Quasi-integrability from PT-symmetry

TL;DR

This work shows that ${\cal PT}$-symmetry provides the natural mechanism for quasi-integrability in deformed integrable models. In the symmetric PT phase, the Lax pair is ${\cal PT}$-odd and the anomaly-turned-contribution to charge evolution is ${\cal PT}$-odd as well, yielding vanishing net changes of quasi-conserved charges $Q^n$ in the far past and future. The authors develop PT-structured quasi-deformations of ${\text{KdV}}$, ${\text{NLS}}$, and nonlocal ${\text{NLS}}$, derive the corresponding quasi-conservation laws, and show that the Abelianization procedure preserves the ${\cal PT}$-symmetry, extending the quasi-conservation to the constructed charges. This identifies a general mechanism by which non-Hermitian, PT-symmetric deformations can sustain quasi-integrability, with potential implications for PT-symmetric nonlinear systems and robust localized excitations.

Abstract

Parity and time-reversal (PT) symmetry is shown as the natural cause of quasi-integrability of deformed integrable models. The condition for asymptotic conservation of quasi-conserved charges appear as a direct consequence of the PT-symmetric phase of the system, ensuring definite PT-properties of the corresponding Lax pair as well as that of the anomalous contribution. This construction applies to quasi-deformations of multiple systems such as KdV, NLSE and non-local NLSE.