We consider a real scalar field equation in dimension 1 + 1 with an even, positive self- interaction potential
having two non-degenerate zeros (vacua) 1 and -1. Such a model admits non-trivial static solutions called kinks and
antikinks. We define a kink n-cluster to be a solution approaching, for large positive times, a superposition of n alternating
kinks and antikinks whose velocities converge to 0. They can be equivalently characterized as the solutions of minimal
possible energy containing n-1 transitions between the vacua, or as the solutions whose kinetic energy decays to 0
in large time.
Our first main result is a determination of the main-order asymptotic behavior of any kink n-cluster. The proof relies
on a reduction, using appropriately chosen modulation parameters, to an n-body problem with attractive exponential
interactions. We then construct a kink n-cluster for any prescribed initial positions of the kinks and antikinks, provided
that their mutual distances are sufficiently large. Finally, we show that kink clusters are universal profiles for the
formation/collapse of multikink configurations. In this sense, they can be interpreted as forming the stable/unstable manifold
of the multikink state given by a superposition of n infinitely separated alternating kinks and antikinks.
This is a joint work with Andrew Lawrie.
