Date: 9 June 2025 - 13 June 2025
Abstract:
Stokes waves are among the most famous global in time solutions of the pure gravity water waves equations. They have the form of a space-periodic profile traveling at constant speed, and were discovered almost two centuries ago by Sir George Stokes. A classical question in fluid mechanics is to study their stability vs instability. In the ’60, Benjamin, Feir, Whitham, Lighthill and Zakharov discovered, through experiments and formal arguments, that small amplitude periodic Stokes waves in sufficiently deep water are unstable when subjected to long-wave perturbations and proposed a heuristic mechanism that leads to the disintegration of wave trains. From a mathematical point of view this amount to study the spectral bands that leave the imaginary axis of a Hamiltonian and reversible pseudo-differential matrix valued operator with periodic coefficients. In this course I will present recent mathematically rigorous results about the band spectrum of the water waves equations linearized at the Stokes solitary waves. This material is based on joint works with M. Berti, P. Ventura and L. Corsi.