The Universal Floquet Modes of (Quasi)-Breathers and Oscillons

TL;DR

This paper demonstrates that in a (1+1)-dimensional relativistic scalar field theory, all breathers, quasi-breathers, and oscillons share identical nonrelativistic Floquet modes at leading order, independent of the potential, revealing a universal structure.

Contribution

It establishes the universality of Floquet modes for various localized oscillatory solutions in relativistic scalar field theories at leading order.

Findings

Floquet modes depend only on b5 and m, not on the potential.

A continuum of Floquet modes exists for each real momentum.

No discrete shape modes are present.

Abstract

Just as linearized perturbations of time-independent configurations can be decomposed into normal modes, those of periodic systems can be decomposed into Floquet modes, which each evolve by a fixed phase over one period. We show that in the case of a (1+1)-dimensional relativistic field theory with a single scalar of mass $m$, all breathers, quasi-breathers and oscillons of length $1/\epsilon$ have identical nonrelativistic Floquet modes at leading order in an $\epsilon/m$ expansion. More precisely, these Floquet modes depend only on $\epsilon$ and $m$, and are independent of the potential of the theory. In particular, there is a continuum of Floquet modes corresponding to each real momentum plus four discrete modes corresponding to space translations, time translations, boosts and amplitude changes. There are no discrete shape modes. We provide simple, explicit formulas for these universal leading-order, nonrelativistic Floquet modes.