  In this paper we study the Cauchy problem for the generalized Boussinesq
equation with initial data in modulation spaces
$M^{s}_{p^\prime,q}(\mathbb{R}^n),$ $n\geq 1.$ After a decomposition of the
Boussinesq equation in a $2\times 2$-nonlinear system, we obtain the existence
of global and local solutions in several classes of functions with values in $
M^s_{p,q}\times D^{-1}JM^s_{p,q}$ spaces for suitable $p,q$ and $s,$ including
the special case $p=2,q=1$ and $s=0.$ Finally, we prove some results of
scattering and asymptotic stability in the framework of modulation spaces.

Analysis of PDEs