We study numerically solutions to the Korteweg-de Vries and Camassa-Holm
equation close to the breakup of the corresponding solution to the
dispersionless equation. The solutions are compared with the properly rescaled
numerical solution to a fourth order ordinary differential equation, the second
member of the Painlev\'e I hierarchy. It is shown that this solution gives a
valid asymptotic description of the solutions close to breakup. We present a
detailed analysis of the situation and compare the Korteweg-de Vries solution
quantitatively with asymptotic solutions obtained via the solution of the Hopf
and the Whitham equations. We give a qualitative analysis for the Camassa-Holm
equationComment: 17 pages, 13 figure
