Description and Classification of 2-Solitary Waves for Nonlinear Damped Klein–Gordon Equations

We describe completely 2-solitary waves related to the ground state of nonlinear damped Klein–Gordon equation $$\begin{aligned} \partial _{tt}u+2\alpha _{t}u-\Delta u+u-|u|^{p-1}u=0 \end{aligned}$$ on $$\mathbb {R}^N$$ , for $$1\leqslant N\leqslant 5$$ and energy subcritical exponents $$p>2$$ . The description is twofold. First, we prove that with same sign do not exist. Second, construct classify full family in case opposite signs. Close sum two remote solitary waves, it turns out only components initial data unstable direction each are relevant large time asymptotic behavior solution. In particular, show have a universal behavior: distance between $$\log t$$ as $$t\rightarrow \infty $$ This due damping combined strong interactions waves.