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.

 

[tacman]

The KdV equation, or the Korteweg-de Vries equation, is a nonlinear partial differential equation that describes the evolution of a one-dimensional shallow water wave. It is valid in a specific region of physical space, which is when the shallow water parameter a is close to zero and time t is much larger than one, and the spatial variable x is of the same order as time (t+O(1)). This means that the KdV equation is applicable to shallow water waves that are small in amplitude and propagate over a long distance.

In this context, the "initial value problem" refers to the initial conditions of the wave, such as its shape and velocity, which are used to solve the KdV equation. The large time limit mentioned by the author means that the solution to the KdV equation will approach a solitary wave, or a single localized disturbance, as time goes to infinity. However, this does not necessarily mean that it will take a long time for a solitary wave to form. In fact, the KdV equation can accurately describe the evolution of a solitary wave over a relatively short period of time.

Overall, the region where the KdV equation is valid is important to consider when studying shallow water waves. It allows us to understand the behavior of these waves in a simplified and manageable way, and provides insight into the formation and evolution of solitary waves.