In the defocusing case, the Miura transformation transforms a solution of defocussing mKdV to a solution of KdV
Thus one expects the LWP and GWP theory for mKdV to be one derivative higher than that for KdV.
In the focusing case, the Miura transform is now . This transforms focussing mKdV to complex-valued KdV, which is a slightly less tractable equation. (However, the transformed solution v is still real in the highest order term, so in principle the real-valued theory carries over to this case.
The Miura transformation can be generalized. If v and w solve the system
Then is a solution of KdV. In particular, if a and b are constants and v solves
then solves KdV (this is the Gardener transform).