Miura transform

From DispersiveWiki

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).