Kink networks for scalar fields in dimension 1+1

We consider a scalar field equation in dimension 1+1 with a positive external potential having non-degenerate isolated zeros. We construct weakly interacting pure multi-solitons, that is solutions converging exponentially in time to a superposition of Lorentz-transformed kinks, in the case of distinct velocities. We find that these solutions form a 2K-dimensional smooth manifold in the space of solutions, where K is the number of the kinks. We prove that this manifold is invariant under the transformations corresponding to the invariances of the equation, that is space–time translations and Lorentz boosts.