Solving an integro differential equation involving hilbert transform

Please help me in solving this integro differential equation. I am unable write a code for this.

∂f/∂t−(H(f)(∂f/∂x)=0

where [H(f)] is hilbert transform of 'f.'

and f=f(x,t) and initial condition is f(x,0)=cos(x) and also has periodic boundary conditions given by F{H{f(x′,t)}}=i⋅sgn(k)F{f(x,t)}, where F(f(x,t) is fourier transform of f(x,t).

and here ''t'' runs from 0 to 1.3 seconds

so I think we have to use iterations on basis of 't' while solving this equation.

And suggestions are highly appreciated.