  In this paper, we study the generalized Boussinesq equation as a model for
the water wave problem with surface tension. Initially, we investigate the
initial value problem within Sobolev spaces, deriving conditions under which
solutions are either global or experience blow-up in time. Subsequently, we
extend our analysis to Bessel potential and modulation spaces, determining the
asymptotic behavior of solutions. We establish the non-existence of solitary
waves for certain parameters using Pohozaev-type identities. Additionally, we
numerically generate solitary wave solutions of the generalized Boussinesq
equation through the Petviashvili iteration method. To further examine the time
evolution of solutions, we propose employing the Fourier pseudo-spectral
numerical method. Our investigation extends to the gap interval, where neither
global existence nor blow-up results have been theoretically established. We
find that our numerical results effectively fill these gaps, supplementing the
theoretical findings.
