# Miura transform

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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*).