Modified Korteweg-de Vries equation

The (defocusing) modified Korteweg-de Vries (mKdV) equation is

$\partial _{t}u+\partial _{x}^{3}u=6u^{2}\partial _{x}u$

It is completely integrable, and has infinitely many conserved quantities. Indeed, for each non-negative integer k, there is a conserved quantity which is roughly equivalent to the H^k norm of u. This equation has been studied on the line, on the circle, and on the half-line.

The focussing mKdV

$\partial _{t}u+\partial _{x}^{3}u=-6u^{2}\partial _{x}u$

is very similar, but admits soliton solutions.

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 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

v_t + v_xxx = 6(a^2 v^2 + bv) v_x

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