Modified Korteweg-de Vries Hierachies in Multiple-Times Variables and the Solutions of Modified Boussinesq Equations

We study solitary-wave and kink-wave solutions of a modified Boussinesq equation through a multiple-time reductive perturbation method. We use appropriated modified Korteweg-de Vries hierarchies to eliminate secular producing terms in each order of the perturbative scheme. We show that the multiple-time variables needed to obtain a regular perturbative series are completely determined by the associated linear theory in the case of a solitary-wave solution, but requires the knowledge of each order of the perturbative series in the case of a kink-wave solution. These appropriate multiple-time variables allow us to show that the solitary-wave as well as the kink-wave solutions of the modified Botussinesq equation are actually respectively a solitary-wave and a kink-wave satisfying all the equations of suitable modified Korteweg-de Vries hierarchies.