Kink Solutions With Power Law Tails

TL;DR

This paper reviews various types of kink solutions with power law tails in field theories, discussing known properties, models with explicit solutions, and partially understood interactions like forces and collisions.

Contribution

It introduces new higher-order field theory models with implicit and explicit kink solutions exhibiting power law tails, and discusses the partially understood kink interactions.

Findings

Models with power law tail kinks are presented.

Discontinuities in potential derivatives are noted in polynomial models.

Open questions on kink interactions and collisions are identified.

Abstract

We present a comprehensive review about the various facets of kink solutions with a power law tail which have received considerable attention during the last few years. This area of research is in its early stages and while several aspects have become clear by now, there are a number of issues which have only been partially understood or not understood at all. We first discuss the aspects which are reasonably well known and then address in some detail the issues which are only partially or not understood at all. We present a wide class of higher (than sixth) order field theory models admitting implicit kink as well as mirror kink solutions where the two tails facing each other have a power law or a power-tower type fall off while the other two ends not facing each other could have either an exponential or a power law tail. The models admitting implicit kink solutions where the two ends facing each other have an exponential tail while the other two ends have a power law tail are also discussed. Moreover, we present several field theory models which admit explicit kink solutions with a power law fall off. We note that in all the polynomial models while the potential $V(\phi)$ is continuous, its derivative is discontinuous. We also discuss one of the most important and only partially understood issues of the kink-kink and the kink-antikink forces in case the tails facing each other have a power law fall off. Finally, we briefly discuss the kink-antikink collisions at finite velocity and present some open questions.