Multiple Soliton Solutions of Alice-Bob Boussinesq Equations

Three Alice Bob Boussinesq (ABB) nonlocal systems with shifted parity ((P) over cap (s)), delayed time reversal ((T) over cap (d)) and (P) over cap (s)(T) over cap (d) nonlocalities are investigated. The multi-soliton solutions of these models are systematically found from the (P) over cap (s), (T) over cap (d) and (P) over cap (s)(T) over cap (d) symmetry reductions of a coupled local Boussinesq system. The result shows that for ABB equations with (P) over cap (s) and/or (T) over cap (d) nonlocality, an odd number of solitons is prohibited. The solitons of the (P) over cap (s) nonlocal ABB and (T) over cap (d) nonlocal ABB equations must be paired, while any number of solitons is allowed for the (P) over cap (s)(T) over cap (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.