What Are the Physical Implications of Transformations in the KdV Equation?

[hanson]

Hi all.
I have seen a lot of different forms of the KdV equation...
The derivation of it results in a form like
Ut+Ux+epsilon(UUx+Uxxx)=0
and after some transformation, the epsilon is removed the equation becomes
Ut+Ux+UUx+Uxxx=0, and, still, after some sort of transformation, it becomes the standard KdV..
Ut+6UUx+Uxxx=0

I just don't know what the physical meaning of these transformation is...
Sometimes the times scale is magnified, what does it mean? does it mean the solitary wave have to be observed in a fast motion or what?
So many doubts.
I am wondering if someone can write a full paragraph explaining the meanings of individual transformation and the form of KdV arrived by that transformation.

Please give me some clues...

 

[Astronuc]

Here are some references, but I didn't find a treatment of Ut+Ux+epsilon(UUx+Uxxx)=0 specifically. This is not an area with which I am familiar.

http://mathworld.wolfram.com/Korteweg-deVriesEquation.html

http://en.wikipedia.org/wiki/Korteweg-de_Vries_equation
http://en.wikipedia.org/wiki/Inverse_scattering_transform

http://math.stanford.edu/~dlevy/papers/particle-kdv.pdf

 

[Astronuc]

More on the KdV equation

The General Analytical Solution for the Korteweg- de Vries Equation
http://home.usit.net/~cmdaven/korteweg.htm

The Korteweg-de Vries Equation:
History, exact Solutions, and graphical Representation
http://people.seas.harvard.edu/~jones/solitons/pdf/025.pdf

Should be interesting to compare the first with the second.


R. VICTOR JONES - http://people.seas.harvard.edu/~jones/#biblio
Robert L. Wallace Research Professor of Applied Physics in the Division of Engineering and Applied Sciences , Harvard University
Teaching since 1957! WOW!

RVJ's Soliton page - http://people.seas.harvard.edu/~jones/solitons/solitons.html


Then there is - http://planetmath.org/encyclopedia/KortewegDeVriesEquation.html

http://www.ams.org/tran/2005-357-05/S0002-9947-04-03726-2/home.html - need to register with American Mathematical Society