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Filamentation of Fast Radio Bursts in magnetar winds

Magnetars are the most promising progenitors of Fast Radio Bursts (FRBs). Strong radio waves propagating through the magnetar wind are subject to non-linear effects, including modulation/filamentation instabilities. We derive the dispersion relation for modulations of strong waves propagating in magnetically-dominated pair plasmas focusing on dimensionless strength parameters $a_0\lesssim 1$, and discuss implications for FRBs. As an effect of the instability, the FRB radiation intensity develops sheets perpendicular to the direction of the wind magnetic field. When the FRB front expands outside the radius where the instability ends, the radiation sheets are scattered due to diffraction. The FRB scattering timescale depends on the properties of the magnetar wind. In a cold wind, the typical scattering timescale is $τ_{\rm sc}\sim{\rm\; μs-ms}$ at the frequency $ν\sim 1{\rm\; GHz}$. The scattering timescale increases at low frequencies, with the scaling $τ_{\rm sc}\proptoν^{-2}$. The frequency-dependent broadening of the brightest pulse of FRB 181112 is consistent with this scaling. From the scattering timescale of the pulse, one can estimate that the wind Lorentz factor is larger than a few tens. In a warm wind, the scattering timescale can approach $τ_{\rm sc}\sim{\rm\; ns}$. Then scattering produces a frequency modulation of the observed intensity with a large bandwidth, $Δν\sim 1/τ_{\rm sc}\gtrsim 100{\rm\; MHz}$. Broadband frequency modulations observed in FRBs could be due to scattering in a warm magnetar wind.