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

• 成果类型: Article
• 署名作者: Berti, Massimiliano; Maspero, Alberto; Ventura, Paolo
• 署名单位: International School for Advanced Studies (SISSA)
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-022-01130-z
• 发表日期: 2022
• 页码: 651-711
• 关键词: small periodic-waves; modulational instability; numerical-simulation; STABILITY; surface; analyticity; EQUATIONS

摘要

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 8, 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's theory of similarity transformation to reduce the problem to determine the eigenvalues of a 4 x 4 complex Hamiltonian and reversible matrix. Applying a procedure inspired by KAM theory, we block-diagonalize such matrix into a pair of 2x2 Hamiltonian and reversible matrices, thus obtaining the full description of its eigenvalues.