Three Solitons Solution - the KdV Equation

KdV equation, u_t - 6uu_x + u_xxx = 0
Initial condition, u(x,0) = -12sech^2(x)
Discretization, h_t = h_x^3

Check the unstable solution in http://www.youtube.com/watch?v=5z5SylS2QHE

Simple Matlab code:

function kdvsimple
dx=0.1;
xmin = -8;
xmax = +8;
ymin = -2;
ymax = +20;
x=(xmin+dx:dx:xmax)';

k = dx^3;
nsteps = 0.5/k;
mesh2d = [];

u = -12 * sech(x).^2;

for ii = 1:nsteps
k1 = k * kdvequ(u,dx);
k2 = k * kdvequ(u+k1/2,dx);
k3 = k * kdvequ(u+k2/2,dx);
k4 = k * kdvequ(u+k3,dx);
u = u + k1/6 + k2/3 + k3/3 + k4/6;
mesh2d(ii,:) = -u;

if mod(ii,10) == 0
plot(x,-u,'b-','LineWidth',2);
axis([xmin,xmax,ymin,ymax])
drawnow;
end
end

function dudt=kdvequ(u,dx)
u = [u(end-1:end); u; u(1:2)];
dudt = 6*(u(3:end-2)).*(u(4:end-1)-u(2:end-3))/(2*dx) - (u(5:end)-2*u(4:end-1)+2*u(2:end-3)-u(1:end-4))/(2*dx^3);

Category:

Science & Technology

Tags:

• KdV
• Soliton

[tushar661]

hello

its nice. i am learning. could u please send me its program if possible. i shall be gratful to u

thanx

[smakadir]

hi, check the description now

[IgorMele]

nice, I had problems with only one soliton. which program did you use for solving kdv equation?

[smakadir]

I used a Matlab program. I just implemented a simple program based on Runge-Kutta method. Which program are you using?

[compxci]

By the way, great job. What are the arguments of the function Kdvsimple? How did you arrive at the IC??

[compxci]

...and solving real and imaginary parts in the wave equation/eigenvalue problem. So now that I see I have been beating a dead horse for an answer when in fact, the proper equation I should have used was the KdV form using matlab, better to have found solution here than elsewhere and be wrong with my conclusion. :)

[compxci]

I am also learning and from a very novice position. My question is how did you know to use the R/K method to arrive at a usable solution? I have been working on a problem similar to this for months using Flex PDE 6.0 to arrive with usable data and so far I have not been successful. I was attempting to solve the PDE so that I may arrive and discover the resonance patterns that were taken during the experiment I was also considering using the Q/M approach, expressing the wave in complex form.

[ScottMcSanchez]

For those interested:

Analytical soliton solutions do exist for the KDV equation. Using an algorithm provided by Hirota's Bilinear method, the inverse scattering transform, etc. you can actually come up with an expression for soliton solutions to the KDV equation.

Getting there, however, is tricky. The math is computationally extensive, so I recommend using Mathematica for most of the number crunching. These methods also work for the x-y analog, the KP equation.

[thomaskoo]

A lot of thanks for sharing the program. Very useful for starters.