Steady waves in local and nonlocal models for water waves

We study the steady Euler equations for inviscid, incompressible, and irrotational water waves of constant density. The thesis consists of three papers. The first paper approaches the Euler equations through a famous nonlocal model equation for gravity waves, namely the Whitham equation. We prove the existence of a highest gravity solitary wave which reaches the largest amplitude and forms a $C^{1/2}$ cusp at its crest. This confirms a 50-year-old conjecture by Whitham in the case of solitary waves, that the full linear dispersion in the Whitham equation would allow for high-frequency phenomena such as highest waves. In the second paper, we use a recently developed center manifold theorem for nonlocal and nonlinear equations to study small-amplitude gravity--capillary generalized and modulated solitary waves in a Whitham equation with small surface tension. The last paper treats the steady Euler equations directly. Here, the gravity and capillary coefficients are fixed but arbitrary, and for simplicity we place a non-resonance condition on the problem. We address the transverse dynamics of two-dimensional gravity--capillary periodic waves using a spatial dynamics technique, followed by a perturbation argument