On the long time asymptotic behaviour of the modified Korteweg de Vries equation with step-like initial data

We study the long time asymptotic behaviour of the solution q(x,t) of the modified Korteweg de Vries equation (MKdV) with step-like initial datum asymptotic to c_+ at +infinity and to c_- at -infinity. We show that the solution for long times decomposes in the (x,t) plane in three main regions:
1. a region where solitons and breathers travel with positive velocities on a constant background c_+;
2. an expanding oscillatory region {\color{black} (that generically contains breathers)};
3. a region of breathers travelling with negative velocities on the constant background c_-.

When the oscillatory region does not contain breathers, the form of the asymptotic solution coincides up to a phase shift with the dispersive shock wave solution obtained for the 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.