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The Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion of order
These oscillations are approximately described by the elliptic solution of KdV where the amplitude, wave-number and frequency are not constant but evolve according to the Whitham equations. In this manuscript we give a quantitative analysis of the discrepancy between the numerical solution of the KdV equation in the small dispersion limit and the corresponding approximate solution for values of
The numerical results are compatible with a difference of order