Miura transform

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In the defocusing case, the Miura transformation v = \partial_x u + u^2 transforms a solution of defocussing mKdV to a solution of KdV

\partial_t v + \partial_x^3 v = 6 v \partial_x v.

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 v = \partial_x u + i u^2. 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

\partial_t v + \partial_x^3 v = 6(v^2 + w) \partial_x v
\partial_t w + \partial_x^3 w = 6(v^2 + w) \partial_x w

Then u = v^2 + \partial_x v + w is a solution of KdV. In particular, if a and b are constants and v solves

\partial_t v + \partial_x^3 v = 6(a^2 v^2 + bv) \partial_x v

then u = a^2 v^2 + a \partial_x v + bv solves KdV (this is the Gardener transform).

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