We study the Cauchy problem of the modified KdV equation in the class of step-like initial data and we derive the asymptotic behaviour over long time of the solution.

We show that in the $(x,t$) plane, there are three main regions where the solution has the following behaviour:

We study the Cauchy problem of the modified KdV equation in the class of step-like initial data and we derive the asymptotic behaviour over long time of the solution.

We show that in the $(x,t$) plane, there are three main regions where the solution has the following behaviour:

[1.] A soliton and a breather region on a constant background;

[2.] A dispersive shock wave region that up to a phase shift coincides with the dispersive shock wave generated by the step initial data;

[3.] A breather region on a constant background.

Finally we consider the behaviour over long times of the initial value problem with an infinite number of solitons, the so called soliton gas.