An introduction to the Zakharov equation for modelling deep water waves

Abstract

The Hamiltonian formulation of the water wave problem due to Zakharov, and the reduced Zakharov equation derived therefrom, have great utility in understanding and modelling water waves. Here we set out to review the cubic Zakharov equation and its uses in understanding deterministic waves in deep water. The background of this equation is developed and several applications are explored. Chief among these is an understanding of dispersion corrections and the energy exchange among modes. It is hoped that readers will be motivated to explore this powerful reformulation of the cubically nonlinear water wave problem for themselves.

References in corpus (4)

• Hamiltonian approach to modelling interfacial internal waves over variable bottom
• The nonlinear Benjamin-Feir instability -- Hamiltonian dynamics, primitive breathers, and steady solutions
• On coupled envelope evolution equations in the Hamiltonian theory of nonlinear surface gravity waves
• Wave-by-wave forecasts in directional seas using nonlinear dispersion corrections