# Soliton wave transmission and experiments

1. What are Solitons?

2. Does energy transfer without interference in Solitons?

I read first about in connection with Breather surface of constant negative Gauss curvature $K$.

1. Are there physical laws and experiments to demonstrate their propagation?
• For "What are solitons?", have you read the Wikipedia article? Please ask something more specific than "What is X?". Commented Apr 10, 2015 at 18:12
• Read the Wiki before. As mentioned above I read about math of it, but wanted to know more about its physics and experimental parts that should come first. Commented Apr 10, 2015 at 18:36

I'm TAing a nonlinear waves course this quarter, so I'll bite on this question.

Throughout this discussion, let's take the Korteweg de Vries (KdV) equation as our model system, that is

$$\eta_t+\eta\eta_x +\eta_{xxx} = 0$$

where $\eta$ is the (non-dimensionalized) free surface displacement. This equation can be derived in a variety of ways, in particular it can be found from the water wave equations in shallow water. Heuristically, $\eta$ is governed by a tendency for the system to steepen (due to the nonlinear term $\eta \eta_x$), and disperse (due to the term $\eta_{xxx}$).

A soliton is a solitary wave that propagates at constant speed with out change of form and is a balance between nonlinearity and dispersion. The functional form can be found by looking for solutions of the form

$$\eta = \eta_o \zeta(X)$$

where $X= x-\alpha^2 t$ for some constants $\eta_o, \alpha$. For compact solutions, one can solve the resulting ODE to find

$$\eta = 3 \alpha^2 sech^2 \left( \frac{\alpha}{2}( x -\alpha^2 t)\right).$$

Note, the amplitude and the speed of propagation are related in this case (this is not true, for instance, for the NLSE). An example of this solution is shown in figure 1 .

We can see the form of the wave remains constant as the wave propagates.

Now, this is all kind of interesting, but where things get surprising is when we look at the interaction of two solitons, eg, figure 2 .

In this case, we see that the two waves interact, and that after their interaction the only difference is a phase shift (related to the amplitudes of the waves). This is very surprising, as this nonlinear system is behaving analogously to a linear system (where superposition holds).

Note, the energy is the integral of the amplitude squared, and it's density clearly travels with the waves. It, along with an infinite number of other quantities, are conserved quantities of the system.

Finally, you can generate these things pretty easily. We had a tabletop demo, which I now wish I took a video of. I've seen them when drinking out a nalgene bottle. There are countless videos online of the phenomenon, so google can help you there.

Notes: Solitons were first observed by John Scott Russell (1844), when he witnessed a barge being quickly stopped in a canal in England. He then saw the generation of a soliton, which he followed on horseback for a few miles down the canal.

Zabusky and Kruskal (1965) numerically solved the KdV equation and observed the interaction properties of solitons.

Gardener et al. (1967) showed the KdV equation is integrable through the inverse scattering transform, which is a very strong statement about the solutions to the KdV equation given particular initial data.

Trefethen (Spectral Methods) provides a very short matlab code to solve the kdv equation (google p27.m). For completeness, I'll post it here:

% p27.m - Solve KdV eq. u_t + uu_x + u_xxx = 0 on [-pi,pi] by
%         FFT with integrating factor v = exp(-ik^3t)*u-hat.

% Set up grid and two-soliton initial data:
N = 256; dt = .4/N^2; x = (2*pi/N)*(-N/2:N/2-1)';
A = 25; B = 16; clf, drawnow, set(gcf,'renderer','zbuffer')
u = 3*A^2*sech(.5*(A*(x+2))).^2 + 3*B^2*sech(.5*(B*(x+1))).^2;
v = fft(u); k = [0:N/2-1 0 -N/2+1:-1]'; ik3 = 1i*k.^3;

% Solve PDE and plot results:
tmax = 0.006; nplt = floor((tmax/25)/dt); nmax = round(tmax/dt);
udata = u; tdata = 0; h = waitbar(0,'please wait...');
for n = 1:nmax
t = n*dt; g = -.5i*dt*k;
E = exp(dt*ik3/2); E2 = E.^2;
a = g.*fft(real( ifft(     v    ) ).^2);
b = g.*fft(real( ifft(E.*(v+a/2)) ).^2);     % 4th-order
c = g.*fft(real( ifft(E.*v + b/2) ).^2);     % Runge-Kutta
d = g.*fft(real( ifft(E2.*v+E.*c) ).^2);
v = E2.*v + (E2.*a + 2*E.*(b+c) + d)/6;
if mod(n,nplt) == 0
u = real(ifft(v)); waitbar(n/nmax)
udata = [udata u]; tdata = [tdata t];
end
end
waterfall(x,tdata,udata'), colormap(1e-6*[1 1 1]); view(-20,25)
xlabel x, ylabel t, axis([-pi pi 0 tmax 0 2000]), grid off
set(gca,'ztick',[0 2000]), close(h), pbaspect([1 1 .13])

• @ Nick P: Fascinating indeed. What range of $\alpha$ can produce such plots?; I obtained 3D plots of Sine-Gordon Fishnet for surfaces of revolution. Is it possible to find solution of KdV using cylindrical coords surfaces of revolution? I,e.,how to convert or correlate linear $\ sech^2..$ to cylindrical boundary waves? Soliton travelling waves given above are to be replaced with multiply humped (cusped) pseudospherical surfaces $(r,\theta,z,t )$.The Beltrami pseudosphere has infinite wave length, so should be left out,I guess. If the question is vague, shall further expand on it. Commented Apr 11, 2015 at 3:51
• In the plots above $\alpha =5$ but in general it's true for all $\alpha \in \mathbb{R}$. I'm not sure what the rest of your comments mean. Commented Apr 12, 2015 at 0:26
• @NickP Very cool! I am just not quite sure it is fair to say that when two waves interact the system is behaving analogously to a linear system. It is true that there is no change of form after the waves interact. But, as you said, there is a phase shift which depends on the wave amplitudes of the interacting waves. Where after interaction the larger wave emerges slightly ahead of where it would be had it not interacted with another wave. And similarly, the smaller wave emerges slightly before of where it would have otherwise. IMHO this is a manifestation of nonlinear behavior. Commented Apr 16, 2015 at 0:06
• @IsopycnalOscillation I agree that the phase shift is indicative of the nonlinear behavior, and hopefully did not convey otherwise. The fact that there is structure in the system that seems to be building blocks for the general behavior is to me one of the very surprising results of this study, and is the point I was trying to highlight. Commented Apr 16, 2015 at 3:49
• @IsopycnalOscillation Thanks. Waves in fluids are very interesting (and very challenging to describe). Commented Apr 16, 2015 at 20:23