Multifrequency parametric resonance in a non-linear wave equation

We consider the boundary-value problem
$$ u_{tt}+\varepsilon u_t+\biggl(1+\varepsilon\sum_{k=1}^m\alpha_k\cos 2\varphi_k\biggr)u=a^2u_{xx}-u^2u_t,\qquad u\big|_{x=0}=u\big|_{x=\pi}=0, $$
where $0<\varepsilon\ll 1$, $a>0$, $\varphi_k=\sigma_kt+c_k$, $k=1,\dots,m$.
We show that a suitable choice of a positive integer $m$ and real parameters $\alpha_k$, $\sigma_k$, $k=1,\dots,m$, enables us to make this problem have any prescribed number of exponentially stable time-quasiperiodic solutions bifurcating from zero.