Non-local defect interaction in one-dimension: Weak versus strong non-locality

Defect interaction (kink-antikink interaction) is studied for a prototypical model for non-local interaction. Mathematically, it is a bistable integrodifferential model, where the non-local interaction is performed due to an integral kernel. The system is able to establish domains where it is in one of its two equilibria, separated by defects. It is shown that the defect interaction depends on the asymptotical behavior of the integral kernel. In the weak non-local regime, when the integral kernel decays faster than an exponential at infinitum, the defect interaction is exponentially weak. Hence, this case is qualitatively similar to the local one. On the other hand, in the strong non-local regime, when the integral kernel decays slower than an exponential at infinitum, the defect interaction is ruled by the asymptotical behavior of the integral kernel. In this case, the defect interaction is stronger, and could be characterized, for instance, by a power law. The effect of this transition (from the weak to strong non-locality) on the domain dynamics is discussed.