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 that 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 that 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 soli tons 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.