Solving a PDE involving Hilbert transform

While solving one research paper published in Physical Review Letters, I came across the following equation and I am unable to solve it.

∂f/∂t−(H(f)(∂f/∂x)=0

where [H(f)] is hilbert transform

and f=f(x,t) and initial condition is f(x,0)=cos(x)

In the paper it is given that the solution of the above mentioned equation is obtained with periodic boundary conditions using pseudospectral method given below,

F{H{f(x′)}}=i⋅sgn(k)F{f(x)}

where

F{f(x)} is Fourier transform of f(x)

sgn(k) is signum function of k

So I am thinking of application of Fourier transforms on both sides of the equation but I am unable proceed forward. Please solve the equation and can give the code for the same in Matlab