########## 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.