  The chiral magnetic wave (CMW) is a macroscopic quantum phenomenon that
arises due to the mixing of the electric and chiral charge oscillations induced
by the chiral anomaly. In this study we report the first quantum simulation (on
classical hardware) of the real-time dynamics of CMWs in Schwinger model. Our
quench protocol is the following: at $t=0$ we suddenly place an electric dipole
at the middle of our lattice. Due to chiral anomaly, this dipole excites the
CMW that propagates towards the edges of the lattice. In Schwinger model tuned
to the conformal critical point (at $\theta = \pi$, $m/g \simeq 0.2$), we find
a gapless linear CMW that propagates with the speed of light. For massless
Schwinger model ($\theta =0, m=0$), we find a gapped linear CMW, in accord with
previous analytical analyses. For massive Schwinger model (that is dual to
strongly interacting bosonic theory), we enter the new regime of nonlinear
CMWs, where we find a surprise. Specifically, for $m/g > 1$, the frequency of
electric charge oscillations becomes much smaller than the frequency of the
oscillations of the chiral charge. For $m/g =4$, we find a solution
corresponding to a nearly static electric dipole with fast oscillations of the
chiral charge confined within. We call this solution a "thumper" and study its
properties in detail.
