Modified Korteweg-de Vries equation

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

$\partial_t u + \partial_x^3 u = 6 u^2 \partial_x u$

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

$\partial_t u + \partial_x^3 u = - 6 u^2 \partial_x u$