Classification and long-time asymptotics of $(2,2)$-sign solitons for the damped nonlinear Klein-Gordon equation

RESUMO

We consider the damped nonlinear Klein-Gordon equation \begin{align*} \partial_t^2u-\Delta u+2\alpha\partial_tu+u-|u|^{p-1}u=0 \end{align*} on $\mathbb{R}^d$, where $\alpha>0$, $2\leq d\leq5$, and $p>2$ is in the energy-subcritical range. We classify global solutions that converge to a superposition of two positive and two negative translates of the ground state, without imposing any symmetry, coplanarity, or a priori geometric condition on their centers. We prove that every such four-soliton configuration is asymptotically either an alternating collinear configuration or an expanding rhombus with alternating signs. We further determine the precise long-time asymptotics of all four centers.