arXiv subjects

Alberto Maspero

Publications and source records attributed to Alberto Maspero.

At least 19 recordsLinked to original sources

arXiv2026

McLean resonances and $3d$ spectral instability of Stokes waves

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.

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arXiv2026

Transfer of energy for pure-gravity water waves with constant vorticity

We consider two-dimensional periodic gravity water waves with constant nonzero vorticity $\gamma$, in infinite depth and with periodic boundary conditions. We prove that, if the characteristic wave number $\frac{\gamma^2}{g}$ is rational, the system admits smooth small-amplitude solutions whose high Sobolev norms grow arbitrarily large while lower-order norms remain arbitrarily small, thereby exhibiting a genuine transfer of energy toward high frequencies. This yields the first rigorous construction of weakly turbulent solutions for a quasilinear hydrodynamic wave system, in a regime where the flow remains smooth. Moreover, the growth occurs simultaneously in the free surface and in the vertical component of the velocity at the interface, showing that the instability involves the full hydrodynamic evolution. The proof relies on a new mechanism for generating energy cascades in quasilinear dispersive PDEs with sublinear dispersion and a nonlinear transport structure. A central ingredient is to exploit quasi-resonances from 2-wave interactions to produce a transport operator that drives energy to high modes and causes Sobolev norm growth. A virial-type argument then shows that the resulting instability affects both the free surface elevation and the velocity field.

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arXiv2025

Full Benjamin-Feir instability of capillary-gravity Stokes waves in finite depth

We study the two-dimensional gravity-capillary water waves equations for a fluid of finite depth $\mathtt{h}>0$ under the combined effects of gravity and surface tension $\kappa \geq 0$. We analyze the linear stability and instability of small-amplitude, $2\pi$-periodic Stokes wave solutions, under the effect of longitudinal long-wave perturbations. The corresponding linearized operator has periodic coefficients and a defective zero eigenvalue of multiplicity four. Using Bloch-Floquet theory, we investigate the associated family of periodic eigenvalue problems. For all surface tension values $\kappa \geq 0$ and depths $\mathtt{h} > 0$, we establish the complete splitting of the four eigenvalues near zero when both the wave amplitude and the Floquet parameter are small. Specifically, we rigorously prove that in the regions of unstable depth and capillarity identified formally by Djordjevic-Redekopp and Ablowitz-Segur in the 1970's, the spectrum of the linearized operator near the origin depicts a "figure 8" pattern.

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arXiv2025

Rogue waves and large deviations for 2D pure gravity deep water waves

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 rigorously proves 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 over very long times, even in the absence of (quasi-)invariant measures.

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arXiv2025

On higher order isolas of unstable Stokes waves

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 $\beta_1^{(\mathtt{p})}(\mathtt{h}) $ is not zero. In [3] it is proved that the map $ \mathtt{h} \mapsto \beta_1^{(\mathtt{p})}(\mathtt{h}) $ is not identically zero for any $ \mathtt{p} \geq 2 $ by showing that $ \lim_{\mathtt{h} \to 0^+}\beta_1^{(\mathtt{p})}(\mathtt{h}) = - \infty $. In this manuscript we compute the asymptotic expansion of $\beta_1^{(\mathtt{p})}(\mathtt{h}) $ in the deep-water limit $ \mathtt{h} \to + \infty $ -- it vanishes exponentially fast to zero -- for $\mathtt{p}=2$, $3$, $4$.

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arXiv2024

Growth of Sobolev norms for completely resonant quantum harmonic oscillators on $\mathbb{R}^2$

We consider time dependently perturbed quantum harmonic oscillators in $\mathbb{R}^2$: $$ {\rm i} \partial_t u=\frac12(-\partial_{x_1}^2-\partial_{x_2}^2 + x_1^2+x_2^2)u +V(t, x, D)u, \qquad \ x \in \mathbb{R}^2, $$ where $V(t, x, D)$ is a selfadjoint pseudodifferential operator of degree zero, $2\pi$ periodic in time. We identify sufficient conditions on the principal symbol of the potential $V(t, x, D)$ that ensure existence of solutions exhibiting unbounded growth in time of their positive Sobolev norms and we show that the class of symbols satisfying such conditions is generic in the Fr\'echet space of classical $2\pi$- time periodic symbols of order zero. To prove our result we apply the abstract Theorem of arXiv:2101.09055v1 : the main difficulty is to find a conjugate operator $A$ for the resonant average of $V(t,x, D)$. We construct explicitly the symbol of the conjugate operator $A$, called escape function, combining techniques from microlocal analysis, dynamical systems and contact topology.

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arXiv2024

One dimensional energy cascades in a fractional quasilinear NLS

We consider the problem of transfer of energy to high frequencies in a quasilinear Schr\"odinger equation with sublinear dispersion, on the one dimensional torus. We exhibit initial data undergoing finite but arbitrary large Sobolev norm explosion: their initial norm is arbitrary small in Sobolev spaces of high regularity, but at a later time becomes arbitrary large. We develop a novel mechanism producing instability, which is based on extracting, via paradifferential normal forms, an effective equation driving the dynamics whose leading term is a non-trivial transport operator with non-constant coefficients. We prove that such operator is responsible for energy cascades via a positive commutator estimate inspired by Mourre's commutator theory.

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arXiv2024

Infinitely many isolas of modulational instability for Stokes waves

This paper proves long-standing conjectures regarding the existence of infinitely many high-frequency modulational instability ``isolas" for a Stokes wave in arbitrary depth $ \mathtt{h} > 0 $, under longitudinal perturbations. We provide a complete characterization of the unstable spectral bands in the $L^2(\mathbb{R})$-spectrum of the water wave equations linearized around a Stokes wave of sufficiently small amplitude $\epsilon$. The unstable spectrum is the union of isolated ``isolas" of elliptical shape, indexed by integers $ \mathtt{p}\geq 2 $, each with semiaxis of size $ |\beta_1^{(\mathtt{p})} (\mathtt{h})| \epsilon^\mathtt{p}+ O(\epsilon^{\mathtt{p}+2} )$. As first key achievement, we obtain an explicit formula for the coefficient $ \beta_1^{(\mathtt{p})} (\mathtt{h}) $ for any $ \mathtt{p} \geq 2 $, that remarkably depends solely on the maximal Taylor-Fourier coefficients of the Stokes wave. We provide simple expressions of the asymptotic expansion of such coefficients in the shallow-water limit $ \mathtt{h} \to 0^+ $, for any $ \mathtt{p} \geq 2 $. This allows to establish that the analytic function $\beta_1^{(\mathtt{p})}(\mathtt{h})$ is not zero for any $\mathtt{p} \geq 2$, by verifying that a combinatorial sum is not zero; this relies on a crucial combinatorial identity due to Koutschan, van Hoeij, and Zeilberger.

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arXiv2024

Full description of Benjamin-Feir instability for generalized Korteweg-de Vries equations

In this paper we consider a family of generalized Korteweg-de Vries equations and study the linear modulational instability of small amplitude traveling waves solutions. Under explicit non-degeneracy conditions on the dispersion relation, we completely describe the spectrum near the origin of the linearized operator at such solutions and prove that the unstable spectrum (when present) is composed by branches depicting always a closed figure ''8''. We apply our abstract theorem to several equations such as the Whitham, the gravity-capillary Whitham and the Kawahara equations, confirming that the unstable spectrum of the corresponding linearized operators exhibits a figure ''8'' instability, as it was observed before only numerically. Our method of proof uses a symplectic version of Kato's theory of similarity transformation to reduce the problem to determine the eigenvalues of a $3 \times 3$ complex Hamiltonian and reversible matrix. Then, via a block-diagonalization procedure, we conjugate such matrix into a block-diagonal one composed by a $2\times 2$ Hamiltonian and reversible matrix, describing the unstable spectrum, and a single purely imaginary element describing the stable eigenvalue.

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arXiv2024

Longtime dynamics for the Landau Hamiltonian with a time dependent magnetic field

We consider a modulated magnetic field, $B(t) = B_0 +\varepsilon f(\omega t)$, perpendicular to a fixed plane, where $B_0$ is constant, $\varepsilon>0$ and $f$ a periodic function on the torus ${\mathbb T}^n$. Our aim is to study classical and quantum dynamics for the corresponding Landau Hamiltonian. It turns out that the results depend strongly on the chosen gauge. For the Landau gauge the position observable is unbounded for "almost all" non resonant frequencies $\omega$. On the contrary, for the symmetric gauge we obtain that, for "almost all" non resonant frequencies $\omega$, the Landau Hamiltonian is reducible to a two dimensional harmonic oscillator and thus gives rise to bounded dynamics. The proofs use KAM algorithms for the classical dynamics. Quantum applications are given. In particular, the Floquet spectrum is absolutely continuous in the Landau gauge while it is discrete, of finite multiplicity, in symmetric gauge.

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arXiv2024

First isola of modulational instability of Stokes waves in deep water

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.

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arXiv2023

Stokes waves at the critical depth are modulational unstable

This 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 $ \mathtt{h}_{\scriptscriptstyle WB} = 1.363... $ and nearby values. We prove that Stokes waves of small amplitude $ \mathcal{O}( \epsilon ) $ are, at the critical depth $ \mathtt{h}_{\scriptscriptstyle WB} $, linearly unstable under long wave perturbations. This is also true for slightly smaller values of the depth $ \mathtt{h} > \mathtt{h}_{\scriptscriptstyle WB} - c \epsilon^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. In order to resolve 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 arXiv:2204.00809v2 at a higher degree of accuracy, derived by 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 $ \mathtt{h} > \mathtt{h}_{\scriptscriptstyle WB} $, as the Floquet exponent varies.

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arXiv2022

Hamiltonian Birkhoff normal form for gravity-capillary water waves with constant vorticity: almost global existence

We prove an almost global in time existence result of small amplitude 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 and any surface tension belonging to a full measure set. The proof demands a Hamiltonian paradifferential Birkhoff normal form reduction for quasi-linear PDEs in presence of resonant wave interactions: the normal form may be not integrable but it preserves the Sobolev norms thanks to its Hamiltonian nature. A major difficulty is that usual paradifferential calculus used to prove local well posedness (as the celebrated Alinhac good unknown) does not preserve the Hamiltonian structure. A major novelty of this paper is to develop an algorithmic perturbative procedure \`a la Darboux to correct usual paradifferential transformations to symplectic maps, up to an arbitrary degree of homogeneity. The symplectic correctors turn out to be smoothing perturbations of the identity, and therefore only slightly modify the paradifferential structure of the equations. The Darboux procedure which recovers the nonlinear Hamiltonian structure is written in an abstract functional setting, in order to be applicable also in other contexts.

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arXiv2022

Reducibility and nonlinear stability for a quasi-periodically forced NLS

Motivated by the problem of long time stability vs. instability of KAM tori of the Nonlinear cubic Schr\"odinger equation (NLS) on the two dimensional torus $\mathbb T^2:= (\mathbb R/2\pi \mathbb Z)^2$, we consider a quasi-periodically forced NLS equation on $\mathbb T^2$ arising from the linearization of the NLS at a KAM torus. We prove a reducibility result as well as long time stability of the origin. The main novelty is to obtain the precise asymptotic expansion of the frequencies which allows us to impose Melnikov conditions at arbitrary order.

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arXiv2022

Benjamin-Feir instability of Stokes waves in finite depth

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 $\mathtt h$ is larger than a critical threshold $\mathtt h_{WB} \approx 1.363$. In this paper we completely describe, for any value of $\mathtt h > 0$, the four eigenvalues close to zero of the linearized equations at the Stokes wave, as the Floquet exponent $\mu$ is turned on. We prove in particular the existence of a unique depth $\mathtt h_{WB}$, which coincides with the one predicted by Whitham and Benjamin, such that, for any $0 < \mathtt h < \mathtt h_{WB}$, the eigenvalues close to zero remain purely imaginary and, for any $\mathtt h > \mathtt h_{WB}$, a pair of non-purely imaginary eigenvalues depicts a closed figure "8", parameterized by the Floquet exponent. As $\mathtt h \to \mathtt h_{WB}^+$ this figure "8" collapses to the origin of the complex plane. The proof combines a symplectic version of Kato's perturbative theory to compute the eigenvalues of a $4 \times 4$ Hamiltonian and reversible matrix, and KAM inspired transformations to block-diagonalize it. The four eigenvalues have all the same size $O(\mu)$ - unlike the infinitely deep water case in [6]- and the correct Benjamin-Feir phenomenon appears only after one non-perturbative block-diagonalization step. In addition one has to track, along the whole proof, the explicit dependence of the entries of the $4 \times 4$ reduced matrix with respect to the depth $\mathtt h$.

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arXiv2022

Generic transporters for the linear time dependent quantum Harmonic oscillator on $\mathbb R$

In this paper we consider the linear, time dependent quantum Harmonic Schr\"odinger equation $i \partial_t u= \frac{1}{2} ( - \partial_x^2 + x^2) u + V(t, x, D)u$, $x \in \mathbb R$, where $V(t,x,D)$ is classical pseudodifferential operator of order 0, selfadjoint, and $2\pi$ periodic in time. We give sufficient conditions on the principal symbol of $V(t,x,D)$ ensuring the existence of weakly turbulent solutions displaying infinite time growth of Sobolev norms. These conditions are generic in the Frechet space of symbols. This shows that generic, classical pseudodifferential, $2\pi$-periodic perturbations provoke unstable dynamics. The proof builds on the results of [36] and it is based on pseudodifferential normal form and local energy decay estimates. These last are proved exploiting Mourre's positive commutator theory.

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arXiv2022

On the analyticity of the Dirichlet-Neumann operator and Stokes waves

We prove an analyticity result for the Dirichlet-Neumann operator under space periodic boundary conditions in any dimension in an unbounded domain with infinite depth. We derive an analytic bifurcation result of analytic Stokes waves -- i.e. space periodic traveling solutions -- of the water waves equations in deep water.

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arXiv2021

Full description of Benjamin-Feir instability of Stokes waves in deep water

Small-amplitude, traveling, space periodic solutions -- called Stokes waves -- of the 2 dimensional gravity water waves equations in deep water are linearly unstable with respect to long-wave perturbations, as predicted by Benjamin and Feir in 1967. We completely describe the behavior of the four eigenvalues close to zero of the linearized equations at the Stokes wave, as the Floquet exponent is turned on. We prove in particular the conjecture that a pair of non-purely imaginary eigenvalues depicts a closed figure eight, parameterized by the Floquet exponent, in full agreement with numerical simulations. Our new spectral approach to the Benjamin-Feir instability phenomenon uses Kato's theory of similarity transformation to reduce the problem to determine the eigenvalues of a $ 4 \times 4 $ complex Hamiltonian and reversible matrix. Applying a procedure inspired by KAM theory, we block-diagonalize such matrix into a pair of $2 \times 2 $ Hamiltonian and reversible matrices, thus obtaining the full description of its eigenvalues.

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