  We have studied laser pulse channeling in underdense plasma by means of
analytical theory and 3D PIC simulations. The most significant result of this
paper is the demonstration of the single fully evacuated stationary channel
solution can be reached an asymptotic state in PIC simulations. In the lower
density and for a laser power above channeling power, we were able to reproduce
in PIC simulations the analytical curve. We showed that single channels are
stable structures against symmetric perturbations. We also studied the
formation of the ring structure in theory and simulations. An evacuated ring
enclosed by an electron filament was observed in our 3D simulations. However
they always coexist with the main laser mode. The threshold power for ring
structure formation is found. Our studies on stability of the rings against
asymmetric perturbations show that ring structure is not stable against
azimuthal perturbations. The growth rate of the instability is shorter for
higher densities and it is therefore more effective at higher densities.
