Long-time asymptotics of (1,3)-sign solitary waves for the damped nonlinear Klein-Gordon equation

We consider the damped nonlinear Klein-Gordon equation: \begin{align*} \partial_{t}^2u-Δu+2α\partial_{t}u+u-|u|^{p-1}u=0, \ & (t,x) \in $\mathbb{R}$ \times $\mathbb{R}$^d, \end{align*} where $\alpha >0$, $2\le d\le 5$ and energy sub-critical exponents $p>2$. In this paper, we prove that any solution which is asymptotic to a superposition of four solitons with exactly one soliton of opposite sign evolves so that the three like-signed solitons spread out in an equilateral-triangle configuration centered at the oppositely signed soliton.