Comparisons between the
BBM equation and a Boussinesq system
This project aims to cast light on a Boussinesq system of
equations modelling
two-way propagation of surface waves. Included in the study are
existence results, comparisons between the Boussinesq equations
and other wave models, and several numerical simulations. The
existence theory is in fact a local well-posedness result that
becomes global when the solution satisfies a practically reasonable
constraint. The comparison result is concerned with initial
velocities and wave profiles that correspond to unidirectional
propagation. In this circumstance, it is shown that the solution of
the Boussinesq system is very well approximated by an associated
solution of the KdV or BBM equation over a long time scale of order ${1 \over
\epsilon}$, where $\epsilon$ is the ratio of the maximum wave amplitude to the
undisturbed depth of the liquid. This result confirms earlier
numerical simulations and suggests further numerical experiments
which are reported here.
Min Chen (chen@math.purdue.edu)