@article {2023, title = {Benjamin-Feir Instability of Stokes Waves in Finite Depth}, journal = {ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS}, volume = {247}, year = {2023}, month = {OCT}, pages = {91}, type = {Article}, abstract = {

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 h(WB) approximate to 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 mu is turned on. We prove, in particular, the existence of a unique depth h(WB), which coincides with the one predicted by Whitham and Benjamin, such that, for any 0 \< h \< h(WB), the eigenvalues close to zero are purely imaginary and, for any h \> h(WB), a pair of non-purely imaginary eigenvalues depicts a closed figure {\textquoteleft}{\textquoteleft}8{\textquoteright}{\textquoteright}, parameterized by the Floquet exponent. As h -\> h(WB)(+) the {\textquoteleft}{\textquoteleft}8{\textquoteright}{\textquoteright} 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 h -\> +infinity (deep water) and mu -\> 0 (long waves) do not commute. In finite depth, the four eigenvalues have all the same size O(mu), 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 h is carefully tracked.

}, issn = {0003-9527}, doi = {10.1007/s00205-023-01916-2}, author = {Berti, Massimiliano and Maspero, Alberto and Ventura, Paolo} } @article {2023, title = {Stokes waves at the critical depth are modulational unstable}, year = {2023}, author = {Berti, Massimiliano and Maspero, Alberto and Ventura, Paolo} } @article {2022, title = {On the analyticity of the Dirichlet-Neumann operator and Stokes waves}, journal = {RENDICONTI LINCEI-MATEMATICA E APPLICAZIONI}, volume = {33}, year = {2022}, pages = {611-650}, type = {Article}, abstract = {

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.

}, keywords = {Bifurcation theory, Dirichlet-Neumann operator, traveling waves, water waves}, issn = {1120-6330}, doi = {10.4171/RLM/983}, author = {Berti, Massimiliano and Maspero, Alberto and Ventura, Paolo} } @article {2022, title = {Benjamin-Feir instability of Stokes waves}, journal = {RENDICONTI LINCEI-MATEMATICA E APPLICAZIONI}, volume = {33}, year = {2022}, pages = {399-412}, type = {Article}, abstract = {

We present the recent results in Berti et al. [Invent. Math. (2022), to appear] regarding the Benjamin-Feir instability of small amplitude Stokes waves in deep water. We completely describe the behavior of the four eigenvalues close to zero of the linearized water waves equations at the Stokes solution, as the Floquet exponent is turned on, proving the conjecture that a pair of non-purely imaginary eigenvalues depicts a closed figure {\textquoteleft}{\textquoteleft}8{\textquoteright}{\textquoteright}, in full agreement with numerical simulations.

}, keywords = {Kato perturbation theory, modulational instability, traveling waves, water waves}, issn = {1120-6330}, doi = {10.4171/RLM/975}, author = {Berti, Massimiliano and Maspero, Alberto and Ventura, Paolo} } @article {2022, title = {Full description of Benjamin-Feir instability of stokes waves in deep water}, volume = {230}, year = {2022}, month = {2022/11/01}, pages = {651 - 711}, abstract = {

Small-amplitude, traveling, space periodic solutions {\textendash}called Stokes waves{\textendash} 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 {\textquotedblleft}8{\textquotedblright}, parameterized by the Floquet exponent, in full agreement with numerical simulations. Our new spectral approach to the Benjamin-Feir instability phenomenon uses a symplectic version of Kato{\textquoteright}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.

}, isbn = {1432-1297}, url = {https://doi.org/10.1007/s00222-022-01130-z}, author = {Berti, Massimiliano and Maspero, Alberto and Ventura, Paolo} }