# The $\phi^4$ model with the BPS preserving defect

## TL;DR

This paper studies a modified $^4$ model with an impurity that preserves half of the BPS property, revealing unique symmetry features and analyzing scattering behaviors near the BPS regime.

## Contribution

It introduces a BPS-preserving impurity in the $^4$ model and explores its effects on soliton solutions, symmetries, and scattering processes.

## Key findings

- Antikink-impurity bound state remains BPS and zero-pressure.
- Restoration of translational symmetry in the BPS sector.
- Elastic-like scattering near the BPS regime with possible backward scattering.

## Abstract

The $\phi^4$ model is coupled to an impurity in a way that preserves one-half of the BPS property. This means that the antikink-impurity bound state is still a BPS solution, i.e., a zero-pressure solution saturating the topological energy bound. The kink-impurity bound state, on the other hand, does not saturate the bound, in general. We found that, although the impurity breaks translational invariance, it is, in some sense, restored in the BPS sector where the energy of the antikink-impurity solution does not depend on their mutual distance. This is reflected in the existence of a generalised translational symmetry and a zero mode. We also investigate scattering processes. In particular, we compare the antikink-impurity interaction close to the BPS regime, which presents a rather smooth, elastic like nature, with other scattering processes. However, even in this case, after exciting a sufficiently large linear mode on the incoming antikink, we can depart from the close-to-BPS regime. This results, for example, in a backward scattering.