Modified Korteweg-de Vries equation

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

u_t + u_xxx = 6 u^2 u_x.

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

u_t + u_xxx = - 6 u^2 u_x

is very similar, but admits soliton solutions.

Miura transform

In the defocusing case, the Miura transformation v = u_x + u^2 transforms a solution of defocussing mKdV to a solution of [#kdv KdV]

v_t + v_xxx = 6 v v_x.

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 = u_x + 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

v_t + v_xxx = 6(v^2 + w) v_x
w_t + w_xxx = 6(v^2 + w) w_x

Then u = v^2 + v_x + 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).