A Complete Asymptotic Analysis of the Spectral Instabilities of Small-Amplitude Periodic Water Waves

Abstract

Euler's equations govern the behavior of gravity waves on the surface of an incompressible, inviscid, and irrotational fluid (water, in this case). We consider the small-amplitude, periodic traveling-wave solutions of Euler's equations known as the Stokes waves. Our focus is on the instabilities of Stokes waves present in the spectrum of the linearized Euler's equations about these solutions. These instabilities encompass the Benjamin-Feir (or modulational) instability as well as the recently discovered high-frequency instabilities. In this dissertation, we develop a perturbation method to describe the unstable spectral elements associated with each of these instabilities, allowing us to obtain desirable asymptotic properties that connect recent numerical and rigorous studies.