Kortewege de Vries equation

[kurious]

In the Kortewege de Vries equation what is the physical origin of the
linear and non-linear terms? I'm thinking of the explanation of a water wave here.What properties of the water molecules cause the profile and stability of the wave?

 

[Gonzolo]

Not familiar with that equation, but I am guessing that the molecule's dipole has something to do with this.

Edit : A quick search showed how important it was. It seems soliton solutions were actually discovered with it. Is water is the only material substance capable of this? (not counting solitons in light propagation)

 

[humanino]

The KdV equation is very important indeed, and one can find many references on it. Unfortunately, I don't know any which provides an actual physical interpretation for the several terms. At least not in a satisfactory manner. I also think that the exact same story goes for the Navier-Stockes equation. Everybody agree : it is a very important equation. But nobody agree on the interpretation of the several terms.

This is a very interesting question though. I wish someone will soon pop-up with a good answer.

 

[naunzer]

Another eqn which admits stable (no dispersion) wave-like solutions is the Sine-Gordon-Eq:

D'Alembert(theta) + Sin(theta) = 0
(using suitable units)
[theta = theta(space,time), of cours]

This theta can be visualized as follows:

Consider a chain of pendulums, each being elastically connected on the top to its neighbours by springs. Theta measures the amplitude of "swinging"
of each pendulum (therefore function of space and time).
They swing in the usual way in the normal gravitational field: therefore the potential term Sin(theta).
The other term in the Sine-Gordon-EQN, namley the differential operator, comes from the continuum-limit of the elastic force modeled here by springs between the pendulums, as in the usual Klein-Gordon-EQN, or in every other harmonic oszilator.

Maybe the terms in the KdV-eqn have a similar origin.