The shifted parity and delayed time-reversal symmetry-breaking solutions for the (1+1)-dimensional Alice–Bob Boussinesq equation

Abstract

The integrable Alice–Bob system with the shifted parity and delayed time reversal is presented through the Lax pair for the (1 + 1)-dimensional Boussinesq equation. After introducing an extended Bäcklund transformation, this system shows abundant exact solutions with the auxiliary functions consisting of hyperbolic functions or rational functions. The corresponding soliton structures contain line solitons, breathers, and lumps, all which satisfied the shifted parity and delayed time-reversal symmetry for the states of Alice A and Bob B. In particular, some lower-order circumstances can be expressed through their explicit solutions and their dynamic structures.