Nonlinear water wave models with vorticity

The results presented in this thesis consider geophysical nonlinear water waves and small-amplitude gravity waves. New exact and explicit solutions in terms of Lagrangian variables are derived to model geophysical surface and internal water waves and an analysis of small-amplitude waves from the point of view of the dispersion relation is presented. The solutions are Gerstner-like or Pollard-like in their character and prescribe three-dimensional water waves propagating in the horizontal direction. The models presented in this thesis represent various complicated and intricate scenarios which build the dynamics of the ocean. We prove using a rigorous mathematical analysis the validity of these models. In each model a dispersion relation arises as a product of the analysis indicating how the relative wave speed of the wave varies with respect to various physical parameters introduced by a particular model