# Iterated ${\phi}^4$ Kinks

## TL;DR

This paper develops an iterative scheme for solving static ${
m oldsymbol{}^4}$ kink equations with impurities, revealing complex solutions like kink-antikink configurations and connecting fixed points to ${
m oldsymbol{}^6}$ kinks.

## Contribution

It introduces a novel iterative approach to analyze ${
m oldsymbol{}^4}$ kinks with impurities, uncovering new solution structures and potential moduli space dynamics.

## Key findings

- First iteration yields standard kink solution.
- Second iteration produces kink-antikink or bump solutions.
- Fixed points correspond to ${
m oldsymbol{}^6}$ kinks.

## Abstract

A first order equation for a static ${\phi}^4$ kink in the presence of an impurity is extended into an iterative scheme. At the first iteration, the solution is the standard kink, but at the second iteration the kink impurity generates a kink-antikink solution or a bump solution, depending on a constant of integration. The third iterate can be a kink-antikink-kink solution or a single kink modified by a variant of the kink's shape mode. All equations are first order ODEs, so the nth iterate has n moduli, and it is proposed that the moduli space could be used to model the dynamics of n kinks and antikinks. Curiously, fixed points of the iteration are ${\phi}^6$ kinks.