Long time behavior of solutions to the generalized Boussinesq equation

In this paper, we investigate the generalized Boussinesq equation (gBq) as a model for the water wave problem with surface tension. Our study begins with the analysis of the initial value problem within Sobolev spaces, where we derive improved conditions for global existence and finite-time blow-up of solutions, extending previous results to lower Sobolev indices. Furthermore, we explore the time-decay behavior of solutions in Bessel potential and modulation spaces, establishing global well-posedness and time-decay estimates in these function spaces. Using Pohozaev-type identities, we demonstrate the non-existence of solitary waves for specific parameter regimes. A significant contribution of this work is the numerical generation of solitary wave solutions for the gBq equation using the Petviashvili iteration method. Additionally, we propose a Fourier pseudo-spectral numerical method to study the time evolution of solutions, particularly addressing the gap interval where theoretical results on global existence or blow-up are unavailable in the Sobolev spaces. Our numerical results provide new insights by confirming theoretical predictions in covered cases and filling gaps in unexplored scenarios. This comprehensive analysis not only clarifies the theoretical and numerical landscape of the gBq equation but also offers valuable tools for further investigations.