On Symmetric Water Waves with Constant Vorticity

Dispersion relation for water waves with non-constant vorticity

We derive the dispersion relation for linearized small-amplitude gravity waves for various choices of non-constant vorticity. To the best of our knowledge, this relation is only known explicitly in the case of constant vorticity. We provide a wide range of examples including polynomial, exponential, trigonometric and hyperbolic vorticity functions.

Nearly-hamiltonian Structure for Water Waves with Constant Vorticity

We show that the governing equations for two-dimensional gravity water waves with constant non-zero vorticity have a nearly-Hamiltonian structure, which becomes Hamiltonian for steady waves.

A nonlinear Schrödinger equation for water waves on finite depth with constant vorticity

Large-Amplitude Solitary Water Waves with Vorticity

We provide the first construction of exact solitary waves of large amplitude with an arbitrary distribution of vorticity. We use continuation to construct a global connected set of symmetric solitary waves of elevation, whose profiles decrease monotonically on either side of a central crest. This generalizes the classical result of Amick and Toland.

Symmetry of Solitary Water Waves with Vorticity

Symmetry and monotonicity properties of solitary water-waves of positive elevation with supercritical values of parameter are established for an arbitrary vorticity. The proof uses the detailed knowledge of asymptotic decay of supercritical solitary waves at infinity and the method of moving planes.