Multiple Soliton Solutions of Alice–Bob Boussinesq Equations**Supported by the National Natural Science Foundation of China under Grant No 11435005, and the K. C. Wong Magna Fund in Ningbo University.

Abstract

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