########## KdV Equation Info ###########
The KdV Equation
Ut+12UUx+Uxxx=0
U(x,0)=f(x)
is a nonlinear wave equation used to model (among other things) the propagation of waves in a shallow canal.
It is possible to find analytically travelling wave or soliton solutions of the form
(1) U(x,t)=(a/2)^2*sech^2(0.5(ax-(a+a^3)t-d))
where a and d are arbitrary constants.
For a fixed value of t, equation one looks alot like a bell curve - a nice, gently sloping hump centered at x=d/a+(1+a^2)t which fades rapidly to zero as x deviates from the center.
The reader should notice three things about (1). First, everything is squared; that means for real values of x,t,a,d our travelling wave solutions cannot have negative amplitude. Second, our amplitude A=a^2/4 and our wave velocity V=1+a^2 are not independent. In fact,
V=1+a^2 =4(1/4+a^2/4)=1+4A
Thus taller waves travel faster than shorter waves. Notice that wave velocity goes to unity, not zero, as height goes to zero.
Finally, our wave velocity V=1+a^2 is always positive. Hence waves can only move from left to right and not right to left. The reason for this is unclear to me as I do not know how the KdV equation is derived from first principles. Perhaps the canal in question has a current moving along it, constraining waves to move in one direction.
Solutions of the form (1) can be found numerically by plugging in f(x)=0.25*a^2*sech^2(ax+d) as an initial position. Under arbitrary initial conditions, it is found numerically that a curve f(x) will split apart into a series of bell shaped curves, each of which will deform itself into a curve of the form (1) and then travel happily along with no further changes in structure. As an example, the decay of a gaussian curve 4*e^-0.3(x-10)^2 on [0,40] into a soliton train is depicted in a video attached to the link "Gaussian Decay" on this page. The decay of a curve identical to (1) but in which I forgot to divide a by 2 is shown in "WrongCoefficient". Finally, the happy, decay-free travel of the curve (1) where I did not forget to divide by 2 is shown in "RightCoefficient". The KdV equation is very selective indeed about which initial conditions get to form soliton solutions.