KdV.c

Soliton Solutions to the Korteweg-De Vries Equation

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Soliton Solutions to the Korteweg-De Vries Equation

So-called soliton solutions may exist as a special case to special equations. This entails that a solution shape, while it moves through space, does not deform over time. Meaning that observations of such phenomenon may be expected to be either very rare (special solutions to special equations) or very common (since they persist over time for many to observe). In this example we follow the soliton solution as listed by wikipedia page on the Korteweg-De Vries equation. Fortunately, there is a solver to this equation in my sandbox.

#include "grid/multigrid1D.h"
#include "KdV.h"
#include "run.h"

We study two soliton solutions, both described by:

\displaystyle c(x,t) = -\frac{v}{2 \mathrm{cosh}^2\left(\frac{\sqrt{v}}{2}(x - vt - b)\right)}

for v = 1 and v = 2. To omit the most prominent issues at the boundaries, we place them far away from the region of interest.

scalar fast[], slow[], both[];

int main(){
  L0 = 50;
  X0 = -L0/2;
  init_grid(512);
  run();
}

Initialization

We initialize the solution fields and we also open a pipeline for output plots with gnuplot.

FILE * gnuplotPipe;
event init(t = 0){
  double v = 2;
  foreach() {
    fast[] = -v/(2.*sq(cosh(pow(v, 0.5)*0.5*(x + 12))));
    slow[] = -1./(2.*sq(cosh(0.5*(x + 3))));
    both[] = fast[] + slow[];
  }
  DT = 0.01;
  gnuplotPipe = popen ("gnuplot", "w");
  fprintf(gnuplotPipe,
        "set term pngcairo\n"
        "set xr [-15: 25]\n"
        "set yr [-1.8: 0.1]\n"
        "set key bottom left\n"
        "set grid\n"
        "set title 'KdV Solitons'\n"
        "set xlabel 'x'\n"
        "set ylabel 'c'\n");
}

Time integration

User convienience is provided via the KdV()-fuction interface.

event advance(i++) { 
  dt = dtnext (DT);
  KdV({fast, slow, both}, dt);
}

Output many plots

Each two time steps we output a plot that is produced via the pipe.

int frame = 0;
event movie(i += 2){
  fprintf(gnuplotPipe, "set output 'plot%d.png'\n", frame);
  fprintf(gnuplotPipe, "plot \
          '-' w l lw 5 t 'fast', \
          '-' w l lw 5 t 'slow', \
          '-' w l lw 3 t 'both'\n");
  for (scalar s in {fast, slow, both}) {
    foreach()
      fprintf(gnuplotPipe, "%g %g\n",x, s[]);
    fprintf(gnuplotPipe, "e\n");
  }
  frame++;
}

Creation of a movie from plots

We use Gnu/Linux commands and ffmpeg to generate a movie from all our plots.

event stop (t = 15){
  system("rm mov.mp4");
  system("ffmpeg -r 25 -f image2 -i plot%d.png -c:v libx264 -vf format=yuv420p -y mov.mp4");
  system("rm plot*");
  return 1;
}

Result

We can check if the solutions are indeed solitons.

Well done KdV() function

Addional analysis

We check the mean position and the variance of the solitons.

set xlabel 'time'
set ylabel 'position'
set grid
plot 'out' u 1:2 t 'fast' , '' u 1:4 t 'slow' , '' u 1:6 t 'both'

The moments of overtaking coincide (script)

set xlabel 'time'
set ylabel 'Variance'
set grid
plot 'out' u 1:3 t 'fast' , '' u 1:5 t 'slow' , '' u 1:7 t 'both'

The evolution of the variance (script)

event positions (t += 0.1) {
  printf ("%g ", t);
  for (scalar s in {fast, slow, both}) {
    double xp = 0, sm = 0, mean;
    foreach() {
      xp += s[]*x*Delta;
      sm += s[]*Delta;
    }
    printf ("%g ", mean = xp/sm);
    xp = 0;
    foreach() 
      xp += s[]*sq(x - mean)*Delta;
    printf ("%g ", xp/sm);
  }
  putchar ('\n');
}