Antonio Milosh Radakovic

arXiv2026

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.

math.AP↗

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.

math.AP↗