Numerical simulation of the small dispersion limit of the KdV equation, Whitham equations and Painleve' equations

In this work we compare numerically the solution of the small dispersion limit of the KdV equation with the first order asympototics formulas given in terms of the Whitham equations. The first order asymptotics fails to be a good asymptotic in some regions of the (x,t) plane. In these regions a better asympotitic is provided by special solutions of the Painleve' equations.