Numerical study of the generalised Klein–Gordon equations

•Derivation of generalised Klein–Gordon equations is presented in the context of deep water waves.•Accuracy of this approximate model is studied using analytical and numerical methods.•For travelling periodic waves the model is shown to be more accurate than the cubic Zakharov equations.•Dynamics of periodic and localised wave trains is studied numerically.•It is shown numerically that this model can develop Riemann-type wave breaking phenomenon.

In this study, we discuss an approximate set of equations describing water wave propagating in deep water. These generalised Klein–Gordon (gKG) equations possess a variational formulation, as well as a canonical Hamiltonian and multi-symplectic structures. Periodic travelling wave solutions are constructed numerically to high accuracy and compared to a seventh-order Stokes expansion of the full Euler equations. Then, we propose an efficient pseudo-spectral discretisation, which allows to assess the stability of travelling waves and localised wave packets.