Multiple Soliton Solutions of Alice–Bob Boussinesq Equations

Three Alice–Bob Boussinesq (ABB) nonlocal systems with shifted parity (\hatP_\rm s), delayed time reversal (\hatT_\rm d) and \hatP_\rm s\hatT_\rm d nonlocalities are investigated. The multi-soliton solutions of these models are systematically found from the \hatP_\rm s, \hatT_\rm d and \hatP_\rm s\hatT_\rm d symmetry reductions of a coupled local Boussinesq system. The result shows that for ABB equations with \hatP_\rm s and/or \hatT_\rm d nonlocality, an odd number of solitons is prohibited. The solitons of the \hatP_\rm s nonlocal ABB and \hatT_\rm d nonlocal ABB equations must be paired, while any number of solitons is allowed for the \hatP_\rm s\hatT_\rm d nonlocal ABB system. t-breathers, x-breathers and rogue waves exist for all three types of nonlocal ABB system. In particular, different from classical local cases, the first-order rogue wave can have not only four leaves but also five and six leaves.