Vladimir Popov and Ekaterina Leleka

Solitary waves, or solitons were first described by Scott Russell, who noted the phenomenon while riding alongside a canal in 1834. He described a peculiar wave in the canal wave a single well-organized heap that propagated, seemingly without dissipation, for several miles. As a naval designer, Scott recognized that there were important things to be learned from these unusual waves.

After more than 130 years, what was at first a chance observation, and then a scientific oddity, has proven to be an important phenomenon in nature. In the last 30 years or so, the mathematics and physics of solitons have found applications in the understanding the fields of meteorology, elementary particle physics, plasma theory, laser physics, and fiber optics communications, just to name a few.

Investigation of solitons, like the investigation of most physical phenomena, began with an observation followed by directed experiments. While Russel believed that solitary waves or "waves of translation", as he called them, were important, a suitable mathematical description of these solitons did not appear until 1895, when Korteveg and de Vriez derived their equation for the wave propagation in one direction along the surface of the shallow channel.

This lag of more than 60 years between observation and mathematical description indicates the difficulty in developing analytical representations for solitary waves. It is possible to look for numerical solutions by experimentation. One example is the sine-Gordon equation,

diff(diff(f(x,t),x),x)-diff(diff(f(x,t),t),t)=m^2*sin(f(x,t))^2

which plays an important role in studies in the fields of ferromagnetic crystals and superconductivity theory.

Determination of an analytical solution for the sine-Gordon's equation itself doesn't present any difficulties. In fact, after applying the boundary properties

f->0 mod 2*p

double integration yields Kink's equation:

4*arctan(exp(m*sqrt(1/1-v^2)*(x-v*t)+d))

The corresponding surface is shown below

The equation describing of collision of a kink and an anti-kink of sin-Gordon equation is also well-known:

4*arctan(((a+b)(exp(x)-exp(y)))/((a-b)(1+exp(x+y))))

However for representing the head-on collision of two kinks, J.K.Perring and T.H.R.Skyrme have had to resort to numerical experiment. Sin-Gordon's equation solutions quite often describe extremely difficult spatial patterns (regularity).

For example consider the stable kink solution of the double sin-Gordon equation, obtained by numerical calculations

-4*arctan(sqrt(5)*csc(1/2*(sqrt(5)*(x-v*t))/sqrt(1-v^2)))

If we consider two excitations (or breathers) as described by the sine-Gordon equation, one of them superincumbent and the other moving, as in:

4*arctan((sqrt(1-a^2)*sin(a)*(t-v*x)*sech((sqrt(1-a^2)*(t-v*x))/(1-v^2)))/(a*(1-v^2)))

we see that the two waves will pass each other seemingly without mutual interference of lasting effects of one on the other. 2D Maple-based animations of this phenomenon can be found on more than one soliton website.

The real scientific interest for us however, is represented not by the surface itself but by its riffling, fractal-like fine structure. If we slightly rotate the surface shown above, a fine structure is noted that is especially apparent at the crest and trough of the wave.

The pattern that appears along the crest of the wave above, when viewed at sufficient resolution is shown below.

The ability of nVizx to show the fine structure features of the soliton equations are an example of the advantages offered by high quality visualization tools (in this case coupled with numerical analysis).