$PT$ symmetry in quasi-integrable models

Observations of almost stable scattering in nonintegrable models have been reinforced and a framework is proposed to describe quasi-integrability in terms of $PT$ symmetry. This new mechanism can be used to regard $PT$ symmetry in classical field theories as a guiding principle to also select relevant systems when it comes to integrability properties. It turns out that the if a deformed Lax pair is invariant under this symmetry, corresponding to the unbroken $PT$-symmetric regime, quasi-integrable excitations are produced with asymptotically conserved charges. A generic nonlinear field equation is used in order to verify the validity of the assumptions but results for a specific non-integrable class of models are also presented. A set of quasi-integrable excitations is investigated and shown to have spectral functions with appropriate properties, which might lead to the determination of the almost conserved charges. (paper)