Quasi-integrability from $$\mathcal{P}\mathcal{T}$$-symmetry

Parity and time-reversal ($$\mathcal{P}\mathcal{T}$$) symmetry is shown as the natural cause of quasi-integrability of deformed integrable models, crucial to represent real physical systems as they posses various irregularities. The condition for asymptotic conservation of quasi-conserved charges appear as a direct consequence of the $$\mathcal{P}\mathcal{T}$$-symmetric phase of the system, ensuring definite $$\mathcal{P}\mathcal{T}$$-properties of the corresponding Lax pair as well as that of the anomalous contribution, consistent with the Wilson-loop criterion for integrability-like behavior. As a result, the quasi-deformed charge densities always acquire definite $$\mathcal{P}\mathcal{T}$$-properties suitable for the asymptotic conservation, as the Abelianization approach to construct them also preserves the definite $$\mathcal{P}\mathcal{T}$$-behavior of the Lax pair. This $$\mathcal{P}\mathcal{T}$$-symmetry based origin of quasi-conservation is general and has been demonstrated for quasi-deformations of multiple systems such as KdV, NLSE and non-local NLSE.