Numerical study of the generalised Klein–Gordon equations

Abstract

Abstract. In this study, we discuss an approximate set of equations describing water wave propagating in deep water. These generalized 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.

Efficient computation of capillary–gravity generalised solitary waves

“…The numerical procedure for the computation should also change in some suitable way. The code employed to obtain all the numerical results presented in this manuscript (but also the results of our previous work [19]) is freely available to download at the following URL address [22]:…”

“…Finally, the main conclusions and perspectives are outlined in Section 5. With the aim of involving a wider audience, the numerical code used in computations below is available to download as open source [22]. Thus, the claims made in this study can be easily verified by interested readers.…”

Serre-type Equations in Deep Water

“…The reasons why these equations can be considered as the analogue of nonlinear shallow water (or Saint-Venant [20]) equations in deep water are explained in [15]. The gKG equations were extensively studied in [21]. In particular, it was shown that these equations possess the canonical symplectic (Hamiltonian) and multi-symplectic structures additionally to the variational structure incorporated into the Lagrangian density (4.2).…”

Modeling Water Waves Beyond Perturbations

“…This means, in particular, that the gKG model is valid for spectra narrow-banded around the wavenumber κ. Further details and properties of the gKG are given in [4] (section 4.2) and in [6].…”