  Fast radio bursts (FRBs) are bright, short-duration radio transients with
very high brightness temperatures implying highly coherent emission. We suggest
that the FRBs are caused by the self-focusing of an electron beam interacting
with an ambient plasma right beyond the light cylinder radius of a neutron
star. The magnetic field at the light cylinder radius is relatively high which
can accommodate both young Crab-like systems and old millisecond pulsars
addressing the diverse environments of FRBs. At the first stage, the intense
pulsed-beam passing through the background plasma causes instabilities such
that the trapped particles in local Buneman-type cavitons saturate the local
field. The beam is then radially self-focused due to the circular electric
field developed by the two-stream instability which leads to Weibel instability
in the transverse direction. Finally, the non-linear saturation of the Weibel
instability results in the self-modulational formation of solitons due to
plasmoid instability. The resonant solitary waves are the breather-type
solitons hosting relativistic particles with self-excited oscillations. The
analytical solutions obtained for non-linear dispersion and solitons suggest
that, near the current sheets, the relativistic bunches are
accelerated/amplified by klystron-like structures due to self-excited
oscillations by the induced local electric field. Boosted coherent radio
emission propagates through a narrow cone with strong focusing due to radial
electric field and magnetic pinching. The non-linear evolution of solitons and
the stimulated emission are associated with the Buneman instability and the
possibility of the presence of nanosecond shots in FRBs are investigated.
