Long-time soliton dynamics via a coarse-grained space-time method
Phys. Rev. A 113, 012211 – Published 6 January, 2026
DOI: https://doi.org/10.1103/59g7-g9q6
Abstract
We investigate the long-time dynamics of the sine-Gordon (SG) model under a class of perturbations whose quantum field theoretic analog—via bosonization—corresponds to the massive Schwinger model describing dimensions relativistic quantum electrodynamics of Dirac fermions. Classical SG solutions offer critical insight into nonperturbative effects in this quantum theory, but capturing their long-time behavior poses significant numerical challenges. To address this, we extend a coarse-graining method to space-time using a dual-mesh construction based on the Minkowski metric. We first validate the approach against the well-studied variant of the SG model describing magnetic fluxon dynamics in Josephson transmission lines (JTLs), where analytical and numerical benchmarks exist. We then apply the method to the Schwinger-inspired SG model and uncover long-lived bound states—“Schwinger atoms”—in which a soliton is trapped by a fixed central charge. In certain regimes, the system exhibits limit cycles that give rise to positronium-like states of oppositely charged solitons, while in others such formation is suppressed. Accessing such long-time solutions requires a rigorous implementation of outgoing boundary conditions on a finite computational domain that provide radiative dissipation to allow relaxation toward states that exist only in an infinite domain. Here we provide such a construction. Our results also suggest the possibility of analog quantum simulation of relativistic quantum field theories with JTLs. These results demonstrate the utility of spatiotemporal coarse-graining methodology for probing nonperturbative structure formation in nonlinear field theories.
