  The Cauchy problem for the Burgers equation with a small dissipation and an
initial weak discontinuity and the Cauchy problem with a large initial gradient
for a quasilinear parabolic equation and for the Korteweg-de Vries (KdV)
equation are considered. Multiscale asymptotics of solutions corresponding to
shock waves are constructed. Some results can also be applied to rarefaction
waves.
