Does the radio-active phase of XTE~J1810$-$197 recur following the same evolutionary pattern?

Abstract: Magnetars are the most strongly magnetized compact objects known in the Universe and are regarded as one of the primary engines powering a variety of enigmatic, high-energy transients. However, our understanding of magnetars remains highly limited, constrained by observational sample size and radiative variability. XTE~J1810$-$197, which re-entered a radio-active phase in 2018, is one of only six known radio-pulsating magnetars. Leveraging the distinctive capability for simultaneous dual-frequency observations, we utilized the Shanghai Tianma Radio Telescope (TMRT) to monitor this magnetar continuously at both 2.25 and 8.60~GHz, capturing its entire evolution from radio activation to quenching. This enabled precise characterization of the evolution in its integrated profile, spin frequency, flux density, and spectral index ($α$, defined by $S \propto f^α$). The first time derivative of its spin frequency $\dotν$ passed through four distinct phases -- rapid decrease, violent oscillation, steady decline, and stable recovery -- before returning to its pre-outburst value concomitant with the cessation of radio emission. Remarkably, both the amplitudes and the characteristic time-scales of these $\dotν$ variations match those observed during the previous outburst that began in 2003, providing the first demonstration that post-outburst rotational evolution and radiative behavior in a magnetar are repeatable. A twisted-magnetosphere model can qualitatively account for this repeatability as well as for the progressive narrowing and abrupt disappearance of the radio pulse radiation, thereby receiving strong observational support.