Burst-with-memory events are potential transient gravitational wave (GW) sources for the maturing pulsar timing array (PTA) efforts. We provide a computationally efficient prescription to model pulsar timing residuals induced by supermassive black hole pairs in general relativistic hyperbolic trajectories employing a Keplerian-type parametric solution. Injection studies have been pursued on the resulting bursts with linear GW memory (LGWM) events with simulated datasets to test the performance of our pipeline, followed by its application to the publicly available NANOGrav 12.5-year (NG12.5) dataset. Given the absence of any evidence of LGWM events within the real NG12.5 dataset, we impose 95% upper limits on the PTA signal amplitude as a function of the sky location of the source and certain characteristic frequency ($n$) of the signal. The upper limits are computed using a signal model that takes into account the presence of intrinsic timing noise specific to each pulsar, as well as a common, spatially uncorrelated red noise, alongside the LGWM signal. Our investigations reveal that the 95% upper limits on LGWM amplitude, marginalized over all other parameters, is $3.48\pm 0.51\mathrm{\mu }\mathrm{s}$ for $n>3.16\mathrm{nHz}$. This effort should be relevant for constraining both burst and memory events in the upcoming International Pulsar Timing Array data releases.

• Received 2 February 2024
• Accepted 11 April 2024

DOI:https://doi.org/10.1103/PhysRevD.109.103018