  The magnetic energy of nonlinear Alfven waves in compressible plasmas may be
ponderomotively coupled only to ion-acoustic quasi-modes which modulate the
wave phase velocity and cause wave-front steepening. In the collisionless
plasma with $\beta\not=0$, the dynamics of nonlinear Alfven wave is also
affected by the resonant particle-wave interactions. Upon relatively rapid
evolution (compared to the particle bounce time), the quasi-stationary wave
structures, identical to the so called (Alfvenic) Rotational Discontinuities,
form, the emergence and dynamics of which has not been previously understood.
Collisionless (Landau) dissipation of nonlinear Alfven waves is also a
plausible and natural mechanism of the solar wind heating. Considering a
strong, compressible, Alfvenic turbulence as an ensemble of randomly
interacting Alfvenic discontinuities and nonlinear waves, it is shown that
there exist two distinct phases of turbulence. What phase realizes depends on
whether this collisionless damping is strong enough to provide adequate energy
sink at all scales and, thus, to support a steady-state cascade of the wave
energy. In long-time asymptotics, however, the particle distribution function
is affected by the wave magnetic fields. In this regime of nonlinear Landau
damping, resonant particles are trapped in the quasi-stationary Alfvenic
discontinuities, giving rise to a formation of a plateau on the distribution
function and quenching collisionless damping. Using the virial theorem for
trapped particles, it is analytically demonstrated that their effect on the
nonlinear dynamics of such discontinuities is non-trivial and forces a
significant departure of the theory from the conventional paradigm.
