Benjamin-Feir instability of Stokes waves

This thesis presents significant contributions to the mathematical comprehension of the Benjamin-Feir instability in water waves in full agreement with the existing numerical findings and extending the analytical literature. The study is divided into three distinct cases, each providing valuable insights into this complex phenomenon. In the first part, we investigate the infinite-depth scenario, providing a full description of the entire linear instability phenomenon. This foundational analysis forms a crucial basis for subsequent inquiries. In the finite-depth part, our analysis is able to recover the shift between shallow and deep water regime. In shallow water our findings reveal a lack of instability emergence for the part of the spectrum of the linearized operator near zero, whereas in deep water we prove a comprehensive instability result in continuity with the infinite-depth case. The critical transition between the two regimes is identified as the depth surpasses a pivotal threshold, termed the "Whitham-Benjamin critical depth". In the final part of the thesis, we focus on the critical case, demonstrating that the Stokes waves exhibit modulation instability at the Whitham-Benjamin critical depth. In the appendix, we provide a rigorous proof of the existence of Stokes waves within high-regularity spaces of analytic functions. Additionally, we present the Taylor expansion of the Stokes waves up to the fourth order, on which the main results of the thesis rely.