Korteweg-de Vries equation

The Korteweg-de Vries (KdV) equation is

$u_{t}+u_{x}xx+6uu_{x}=0.$

The factor of 6 is convenient for reasons of complete integrability, but can easily be scaled out if desired.

The equation 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.

The KdV equation has been studied on the line, on the circle, and on the half-line.