Numerical solution of the small dispersion limit of Korteweg de Vries and Whitham equations

Abstract

The Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion of order ${\epsilon }^2$, is characterized by the appearance of a zone of rapid modulated oscillations of wave-length of order $\epsilon$. 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 $\epsilon$ between $10^{-1}$ and $10^{-3}$. The numerical results are compatible with a difference of order $\epsilon$ within the `interior' of the Whitham oscillatory zone, of order ${\epsilon }^{1/3}$ at the left boundary outside the Whitham zone and of order ${\epsilon }^{1/2}$ at the right boundary outside the Whitham zone.

arXiv comment39 pages and 27 figures

Cited by in corpus (3)

• Numerical study of oscillatory regimes in the Kadomtsev-Petviashvili equation
• The double scaling limit method in the Toda hierarchy
• Regularization of Hele-Shaw flows, multiscaling expansions and the Painleve I equation