Construction of periodic solutions of a Boussinnesq type equation using the method of quasi-normal forms

Using the asymptotic method of quasi-normal forms the dynamic characteristics of the following boundary value problem are analyzed:
$$
\begin{array}{c} u_{tt}-u_{xx}-\varepsilon a^2 u_{xxtt} =\varepsilon\alpha u_{xxt}+\varepsilon u_{t}-u^2u_{t}, \\[2mm] \left.u\right|_{x=0}=\left.u\right|_{x=1}=0,\quad \alpha=\mathrm{const}>0,\quad 0<\varepsilon\ll1. \end{array}
$$