Bifurcation solitons and breathers for the nonlocal Boussinesq equations

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.