The weak-nonlinear process of pulse propagation in the cylindrical waveguide was studied for the
refractive index depending quadratically on the radial coordinate within the core, and being constant in the
cladding. A transcendent characteristic equation is stated for determination of the mode propagation constants,
and mode groups are shown to split in the vicinity of cutoff frequency. The nonlinear Schr ¨odinger equation is
derived for the pulse envelope with coefficients depending on the mode indices, and proceeding from it, the
velocities range is estimated for the pulses whose carrier is represented by the modes of a given mode group.