Nonlocal symmetry and explicit solutions from the CRE method of the Boussinesq equation

In this paper, we analyze the integrability of the Boussinesq equation by using the truncated Painleve expansion and the CRE method. Based on the truncated Painleve expansion, the nonlocal symmetry and Backlund transformation of this equation are obtained. A prolonged system is introduced to localize the nonlocal symmetry to the local Lie point symmetry. It is proved that the Boussinesq equation is CRE solvable. The two-solitary-wave fusion solutions, single soliton solutions and soliton-cnoidal wave solutions are presented by means of the Backlund transformations.