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.

翻译：本文研究广义Boussinesq方程作为含表面张力水波问题的模型。首先，我们在Sobolev空间中研究初值问题，推导解全局存在或发生有限时间爆破的条件。随后，将分析扩展至Bessel势空间和调制空间，确定解的渐近行为。利用Pohozaev型恒等式，证明特定参数条件下孤立波的不存在性。此外，通过Petviashvili迭代法数值生成广义Boussinesq方程的孤立波解。为深入考察解的时间演化，我们提出采用傅里叶伪谱数值方法。研究还涉及间隙区间——该区间内全局存在性与爆破结果均未获得理论证明。我们发现数值结果有效填补了这些理论空白，对理论发现进行了补充。