| Abstract: |
| We will investigate the fundamental issue of
determining
conditions on the initial data that ensure the solution
exists for all time for the Boussinesq system
$$
\eta_t+u_x + (\eta u)_x +au_{xxx} -b \eta_{xxt}=0,
$$
$$
u_t+ \eta_x +\frac 12 (u^2)_x+c\eta_{xxx} -d u_{xxt}=0.
$$
This system has been used in theory and practice
as a model for small-amplitude, long-crested water waves.
The investigation proceeds by way of numerical simulations using
a computer code based on a
a semi-implicit, pseudo-spectral code.
It turns out that larger amplitudes or velocities do seem to lead to singularity
formation in finite time, indicating that the problem is not globally well posed. |
|