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## Main Question or Discussion Point

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:

∂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:

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}
```