Multiple Soliton Solutions of Alice–Bob Boussinesq Equations*

Abstract

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