Solving 2-D partial integro-differential equation

[semivermous]

While reproducing a research paper, I came across the following equation,
∂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.
Please help me in solving this integro differential equation(PDE). I am unable write a code for this(Matlab/Mathematica/Maple)
And suggestions are highly appreciated.
Link for research paper: https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.75.4614
After solving the given equation, we have to get the figure-1 of the paper

Code I have written is:
[CODE title="Mathematica code"]L = Pi; tmax = 1.2; sys = {D[u[x, t], t] -
1/(Pi)*int[u[x, t], x, t]*D[u[x, t], x] == 0, u[0, t] == 1,
u[x, 0] == Cos[x]};
int[u_, x_?NumericQ, t_ /; t == 0] :=
NIntegrate[Cos[xp]/(x - xp), {xp, 0, x, x + 2 L},
Method -> {"InterpolationPointsSubdivision",
Method -> "PrincipalValue"}, MaxRecursion -> 20];
PrintTemporary@Dynamic@{foo, Clock[Infinity]};
Internal`InheritedBlock[{MapThread}, {state} =
NDSolve`ProcessEquations[sys, u, {x, 0, 2 L}, {t, 0, tmax},
StepMonitor :> (foo = t)];
NDSolve`Iterate[state, {0, tmax}];
sol = NDSolve`ProcessSolutions[state]] // AbsoluteTiming
{Plot3D[u[x, t] /. sol, {x, 0, 2 Pi}, {t, 0., 1.}, Mesh -> None,
ColorFunction -> Hue, AxesLabel -> Automatic] // Quiet,
Plot[Evaluate[Table[u[x, t] /. sol, {t, 0., 1., .2}]], {x, -Pi,
Pi}] // Quiet}
[/CODE]

 

[jedishrfu]

Hopefully someone here knows mathematica sufficiently to help.

In the meantime, I will look for some comparable examples using the same functions.

 

[jedishrfu]

I found these references with the first one suspiciously looking a lot like your code:

https://mathematica.stackexchange.c...erential-equation-involving-hilbert-transform
https://mathematica.stackexchange.com/questions/73150/solving-a-nasty-partial-differential-equation
https://www.wolfram.com/language/11...o-differential-equation.html?product=language

 

[semivermous]

The question in first link was posted by myself but the solution is wrong.

 

[jedishrfu]

Ahh okay. Well it may still hope others who read this thread.

One problem with this kind of question is the specialized knowledge needed to answer it. This would happen to me often in grad school before the internet. There was simply no place to turn to get help.