic vortex patch solutions of the 2 $d$ -Euler equations in $\mathbb{R}^{2} $ , close to uniformly rotating Kirchhoff elliptical vortices, with aspect ratios belonging to a set of asymptotically full Lebesgue measure. The problem is reformulated into a quasi-linear Hamiltonian equation for a radial displacement from the ellipse. A major difficulty of the KAM proof is the presence of a zero normal mode frequency, which is due to the conservation of the angular momentum. The key novelty to overcome this degeneracy is to perform a perturbative symplectic reduction of the angular momentum, introducing it as a symplectic variable in the spirit of the Darboux-Carathéodory theorem of symplectic rectification, valid in finite dimension. This approach is particularly delicate in a infinite dimensional phase space: our symplectic change of variables is a nonlinear modification of the transport flow generated by the angular momentum itself. This is the first time such an idea is implemented in KAM for PDEs. Other difficulties are the lack of rotational symmetry of the equation and the presence of hyperbolic/elliptic normal modes. The latter difficulties -as well as the degeneracy of a normal frequency- are absent in other vortex patches problems which have been recently studied using the formulation introduced in this paper.

Whitham and Benjamin predicted in 1967 that small-amplitude periodic traveling Stokes waves of the 2d-gravity water waves equations are linearly unstable with respect to long-wave perturbations, if the depth 𝚑 is larger than a critical threshold _WB≈ 1.363 . In this paper, we completely describe, for any finite value of 𝚑 >0 , the four eigenvalues close to zero of the linearized equations at the Stokes wave, as the Floquet exponent μ is turned on. We prove, in particular, the existence of a unique depth _WB , which coincides with the one predicted by Whitham and Benjamin, such that, for any 0< 𝚑 < _WB , the eigenvalues close to zero are purely imaginary and, for any 𝚑 > _WB , a pair of non-purely imaginary eigenvalues depicts a closed figure “8”, parameterized by the Floquet exponent. As 𝚑→_WB^ + the “8” collapses to the origin of the complex plane. The complete bifurcation diagram of the spectrum is not deduced as in deep water, since the limits → +∞ (deep water) and μ→ 0 (long waves) do not commute. In finite depth, the four eigenvalues have all the same size 𝒪(μ ) , unlike in deep water, and the analysis of their splitting is much more delicate, requiring, as a new ingredient, a non-perturbative step of block-diagonalization. Along the whole proof, the explicit dependence of the matrix entries with respect to the depth is carefully tracked.

We consider the gravity water waves system with a periodic one‐dimensional interface in infinite depth and give a rigorous proof of a conjecture of Dyachenko‐Zakharov [16] concerning the approximate integrability of these equations. More precisely, we prove a rigorous reduction of the water waves equations to its integrable Birkhoff normal form up to order 4. As a consequence, we also obtain a long‐time stability result: periodic perturbations of a flat interface that are initially of size ε remain regular and small up to times of order ε−3 . This time scale is expected to be optimal. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.