Numerical study of the small dispersion limit of the Korteweg-de Vries equation and asymptotic solutions

We study numerically the small dispersion limit for the Korteweg-de Vries (KdV) equation $u_t+6uu_x+\epsilon^{2}u_{xxx}=0$ for $\epsilon\ll1$ and give a quantitative comparison of the numerical solution with various asymptotic formulae for small $\epsilon$ in the whole $(x,t)$-plane. The matching of the asymptotic solutions is studied numerically.