Perturbations of the signum-Gordon model

TL;DR

This paper studies how small perturbations to the signum-Gordon scalar field model affect its solutions, revealing periodic behaviors and symmetry breaking effects for self-similar initial data.

Contribution

It introduces the signum-Klein-Gordon model as a perturbation of the signum-Gordon model and analyzes the resulting solution dynamics.

Findings

Solutions exhibit periodic escape and return to initial data.

Breaking of scaling symmetry leads to new solution behaviors.

Self-similar initial data are key to understanding perturbation effects.

Abstract

We investigate a perturbation of a scalar field model (called here the signum-Gordon model) with the potential $V(f)=|f|$. The perturbation generalizes the signum-Gordon model to the signum-Klein-Gordon model i.e. to the case $V(f)=|f|-{1/2}\lambda f^2$, where $\lambda$ is a small parameter. Such a generalization breaks the scaling symmetry of the signum-Gordon model. In this paper we concentrate on solutions for self-similar initial data. Such data are particulary useful for identification of the effects caused by the term that breaks the scaling symmetry. We have found that the behaviour of the solutions is quite interesting - they escape and return periodically to the self-similar initial data.