Nonlinear Hamiltonian waves with constant frequency and surface waves on vorticity discontinuities

Nonlinear Hamiltonian Waves with Constant Frequency and Surface Waves on Vorticity Discontinuities

Waves with constant, nonzero linearized frequency form an interesting class of nondispersive waves whose properties differ from those of nondispersive hyperbolic waves. We propose an inviscid Burgers-Hilbert equation as a model equation for such waves, and give a dimensional argument to show that it models Hamiltonian surface waves with constant frequency. Using the method of multiple scales, w...

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

Hamiltonian formalism for nonlinear waves

Hamiltonian description for nonlinear waves in plasma, hydrodynamics and magnetohydrodynamics is presented. The main attention is paid to the problem of canonical variables introducing. The connection with other approaches of the Hamiltonian structure introducing is presented, in particular, with the help of the Poisson brackets expressed in terms of natural variables. It is shown that the dege...

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.