  We study the long time asymptotic behaviour of the solution $q(x,t) $, to the
modified Korteweg de Vries equation (MKDV) $q_t+6q^2q_x+q_{xxx}=0$ with
step-like initial datum q(x,t=0)->c_- for x->-infinity and q(x,t=0)->c_+ for
x-> +infinity. For the exact shock initial data q(x,t=0)=c_- for x<0 and
q(x,t=0)=c_+ for x>0 the solution develops an oscillatory regions called
dispersive shock wave that connects the two constant regions c_+ and c_-. We
show that the dispersive shock wave is described by a modulated periodic
travelling wave solution of the MKDV equation where the modulation parameters
evolve according to the Whitham modulation equation.
  The oscillatory region is expanding within a cone in the $(x,t) plane defined
as -6c_{-}^2+12c_{+}^2<x/t<4c_{-}^2+2c_{+}^2, with t\gg 1.
  For step like initial data we show that the solution decomposes for long
times in three main regions:
  - a region where solitons and breathers travel with positive velocities on a
constant background c_+;
  - an expanding oscillatory region that can contain generically breathers;
  - a region of breathers travelling with negative velocities on the constant
background c_-.
  When the oscillatory region does not contain breathers, it coincides up to a
phase shift with the dispersive shock wave solution obtained for the exact step
initial data. The phase shift depends on the solitons, breathers and the
radiation of the initial data. This shows that the dispersive shock wave is a
coherent structure that interacts in an elastic way with solitons, breathers
and radiation.
