  Recent studies have reported the detection of the galactic stellar halo wake
and dipole triggered by the Large Magellanic Cloud (LMC), mirroring the
corresponding response from dark matter (DM). These studies open up the
possibility of adding constraints on the global mass distribution of the Milky
Way (MW), and even on the nature of DM itself, with current and upcoming
stellar surveys reigniting the discussion on response modes in dynamical
friction. However, the simulation of such features remains computationally
challenging. We used a collisionless Boltzmann equation (CBE)+Poisson solver
based on an existing method from the literature. We investigated the density
and velocity response modes in simulations of Galactic-type DM halos accreting
LMC-sized satellites, including the dependence on the halo density profile. We
successfully captured both the local wake and the global over- and underdensity
induced in the host halo. We also captured the velocity response. In line with
previous studies, we find that the code can reproduce the core formation in the
cuspy profile and the satellite core stalling. The angular power spectrum (APS)
response is shown to be sensitive to each density profile. The cored Plummer
density profile seems the most responsive, displaying a richness of modes. At
the end of the simulation, the central halo acquires cylindrical rotation. The
CBE description makes it tenable to capture the response modes with a better
handling of noise in comparison to traditional N-body simulations. Hence, given
a certain noise level, BPM has a lower computational cost than N-body
simulations, making it feasible to explore large parameter sets. We anticipate
that stellar spheroids in the MW or external galaxies could show central
cylindrical rotation if they underwent a massive accretion event. The code can
be adjusted to include a variety of DM physics.
