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Surface water waves are considered propagating over highly variable non-smooth topographies. For this three dimensional problem a Dirichlet-to-Neumann (DtN) operator is constructed reducing the numerical modeling and evolution to the two dimensional free surface. The corresponding Fourier-type operator is defined through a matrix decomposition. The topographic component of the decomposition requires special care and a Galerkin method is provided accordingly. One dimensional numerical simulations, along the free surface, validate the DtN formulation in the presence of a large amplitude, rapidly varying topography. An alternative, conformal mapping based, method is used for benchmarking. A two dimensional simulation in the presence of a Luneburg lens (a particular submerged mound) illustrates the accurate performance of the three dimensional DtN operator.

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This article is concerned with the non-linear interaction of homogeneous random ocean surface waves. Under this umbrella, numerous kinetic equations have been derived to study the evolution of the spectral action density, each employing slightly different assumptions. Using analytical and numerical tools, and providing exact formulas, we demonstrate that the recently derived generalized kinetic equation exhibits blow up in finite time for certain degenerate quartets of waves. This is discussed in light of the assumptions made in the derivation, and this equation is contrasted with other kinetic equations for the spectral action density.

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The problem of unidirectional shoaling of a water-wave field with a narrow energy spectrum is treated by using a new Alber equation. The stability of the linear stationary solution to small non-stationary disturbances is analysed; and numerical solutions for its subsequent long-distance evolution are presented. The results quantify the physics which causes the gradual decay in the probability of freak-wave occurrence, when moving from deep to shallow coastal waters.

We develop a general framework to describe the cubically nonlinear interaction of a degenerate quartet of deep-water gravity waves in one or two spatial dimensions. Starting from the discretised Zakharov equation, and thus without restriction on spectral bandwidth, we derive a planar Hamiltonian system in terms of the dynamic phase and a modal amplitude. This is characterised by two free parameters: the wave action and the mode separation between the carrier and the sidebands. For unidirectional waves, the mode separation serves as a bifurcation parameter, which allows us to fully classify the dynamics. Centres of our system correspond to non-trivial, steady-state nearly resonant degenerate quartets. The existence of saddle-points is connected to the instability of uniform and bichromatic wave trains, generalising the classical picture of the Benjamin–Feir instability. Moreover, heteroclinic orbits are found to correspond to discrete, three-mode breather solutions, including an analogue of the famed Akhmediev breather solution of the nonlinear Schrödinger equation.

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