  Kinetic Alfv\'en waves represent an important subject in space plasma
physics, since they are thought to play a crucial role in the development of
the turbulent energy cascade in the solar wind plasma at short wavelengths (of
the order of the proton inertial length $d_p$ and beyond). A full understanding
of the physical mechanisms which govern the kinetic plasma dynamics at these
scales can provide important clues on the problem of the turbulent dissipation
and heating in collisionless systems. In this paper, hybrid Vlasov-Maxwell
simulations are employed to analyze in detail the features of the kinetic
Alfv\'en waves at proton kinetic scales, in typical conditions of the solar
wind environment. In particular, linear and nonlinear regimes of propagation of
these fluctuations have been investigated in a single-wave situation, focusing
on the physical processes of collisionless Landau damping and wave-particle
resonant interaction. Interestingly, since for wavelengths close to $d_p$ and
proton plasma beta $\beta$ of order unity the kinetic Alfv\'en waves have small
phase speed compared to the proton thermal velocity, wave-particle interaction
processes produce significant deformations in the core of the particle velocity
distribution, appearing as phase space vortices and resulting in flat-top
velocity profiles. Moreover, as the Eulerian hybrid Vlasov-Maxwell algorithm
allows for a clean almost noise-free description of the velocity space,
three-dimensional plots of the proton velocity distribution help to emphasize
how the plasma departs from the Maxwellian configuration of thermodynamic
equilibrium due to nonlinear kinetic effects.
