TY - JOUR
T1 - Singular Solutions of a Boussinesq System for Water Waves
AU - Bona , Jerry L.
AU - Chen , Min
JO - Journal of Mathematical Study
VL - 3
SP - 205
EP - 220
PY - 2016
DA - 2016/09
SN - 49
DO - http://doi.org/10.4208/jms.v49n3.16.01
UR - https://global-sci.org/intro/article_detail/jms/999.html
KW - Boussinesq systems, global well-posedness, singular solutions, Fourier spectral method, nonlinear water wave.
AB - <p style="text-align: justify;">Studied here is the Boussinesq system $$η_t+u_x+(ηu)_x+au_{xxx}-bη_{xxt}=0,$$ $$u_t+η_x+\frac{1}{2}(u²)_x+cη_{xxx}-du_{xxt}=0,$$of partial differential equations. This system has been used in theory and practice as a
model for small-amplitude, long-crested water waves. The issue addressed is whether
or not the initial-value problem for this system of equations is globally well posed.<br/>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.</p>