The improved Fan sub-equation method and its application to the Boussinseq wave equation

The improved Fan sub-equation method is used to construct the exact travelling wave solutions to the Boussinseq wave equation in mathematical physics. The key idea of this method is to take full advantage of the Fan sub-equation method and bifurcation method of dynamical systems which can help us obtain the phase portraits and dynamical behavior of the general sub-equations and find more new exact solutions by solving the sub-equations. As a result, more exact travelling wave solutions to the Boussinesq wave equation are obtained which include more general soliton solutions, kink solutions, solitary pattern solutions, triangular solutions, singular periodic solutions and double periodic solutions.