The asymptotic stability of the stable solitons of the generalized good Boussinesq equation

This paper continues the series where we apply a synthesis of the virial inequalities in the tradition of Merle, Martel and their collaborators and of Kato smoothing, to close the virial inequalities. We prove absence of nonzero imaginary eigenvalues for the linearization in unweighted space and asymptotic stability for every velocity $\frac{\sqrt{p-1}}{2}<c<1$ and every $1<p<5$.