Long-range interactions of kinks

Abstract

We present a computational analysis of the long-range interactions of solitary waves in higher-order field theories. Our vehicle of choice is the ${\ensuremath{\varphi}}^{8}$ field theory, although we explore similar issues in example ${\ensuremath{\varphi}}^{10}$ and ${\ensuremath{\varphi}}^{12}$ models. In particular, we discuss the fundamental differences between the latter higher-order models and the standard ${\ensuremath{\varphi}}^{4}$ model. Upon establishing the power-law asymptotics of the model's solutions' approach towards one of the steady states, we make the case that such asymptotics require particular care in setting up multisoliton initial conditions. A naive implementation of additive or multiplicative ans\"atze gives rise to highly pronounced radiation effects and eventually leads to the illusion of a repulsive interaction between a kink and an antikink in such higher-order field theories. We propose and compare several methods for how to ``distill'' the initial data into suitable ans\"atze, and we show how these approaches capture the attractive nature of interactions between the topological solitons in the presence of power-law tails (long-range interactions). This development paves the way for a systematic examination of solitary wave interactions in higher-order field theories and raises some intriguing questions regarding potential experimental observations of such interactions. As an Appendix, we present an analysis of kink-antikink interactions in the example models via the method of collective coordinates.