  Diffraction-free Bessel beams have attracted major interest because of their
stability even in regimes of nonlinear propagation and filamentation. However,
Kerr nonlinear couplings are known to induce significant longitudinal intensity
modulation, detrimental to the generation of uniform plasma or for applications
in the processing of transparent materials. These nonlinear instabilities arise
from the generation of new spatio-spectral components through an initial stage
of continuous spectral broadening followed by four wave mixing. In this paper,
we investigate analytically and numerically these processes and show that
nonlinear instabilities can be controlled through shaping the spatial spectral
phase of the input beam. This opens new routes for suppressing the nonlinear
growth of new frequencies and controlling ultrashort pulse propagation in
dielectrics.
