Pseudo-spectral integration of PDE $f_t - \mathcal{H}[f]\, f_x = 0$ containing Hilbert transform While reading the research paper [1], I came across the following equation and I am unable to solve it.
$$\frac{\partial f}{\partial t}−\mathcal{H}(f)\left(\frac{\partial f}{\partial x}\right)=0 
$$
where $ \displaystyle [\mathcal{H}(f)] \stackrel{\text{def}}{=} \text{p.v.} \frac{1}{\pi} \int_{- \infty}^{\infty} \frac{f(x')}{x - x'} ~ d{x'} $.
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 conditions using pseudospectral method given below,
$$F_k\{H_x\{f(x')\}\}=i \cdot\text{sgn}(k) F_k\{f(x)\}$$
where $$F_k\{f(x)\}=\frac{1}{\sqrt{2 \pi}}\int_{- \infty}^\infty e^{-ikx}f(x)dx $$
So I am thinking of application of Fourier transforms on both sides of the equation but I am unable proceed forward.
Please help me solve the equation.
[1] A. Thess, D. Spirn, B. Jüttner, "Viscous Flow at Infinite Marangoni Number", Physical Review Letters 75(25), 1995. doi:10.1103/PhysRevLett.75.4614
 A: Let us implement the pseudo-spectral method. Assuming periodic conditions, the spatially-periodic solution is expanded as a sum of Fourier harmonics
$$
f(x,t) = \sum_{n} c_n(t)\,\text{e}^{\text i n x}\, ,
\qquad
0\leq x \leq 2 \pi ,
$$
where the complex coefficients $c_n$ need to be determined. This Ansatz is injected in the PDE $\partial_t f = (\mathcal{H}f)\, \partial_x f$ by using
\begin{aligned}
\partial_t f(x,t) &= \sum_{n} c'_n(t)\,\text{e}^{\text i n x} \\
\mathcal{H}f(x,t) &= -\text i \sum_{n} \text{sgn}(n)\,c_n(t)\,\text{e}^{\text i n x} \\
\partial_x f(x,t) &= \text i \sum_{n} n\, c_n(t)\,\text{e}^{\text i n x} \, .\\
\end{aligned}
It remains to truncate the sums, and to solve the first-order differential system satisfied by the coefficients $c_n$. The initial condition $f(x,0) = \cos x$ becomes $c_1 = c_{-1} = \frac12$ and $c_{|n| \neq 1} = 0$ at $t=0$.
For instance, if we use one harmonic only, the PDE becomes
$$
c'_0 + c'_1\, \text e^{\text i x} + c'_{-1}\, \text e^{-\text i x} + \dots = -2\, c_1\, c_{-1} + \dots \, ,
$$
which yields the differential system $c'_0 = -2\, c_1\, c_{-1}$ and $c'_1 =c'_{-1} = 0$.
Solving for the three coefficients $c_n$ with the prescribed initial condition gives $c_1(t) = c_{-1}(t) = \frac12$ and $c_0(t) = -\frac12 t$, i.e. $f(x,t) \simeq \cos x - \frac12 t$ at the first order ($|n| \leq 1$). Of course, one needs to implement this method for a larger number of harmonics to get more accurate results.

Note 1: in the article, the authors claim that the solution is increasing in time, whereas I found a decreasing solution at the first order. There might be a sign mistake somewhere.
Note 2: in the article, the authors are presumably developing their method with spatial Fourier transforms, rather than with spatial Fourier series. Fourier transformation is actually more the standard way of writing pseudo-spectral methods. To do so, we use the note [17] of the article, the convolution theorem, and FFT.
