Asymptotic stability manifolds for solitons in the generalized good Boussinesq equation (Q6176360)

This paper studies the asymptotic behavior of solutions of the good Bousssinesq model \[ \partial_{tt}\phi+\partial_{xx}(\partial_{xx}\phi-\phi +|\phi|^{p-1}\phi)=0 \] with \(p\geq 2\). Setting \(u=\phi\) and \(v=\partial_x^{-1}\partial_t \phi\) it can be written as a system \[ \begin{aligned} \partial_t u&=\partial_x v,\\ \partial_t v&=\partial_x(-\partial_{xx} u+u-|u|^{p-1} u). \end{aligned} \] The stationary solution \(Q\) (standing solitary wave) is known explicitly in this case. The following perturbations of it are considered \[ u(t,x)= Q(x)+w(t,x), \quad v(t,x)=z(t,x) \] with \(w\) even in \(x\) and \(z\) odd in \(x\). The first result of the paper says that if a global trajectory remains for all times near \((Q,0)\) in the energy space then in fact it is locally asymptotically stable. The second result is a construction of a manifold of such trajectories. The methodology used in this paper is based on virial identities.