Echoes of black holes in Einstein-nonlinear electrodynamic theories

Black hole echo is an important observable that can help us better understand gravitational theories. The non-linear electrodynamic black holes can admit multi-horizon, and the destruction of outer horizons does not violate the weak cosmic censorship, which leads to the multi-peak effective potential for the scalar perturbations and give rise to the echoes. Putting the initial wave packet released outside the peaks, we find that the time-domain profile of the echo will split when the peaks of the effective potential change from two to three. This is a distinctive phenomenon of black hole echo and it might be possible to determine the geometric structure of the non-linear electrodynamic black hole. We also analyse the properties of echo produced by different kinds of effective potentials.