Particle paths beneath nonlinear waves

We explore particle trajectories within a two-dimensional flow of an ideal liquid, driven by the passage of a nonlinear gravity surface wave. These particle paths are described by solutions to a dynamical system. In a moving frame that follows a nonlinear traveling wave solution, the system becomes autonomous, allowing the particle paths to be expressed in Hamiltonian form. We provide a comprehensive analysis of the particle trajectories, considering the influence of a background flow in the fluid bulk. Additionally, we address the case of a conducting fluid, bounded above by a dielectric gas and influenced by normal electric fields.