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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?

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# The region in physical spce where KdV equation is valid

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