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

TL;DR

The paper analyzes linear perturbations around small oscillons in a 1+1D scalar field theory with mass $m$, showing that in the long-wavelength, nonrelativistic limit $\epsilon \ll m$ the Floquet modes are universal, depending only on $m$ and $\epsilon$ and independent of the potential $V(\phi)$. By expanding the perturbations and examining resonance conditions, the authors demonstrate that the nonrelativistic Floquet modes form a continuum labeled by real momentum plus four discrete zero modes corresponding to space translations, time translations, boosts, and amplitude changes, with no discrete shape modes in this regime. They provide explicit universal expressions for the continuum modes $G_k(\epsilon x)$ and $H_k(\epsilon x)$ (even/odd sectors) and identify the four zero modes, using the Sine-Gordon breather as a concrete calculational vehicle. Orthogonality relations among the modes are established to enable canonical quantization, highlighting a model-independent Floquet structure for small oscillons across 1+1D theories. The results have implications for the perturbative and resonant dynamics of breathers, quasi-breathers, and oscillons in diverse potentials.

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 $1/ε$ have identical nonrelativistic Floquet modes at leading order in an $ε/m$ expansion. More precisely, these Floquet modes depend only on $ε$ 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.