The region in physical spce where KdV equation is valid

[hanson]

Hi all.
Here is a question again regarding KdV equation.
I am now reading a book "Wave Motion" by J.Billinghm nd A.C. King and also "A modern introduction to the mathematical theory of water waves" by R.S. Johnson.
Both are them mention in the derivation of the KdV equation that the KdV equation is valid when a = O (epislon^2) and t>>1 and x = t + O(1) where a is the shallow water parameter.
The author says: "This leads us to interpret any waveform tht arises as a solution of the KdV equation as the large time limit of an initial value problem"
What does it really mean?
What is the "initial value problem" here?
This puzzle me about the evolution of a solitary wave, the "large time limit" makes me think that a solitary wave will take a long long time to form. Is it?

 

[Chris Hillman]

Hi all.
Here is a question again regarding KdV equation.
I am now reading a book "Wave Motion" by J.Billinghm nd A.C. King and also "A modern introduction to the mathematical theory of water waves" by R.S. Johnson.

I'd recommend that you also seek out Solitons: An Introduction,
by P.G. Drazin and R.S. Johnson (same Johnson you mentioned), Cambridge University Press, 1989. This is a very readable undergraduate textbook which picks up themes from the coauthors other books (P.G. Drazin is author of Solitons, Cambridge University Press, 1983, not the same book I just cited), but in a more user-friendly way.

You still haven't explained what you consider to be a "solitary wave", but I am guessing you are thinking of the classical sech solution of the KdV. This arises as a limiting case of certain periodic waves which can be written in terms of Jacobi elliptic functions. Drazin and Johnson have a very nice explanation of all this, and I think you will find this book very helpful.