Numerical study of the generalised Klein-Gordon equations
Denys Dutykh (LAMA), Marx Chhay (LOCIE), Didier Clamond (JAD)
TL;DR
This paper investigates generalized Klein-Gordon equations for deep water waves, constructing high-accuracy numerical solutions, analyzing their stability, and comparing them to full Euler equations.
Contribution
It introduces a variational, Hamiltonian, and multi-symplectic formulation of the gKG equations and develops an efficient spectral discretisation for stability analysis.
Findings
High-accuracy numerical traveling wave solutions
Comparison with seventh-order Stokes expansion
Assessment of wave stability and localized wave packets
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.
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