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

TL;DR

This paper analyzes the long-term behavior of kink clusters in a 1+1 dimensional scalar field model, providing asymptotic descriptions, construction methods, and demonstrating their universality in multi-kink formation.

Contribution

It offers the first detailed asymptotic analysis of kink clusters, constructs solutions with prescribed initial conditions, and shows their universal role in multi-kink dynamics.

Findings

Asymptotic behavior of kink clusters determined

Constructed kink clusters with prescribed initial positions

Proved universality of kink clusters in multi-kink formation

Abstract

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. It is known that such a model admits non-trivial static solutions called kinks and antikinks. A kink cluster is a solution approaching, for large positive times, a superposition of alternating kinks and antikinks whose velocities converge to 0. They can be equivalently characterised as the solutions of minimal possible energy containing a given number of transitions between the vacua, or as the solutions whose kinetic energy decays to 0 for large time. Our main result is a determination of the main-order asymptotic behaviour of any kink cluster. Moreover, we construct a kink 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 multi-kink configurations. The proofs rely on a reduction, using appropriately chosen modulation parameters, to an n-body problem with attractive exponential interactions.