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I am re-visiting again the origin of the KdV equation.

On one hand, in the theory of small amplitude waves, the amplitude is assumed to be very small and hence the governing equations and the boundary conditions are hence linearised.

On the other hand, having linearised the above equtaions, if there is a very long wave (or very shallow water), then we further obtain a wave solution with no dispersion at all.

So is it correct to say that the KdV equation originates when we are relaxing the assumption of the small amplitude wave approximtionm, making it a small but not very small amplitudes, and hence having the equation WEAKLY nonlinear (no longer linear), and at the same time, we also consider shallow but not very shallow water, and hence the wave is WEAKLY nonlinear, and it is a balance between these two effects that produce the KdV equation?i

In other words, the relaxation of BOTH the small amplitude wave and shallow water approximations are required to get the KdV equation, right?

If we just introduce weakly nonlinear effect with the shallow water parameter still tends to zero, then there won't be a KdV type of balance, right?

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# Shallow water and small amplitude wave?

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