Long-range interactions of kinks

TL;DR

This paper investigates the long-range interactions of kinks in higher-order field theories, highlighting the importance of proper initial conditions and demonstrating the attractive nature of soliton interactions despite power-law tails.

Contribution

It introduces methods to accurately set initial conditions for multi-soliton simulations in higher-order models and compares their effectiveness in capturing attractive interactions.

Findings

Power-law asymptotics influence initial condition setup.

Naive ans"atze can cause radiation and false repulsion.

Proper initial data reveal attractive soliton interactions.

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 $\varphi^8$ field theory, although we explore similar issues in example $\varphi^{10}$ and $\varphi^{12}$ models. In particular, we discuss the fundamental differences between the latter higher-order models and the standard $\varphi^4$ 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 multi-soliton 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.