Multiple Soliton Solutions of Alice–Bob Boussinesq Equations
Hui Li1, S. Y. Lou1,2**
1School of Physical Science and Technology, Ningbo University, Ningbo 315211 2Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai 200062
Abstract：Three Alice–Bob Boussinesq (ABB) nonlocal systems with shifted parity ($\hat{P}_{\rm s}$), delayed time reversal ($\hat{T}_{\rm d}$) and $\hat{P}_{\rm s}\hat{T}_{\rm d}$ nonlocalities are investigated. The multi-soliton solutions of these models are systematically found from the $\hat{P}_{\rm s}$, $\hat{T}_{\rm d}$ and $\hat{P}_{\rm s}\hat{T}_{\rm d}$ symmetry reductions of a coupled local Boussinesq system. The result shows that for ABB equations with $\hat{P}_{\rm s}$ and/or $\hat{T}_{\rm d}$ nonlocality, an odd number of solitons is prohibited. The solitons of the $\hat{P}_{\rm s}$ nonlocal ABB and $\hat{T}_{\rm d}$ nonlocal ABB equations must be paired, while any number of solitons is allowed for the $\hat{P}_{\rm s}\hat{T}_{\rm d}$ 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.
2019, Vol. 36 
