| Résume | 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.
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