Application of the bifurcation method to the modified Boussinesq equation

In this paper, we investigate the modified Boussinesq equation u(tt) - u(xx) - epsilon u(xxxx) - 3(u(2))(xx) + 3(u(2)u(x))(x) = 0. Firstly, we give a property of the solutions of the equation, that is, if 1 + u(x, t) is a solution, so is 1 - u(x, t). Secondly, by using the bifurcation method of dynamical systems we obtain some explicit expressions of solutions for the equation, which include kink-shaped solutions, blow-up solutions, periodic blow-up solutions and solitary wave solutions. Some previous results are extended.