Focusing on multi-solitons for the Klein–Gordon equations, in the first part of this paper, we establish their conditional asymptotic stability. In the second part of this paper, we classify pure multi-solitons which are solutions converging to multi-solitons in the energy space as \(t\rightarrow \infty \) . Using Strichartz estimates developed in our earlier work (Chen and Jendrej in Strichartz estimates for Klein–Gordon equations with moving potentials, 2022) and the modulation techniques, we show that if a solution stays close to the multi-soliton family, then it scatters to the multi-soliton family in the sense that the solution will converge in large time to a superposition of Lorentz-transformed solitons (with slightly modified velocities), and a radiation term which is in main order a free wave. Moreover, we construct a finite-codimension centre-stable manifold around the well-separated multi-soliton family. Finally, given different Lorentz parameters and arbitrary centers, we show that all the corresponding pure multi-solitons form a finite-dimension manifold.
