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In this paper, we study the generalized Boussinesq equation to model thewater wave problem with surface tension. First, we investigate the initialvalue problem in the Sobolev spaces. We derive some conditions under which thesolutions of this equation are global or blow-up in time, and next, we extendour results to the Bessel potential spaces. The asymptotic behavior of thesolutions is also determined. The non-existence of solitary waves for someparameters is proved using Pohozaev-type identities. We generate solitary wavesolutions of generalized Boussinesq equation using the Petviashvili iterationmethod numerically. In order to investigate the time evolution of solutions tothe generalized Boussinesq equation, we propose the Fourier pseudo-spectralnumerical method. After studying the time evolution of the single solitarywave, we focus on the gap interval where neither a global existence nor ablow-up result has been established theoretically. Our numerical resultssuccessfully fill the gaps left by the theoretical ones.

Long time behavior of solutions to the generalized Boussinesq equation in Sobolev spaces