  Focusing on multi-solitons for the Klein-Gordon equations, in 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 \cite{CJ2} 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 at 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.
