KdV + asymptotic solution(Whitham)

 

The small dispersion limit can be asymptotically described in the following way: before a shock of the dispersionless equation, the KdV solution is approximately given by the solution of the Hopf equation. For values of t beyond breakup of the Hopf solution, Gurevitch and Pitaevski suggested the following picture which was rigorously proven by Lax-Levermore and Venakides, see 99/2005 for details and references. An oscillatory zone is identified where the KdV solution is approximately given as the exact elliptic solution of KdV where the branch points of the elliptic surface depend via the Whitham equations on the physical coordinates. Outside this zone, the KdV solution is approximated via the corresponding Hopf solution. The video shows the KdV solution Low-dispersion KdV (blue) and the corresponding asymptotic solution (magenta). The difference of these solutions can be seen in Difference KdV-asymptotic solution (Whitham).

T. Grava and C. Klein