Dynamics of Partial Differential Equations

Volume 15 (2018)

Number 1

Travelling waves over an arbitrary bathymetry: a local stability result

Pages: 81 – 94

DOI: http://dx.doi.org/10.4310/DPDE.2018.v15.n1.a4

Author

Alessandro Fortunati (School of Mathematics, University of Bristol, United Kingdom)

Abstract

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.

Keywords

shallow-water equations, aperiodic time dependence, nearly-integrable Hamiltonian systems

2010 Mathematics Subject Classification

Primary 35C07, 70H08. Secondary 35C10, 37J40.

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This research was supported by ONR Grant No. N00014-01-1-0769.

Paper received on 10 July 2017.