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

Jendrej, Jacek; Lawrie, Andrew

Journées équations aux dérivées partielles (2023), Exposé no. 7, 12 p.

We consider classical scalar fields in dimension $1+1$ with a self-interaction potential being a symmetric double-well. 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 and mutual distances grow to infinity.

The aim of this note is to expose some developments on asymptotic behaviour of kink clusters obtained in our recent preprint [23]. The results are partially inspired by the notion of “parabolic motions” in the Newtonian $n$-body problem. We present this analogy and mention its limitations. We also explain the role of kink clusters as universal profiles for formation of multi-kink configurations.

Classification : 35L71, 35B40, 37K40
Keywords: kink, multi-soliton, nonlinear wave

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