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.

The modified KdV equation is related to the KdV equation via the Miura transform.