Classification of kink clusters for scalar fields on the line

Andy Lawrie, MIT
Fine Hall 314

I will present joint work with Jacek Jendrej. We consider classical scalar fields in dimension 1+1 with a symmetric double-well self-interaction potential. Examples of such equations are the phi-4 model and the sine-Gordon equation. These nonlinear wave equations admit non-trivial static solutions called kinks and antikinks, which are amongst the simplest examples of topological solitons. We define an n-kink cluster to be a solution approaching, for large positive times, a  superposition of n alternating kinks and antikinks whose velocities converge to zero and mutual distances grow to infinity. Our main result is a determination of the leading order asymptotic behavior of any n-kink cluster. This result is used to construct the smooth n-dimensional invariant manifold of n-kink clusters. This analysis is partially inspired by the notion of "parabolic motions" in the Newtonian n-body problem. We explain this analogy and its limitations. We also explain the role of kink clusters as universal profiles for the formation/annihilation of multikink configurations.