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.