The exp(−Φ(η))-expansion method with application in the (1+1)-dimensional classical Boussinesq equations

Periodic and soliton solutions are presented for the (1+1)-dimensional classical Boussinesq equation which governs the evolution of nonlinear dispersive long gravity wave traveling in two horizontal directions on shallow water of uniform depth. The equation is handled via the exp(−Φ(η))-expansion method. It is worth declaring that the method is more effective and useful for solving the nonlinear evolution equations. In particular, mathematical analysis and numerical graph are provided for those solitons, periodic, singular kink and bell type solitary wave solutions to visualize the dynamics of the equation.