We are interested in the nonlinear damped Klein–Gordon equation \( \partial _t^2 u+2\alpha \partial _t u-\Delta u+u-|u|^{p-1}u=0 \) on \(\mathbb {R}^d\) for \(2\leqslant d\leqslant 5\) and energy sub-critical exponents \(2< p < \frac{d+2}{d-2}\) . We construct multi-soliton, that is, solutions which behave for large times as a sum of decoupled solitons, in various configurations with symmetry: this includes multi-soliton whose soliton centers lie at the vertices of an expanding regular polygon (with or without a center), of a regular polyhedron (with a center), of a higher dimensional regular polytope, or on a line. We give a precise description of these multi-solitons, in particular the interaction between nearest neighbour solitons is asymptotic to \(\ln t - \frac{d-1}{2} \ln \ln t\) as \(t \rightarrow +\infty \) . We also prove that in any multi-soliton, the solitons cannot share all the same sign. Both statements generalize and refine the results of [13,14] and are based on the analysis developed in [8,9].
