Singular Solutions of a Boussinesq System for Water Waves (Journal of Mathematical Study, 2022)
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 amodel for small-amplitude, long-crested water waves. The issue addressed is whetheror not the initial-value problem for this system of equations is globally well posed.The investigation proceeds by way of numerical simulations using a computer codebased on a a semi-implicit, pseudo-spectral code. It turns out that larger amplitudesor velocities do seem to lead to singularity formation in finite time, indicating that theproblem is not globally well posed.
Keywords: Boussinesq systems; global well-posedness; singular solutions; Fourier spectral method; nonlinear water wave.