  Nonlinear effects are crucial for the propagation of Fast Radio Bursts (FRBs)
near the source. We study the filamentation of FRBs in the relativistic winds
of magnetars, which are commonly invoked as the most natural FRB progenitors.
As a result of filamentation, the particle number density and the radiation
intensity develop strong gradients along the direction of the wind magnetic
field. A steady state is reached when the plasma pressure balances the
ponderomotive force. In such a steady state, particles are confined into
periodically spaced thin sheets, and electromagnetic waves propagate between
them as in a waveguide. We show that: (i) The dispersion relation resembles
that in the initial homogeneous plasma, but the effective plasma frequency is
determined by the separation of the sheets, not directly by the mean particle
density. (ii) The contribution of relativistic magnetar winds to the dispersion
measure of FRBs could be several orders of magnitude larger than previously
thought. The dispersion measure of the wind depends on the properties of
individual bursts (e.g. the luminosity), and therefore can change significantly
among different bursts from repeating FRBs. (iii) Induced Compton scattering is
suppressed because most of the radiation propagates in near vacuum regions.

High Energy Astrophysical Phenomena