Dynamics of Partial Differential Equations
Volume 15 (2018)
Travelling waves over an arbitrary bathymetry: a local stability result
Pages: 81 – 94
The problem of a travellingwave in a semi-infinite channel over an arbitrary quasi-flat bathymetry is studied in the shallow-water formulation. It is shown how the streamfunction can be cast, in the vicinity of an elliptic equilibrium for the fluid flow, in the form of a nearly-integrable non-autonomous Hamiltonian with a perturbation which is aperiodic in time. The obtained model provides a natural example in the context of the aperiodically time-dependent Hamiltonian systems studied in Fortunati and Wiggins (2016) and the local stability of the perturbed streamlines is obtained as a consequence of this theory.
shallow-water equations, aperiodic time dependence, nearly-integrable Hamiltonian systems
2010 Mathematics Subject Classification
Primary 35C07, 70H08. Secondary 35C10, 37J40.
This research was supported by ONR Grant No. N00014-01-1-0769.
Received 10 July 2017
Published 14 December 2017