The symmetric breathers and lumps of the Boussinesq equation using the Alice-Bob transformation and Hirota's bilinear derivative method

Many two-place physical problems can be explicitly presented as related events model named Alice-Bob systems. In this paper, an integrable Alice-Bob Boussinesq system is introduced via the Boussinesq equation with parameters, which may meet the symmetry transformation of ??????????????????(parity with a shift) and ??????????????????(time reversal with a delay). After constructing an Backlund transformation, the system has rich symmetry solutions with the aid of auxiliary functions. The structures of obtained soliton solutions, such as the breathers, lumps and their hybrids, are all satisfied the ??????????????????or ??????????????????symmetry. To illustrate the symmetric characteristic, some lower-order solutions and the related dynamic structures are explicitly presented. The residual symmetry and its finite transformation for this system are also verified.