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We consider a real scalar field equation in dimension 1+1 with an evenpositive self-interaction potential having two non-degenerate zeros (vacua) 1and -1. It is known that such a model admits non-trivial static solutionscalled kinks and antikinks. A kink cluster is a solution approaching, for largepositive times, a superposition of alternating kinks and antikinks whosevelocities converge to 0. They can be equivalently characterised as thesolutions of minimal possible energy containing a given number of transitionsbetween the vacua, or as the solutions whose kinetic energy decays to 0 forlarge time. Our main result is a determination of the main-order asymptoticbehaviour of any kink cluster. Moreover, we construct a kink cluster for anyprescribed initial positions of the kinks and antikinks, provided that theirmutual distances are sufficiently large. Finally, we show that kink clustersare universal profiles for the formation/collapse of multi-kink configurations.The proofs rely on a reduction, using appropriately chosen modulationparameters, to an n-body problem with attractive exponential interactions.

Dynamics of kink clusters for scalar fields in dimension 1+1