The nonlocal Boussinesq equations (NLBEs) are investigated in this work. The general forms of soliton solutions of the equations are firstly derived via Hirota bilinear method. Subsequently, the first- to fourth-order soliton solutions are obtained by taking auxiliary function in the bilinear form. According to the system parameter, we classify the multiple solitons into two types: stripe-like solitons and breathers. When the stripe-like solitons resonate, there are bifurcation solitons. Further, we find that solitons’ bifurcation behavior is nonlinear by analytical and numerical analysis. It is interesting that there exist three- and four-leaf envelopes for the breathers.

在这项工作中研究了非局部 Boussinesq 方程 (NLBE)。首先通过Hirota双线性方法推导出方程的孤子解的一般形式。随后，通过采用双线性形式的辅助函数获得一阶到四阶孤子解。根据系统参数，我们将多重孤子分为两类：条状孤子和呼吸器。当条纹状孤子共振时，就会出现分叉孤子。此外，我们通过解析和数值分析发现孤子的分岔行为是非线性的。有趣的是，呼吸器存在三叶和四叶封套。