Miura transform

In the defocusing case, the Miura transformation $v=\partial _{x}u+u^{2}$ transforms a solution of defocussing mKdV to a solution of KdV

$\partial _{t}v+\partial _{x}^{3}v=6v\partial _{x}v$.

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 $v=\partial _{x}u+iu^{2}$. 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).